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Applied Bionics and Biomechanics

Review Article Volume 1 Issue 3

Reconstruction of multidimensional data in pattern analysis

Dariusz Jacek Jakobczak

Department of Electronics and Computer Science, Koszalin University of Technology, Poland

Correspondence: Dariusz Jacek Jakobczak, Department of Electronics and Computer Science, Koszalin University of Technology, Sniadeckich 2, 75-453 Koszalin, Poland

Received: October 23, 2017 | Published: October 30, 2017

Citation: Jakóbczak DJ. Reconstruction of multidimensional data in pattern analysis. MOJ App Bio Biomech. 2017;1(3):110–120. DOI: 10.15406/mojabb.2017.01.00017

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Abstract

Proposed method, called Probabilistic Features Combination (PFC), is the method of multi-dimensional data modeling, extrapolation and interpolation using the set of high-dimensional feature vectors. This method is a hybridization of numerical methods and probabilistic methods. Identification of faces or fingerprints need modeling and each model of the pattern is built by a choice of multi-dimensional probability distribution function and feature combination. PFC modeling via nodes combination and parameter γ as N-dimensional probability distribution function enables data parameterization and interpolation for feature vectors. Multi-dimensional data is modeled and interpolated via nodes combination and different functions as probability distribution functions for each feature treated as random variable: polynomial, sine, cosine, tangent, cotangent, logarithm, exponent, arc sin, arc cos, arc tan, arc cot or power function.

Keywords: image retrieval, pattern recognition, data modeling, vector interpolation, pfc method, feature reconstruction, probabilistic modeling

Introduction

The problem of multidimensional data modeling appears in many branches of science and industry. Image retrieval, data reconstruction; object identification or pattern recognition is still the open problems in artificial intelligence and computer vision. The chapter is dealing with these questions via modeling of high-dimensional data for applications of image segmentation in image retrieval and recognition tasks. Handwriting based author recognition offers a huge number of significant implementations which make it an important research area in pattern recognition. There are so many possibilities and applications of the recognition algorithms that implemented methods have to be concerned on a single problem: retrieval, identification, verification or recognition. This chapter is concerned with two parts: image retrieval and recognition tasks. Image retrieval is based on probabilistic modeling of unknown features via combination of N-dimensional probability distribution function for each feature treated as random variable. Handwriting and signature recognition and identification represent a significant problem. In the case of biometric writer recognition, each person is represented by the set of modeled letters or symbols. The sketch of proposed Probabilistic Features Combination (PFC) method consists of three steps: first handwritten letter or symbol must be modeled by a vector of features (N-dimensional data), then compared with unknown letter and finally there is a decision of identification. Author recognition of handwriting and signature is based on the choice of feature vectors and modeling functions. So high-dimensional data interpolation in handwriting identification1 is not only a pure mathematical problem but important task in pattern recognition and artificial intelligence such as: biometric recognition, personalized handwriting recognition,2-4 automatic forensic document examination,5,6 classification of ancient manuscripts.7 Also writer recognition8 in monolingual handwritten texts is an extensive area of study and the methods independent from the language are well-seen.9-12 Proposed method represents language-independent and text-independent approach because it identifies the author via a single letter or symbol from the sample. Writer recognition methods in the recent years are going to various directions13-17 writer recognition using multi-script handwritten texts, introduction of new features, combining different types of features, studying the sensitivity of character size on writer identification, investigating writer identification in multi-script environments, impact of ruling lines on writer identification, model perturbed handwriting, methods based on run-length features, the edge-direction and edge-hinge features, a combination of codebook and visual features extracted from chain code and polygon zed representation of contours, the autoregressive coefficients, codebook and efficient code extraction methods, texture analysis with Gabor filters and extracting features, using Hidden Markov Model18 or Gaussian Mixture Model.19 So hybrid soft computing is essential: no method is dealing with writer identification via N-dimensional data modeling or interpolation and multidimensional points comparing as it is presented in this paper. The paper wants to approach a problem of curve interpolation and shape modeling by characteristic points in handwriting identification.20 Proposed method relies on nodes combination and functional modeling of curve points situated between the basic set of key points. The functions that are used in calculations represent whole family of elementary functions with inverse functions: polynomials, trigonometric, cyclometric, logarithmic, exponential and power function. These functions are treated as probability distribution functions in the range [0;1]. Nowadays methods apply mainly polynomial functions, for example Bernstein polynomials in Bezier curves, splines21 and NURBS. But Bezier curves don’t represent the interpolation method and cannot be used for example in signature and handwriting modeling with characteristic points (nodes). Numerical methods22-24 for data interpolation are based on polynomial or trigonometric functions, for example Lagrange, Newton, Aitken and Hermite methods. These methods have some weak sides and are not sufficient for curve interpolation in the situations when the curve cannot be build by polynomials or trigonometric functions.25 This chapter presents novel Probabilistic Features Combination (PFC) method of high-dimensional interpolation in hybrid soft computing and takes up PFC method of multidimensional data modeling. The method of PFC requires information about data (image, object, and curve) as the set of N-dimensional feature vectors. Proposed PFC method is applied in image retrieval and recognition tasks via different coefficients for each feature as random variable: polynomial, sinusoidal, cosinusoidal, tangent, cotangent, logarithmic, exponential, arc sin, and arc cos, arc tan, arc cot or power. Modeling functions for PFC calculations are chosen individually for every task and they represent probability distribution functions of random variable α i ε[ 0;1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeg7aHTWdamaaBaaabaqcLbmapeGaamyAaaWcpaqabaqc LbsacqaH1oqzjuaGdaWadaGcbaqcLbsapeGaaGimaiaacUdacaaIXa aak8aacaGLBbGaayzxaaaaaa@4271@ for every feature i=1,2,N1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadMgacqGH9aqpcaaIXaGaaiilaiaaikdacaGGSaGaeyOj GWRaamOtaiabgkHiTiaaigdaaaa@3F79@ . So this chapter wants to answer the question: how to retrieve the image using N-dimensional feature vectors and to recognize a handwritten letter or symbol by a set of high-dimensional nodes via hybrid soft computing?

Hybrid multidimensional modeling of feature vectors

The method of PFC is computing (interpolating) unknown (unclear, noised or destroyed) values of features between two successive nodes (N-dimensional vectors of features) using hybridization of probabilistic methods and numerical methods. Calculated values (unknown or noised features such as coordinates, colors, textures or any coefficients of pixels, voxels and doxels or image parameters) are interpolated and parameterized for real number x( c ) or z( c ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG4bWdamaabmaabaWdbiaadogaa8aacaGLOaGaayzkaaWd biaabccacaWGVbGaamOCaiaabccacaWG6bWdamaabmaabaWdbiaado gaa8aacaGLOaGaayzkaaaaaa@411F@   i=1,2,N1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadMgacqGH9aqpcaaIXaGaaiilaiaaikdacaGGSaGaeyOj GWRaamOtaiabgkHiTiaaigdaaaa@3F79@ between two successive values of feature. PFC method uses the combinations of nodes (N-dimensional feature vectors) p 1 =( x 1 , y 1 ,, z 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadchal8aadaWgaaqaaKqzadWdbiaaigdaaSWdaeqaaKqz GeWdbiabg2da9Kqba+aadaqadaGcbaqcLbsapeGaamiEaSWdamaaBa aabaqcLbmapeGaaGymaaWcpaqabaqcLbmapeGaaiilaKqzGeGaamyE aSWdamaaBaaabaqcLbmapeGaaGymaaWcpaqabaqcLbsapeGaaiilai abgAci8kaacYcacaWG6bqcfa4damaaBaaaleaajugWa8qacaaIXaaa l8aabeaaaOGaayjkaiaawMcaaaaa@4EE2@ , p 2 =( x 2 , y 2 ,, z 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadchal8aadaWgaaqaaKqzadWdbiaaikdaaSWdaeqaaKqz GeWdbiabg2da9Kqba+aadaqadaGcbaqcLbsapeGaamiEaKqba+aada WgaaWcbaqcLbmapeGaaGOmaaWcpaqabaqcLbsapeGaaiilaiaadMha l8aadaWgaaqaaKqzadWdbiaaikdaaSWdaeqaaKqzGeWdbiaacYcacq GHMacVcaGGSaGaamOEaSWdamaaBaaabaqcLbmapeGaaGOmaaWcpaqa baaakiaawIcacaGLPaaaaaa@4DB8@ , p n =( x n , y n , z n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadchal8aadaWgaaqaaKqzadWdbiaad6gaaSWdaeqaaKqz GeWdbiabg2da9Kqba+aadaqadaGcbaqcLbsapeGaamiEaSWdamaaBa aabaqcLbmapeGaamOBaaWcpaqabaqcLbsapeGaaiilaiaadMhajuaG paWaaSbaaSqaaKqzadWdbiaad6gaaSWdaeqaaKqzGeWdbiaacYcacq GHMacVcaWG6bWcpaWaaSbaaeaajugWa8qacaWGUbaal8aabeaaaOGa ayjkaiaawMcaaaaa@4DE4@ as h( p 1 , p 2 ,, p m ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadIgajuaGpaWaaeWaaOqaaKqzGeWdbiaadchajuaGpaWa aSbaaSqaaKqzadWdbiaaigdaaSWdaeqaaKqzGeWdbiaacYcacaWGWb WcpaWaaSbaaeaajugWa8qacaaIYaaal8aabeaajugib8qacaGGSaGa eyOjGWRaaiilaiaadchajuaGpaWaaSbaaSqaaKqzadWdbiaad2gaaS WdaeqaaaGccaGLOaGaayzkaaaaaa@4A64@ and m=1,2,n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad2gacqGH9aqpcaaIXaGaaiilaiaaikdacaGGSaGaeyOj GWRaamOBaaaa@3DF5@ to interpolate unknown value of feature (for example y) for the rest of coordinates:
c 1 = a 1 × x k + (1 a 1 )× x k +1 , c N 1 = a N 1 × z k + (1 a N 1 )× z k +1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadogal8aadaWgaaqaaKqzadWdbiaaigdaaSWdaeqaaKqz GeWdbiabg2da9iaadggal8aadaWgaaqaaKqzadWdbiaaigdaaSWdae qaaKqzGeWdbiabgEna0kaadIhal8aadaWgaaqaaKqzadWdbiaadUga aSWdaeqaaKqzGeWdbiabgUcaRiaabccapaGaaiika8qacaaIXaGaey OeI0IaamyyaSWdamaaBaaabaqcLbmapeGaaGymaaWcpaqabaqcLbsa caGGPaWdbiabgEna0kaadIhal8aadaWgaaqaaKqzadWdbiaadUgaaS WdaeqaamaaBaaabaqcLbmapeGaey4kaSIaaGymaaWcpaqabaqcLbsa peGaaiilaiabgAci8kabgAci8kaadogal8aadaWgaaqaaKqzadWdbi aad6eaaSWdaeqaamaaBaaabaqcLbmapeGaeyOeI0IaaGymaaWcpaqa baqcLbsapeGaeyypa0JaamyyaSWdamaaBaaabaqcLbmapeGaamOtaa WcpaqabaWaaSbaaeaajugWa8qacqGHsislcaaIXaaal8aabeaajugi b8qacqGHxdaTcaWG6bWcpaWaaSbaaeaajugWa8qacaWGRbaal8aabe aajugib8qacqGHRaWkcaqGGaWdaiaacIcapeGaaGymaiabgkHiTiaa dggal8aadaWgaaqaaKqzadWdbiaad6eaaSWdaeqaamaaBaaabaqcLb mapeGaeyOeI0IaaGymaaWcpaqabaqcLbsacaGGPaWdbiabgEna0kaa dQhal8aadaWgaaqaaKqzadWdbiaadUgaaSWdaeqaamaaBaaabaqcLb mapeGaey4kaSIaaGymaaWcpaqabaaaaa@84D1@ , k= 1,2,n1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadUgacqGH9aqpcaqGGaGaaGymaiaacYcacaaIYaGaaiil aiabgAci8kaad6gacqGHsislcaaIXaaaaa@403E@ ,

c= ( c 1 ,, c N 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadogacqGH9aqpcaqGGaWdaiaacIcapeGaam4yaKqba+aa daWgaaWcbaqcLbmapeGaaGymaaWcpaqabaqcLbsapeGaaiilaiabgA ci8kaacYcacaWGJbWcpaWaaSbaaeaajugWa8qacaWGobaal8aabeaa daWgaaqaaKqzadWdbiabgkHiTiaaigdaaSWdaeqaaKqzGeGaaiykaa aa@48FD@ , α = ( α 1 ,, a N 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeg7aHjaabccacqGH9aqpcaqGGaWdaiaacIcapeGaeqyS de2cpaWaaSbaaeaajugWa8qacaaIXaaal8aabeaajugib8qacaGGSa GaeyOjGWRaaiilaiaadggal8aadaWgaaqaaKqzadWdbiaad6eaaSWd aeqaamaaBaaabaqcLbmapeGaeyOeI0IaaGymaaWcpaqabaqcLbsaca GGPaaaaa@4A7E@ , γ i = F i ( α i [ 0;1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaajo7al8aadaWgaaqcbasaaKqzGbWdbiaajMgaaKqaG8aa beaajugib8qacaqI9aGaaKOraSWdamaaBaaajeaibaqcLbgapeGaaK yAaaqcbaYdaeqaaKqzGeGaaKika8qacaqIXoWcpaWaaSbaaKqaGeaa jugya8qacaqIPbaajeaipaqabaqcLbsacaqIPaGaaKyTdKqbaoaadm aakeaajugib8qacaqIWaGaaK4oaiaajgdaaOWdaiaawUfacaGLDbaa aaa@4C04@ , i= 1,2,N1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadMgacqGH9aqpcaqGGaGaaGymaiaacYcacaaIYaGaaiil aiabgAci8kaad6eacqGHsislcaaIXaaaaa@401C@

y(c)=γ y k +(1γ) y k+1 +γ(1γ)h( p 1 , p 2 ,..., p m ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG5b GaaiikaiaadogacaGGPaGaeyypa0Jaeq4SdCMaeyyXICTaamyEaSWa aSbaaeaajugWaiaadUgaaSqabaqcLbsacqGHRaWkcaGGOaGaaGymai abgkHiTiabeo7aNjaacMcacaWG5bWcdaWgaaqaaKqzadGaam4Aaiab gUcaRiaaigdaaSqabaqcLbsacqGHRaWkcqaHZoWzcaGGOaGaaGymai abgkHiTiabeo7aNjaacMcacqGHflY1caWGObGaaiikaiaadchalmaa BaaabaqcLbmacaaIXaaaleqaaKqzGeGaaiilaiaadchalmaaBaaaba qcLbmacaaIYaaaleqaaKqzGeGaaiilaiaac6cacaGGUaGaaiOlaiaa cYcacaWGWbqcfa4aaSbaaSqaaKqzadGaamyBaaWcbeaajugibiaacM caaaa@693F@ , (1)

α i ε[ 0;1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeg7aHTWdamaaBaaabaqcLbmapeGaamyAaaWcpaqabaqc LbsacqaH1oqzjuaGdaWadaGcbaqcLbsapeGaaGimaiaacUdacaaIXa aak8aacaGLBbGaayzxaaaaaa@4271@ , γ =F( α ) =F( α 1 ,, a N 1 )ε[ 0;1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNjaabccacqGH9aqpcaWGgbqcfa4damaabmaakeaa jugib8qacqaHXoqyaOWdaiaawIcacaGLPaaajugib8qacaqGGaGaey ypa0JaamOra8aacaGGOaWdbiabeg7aHTWdamaaBaaabaqcLbmapeGa aGymaaWcpaqabaqcLbsapeGaaiilaiabgAci8kaacYcacaWGHbWcpa WaaSbaaeaajugWa8qacaWGobaal8aabeaadaWgaaqaaKqzadWdbiab gkHiTiaaigdaaSWdaeqaaKqzGeGaaiykaiabew7aLLqbaoaadmaake aajugib8qacaaIWaGaai4oaiaaigdaaOWdaiaawUfacaGLDbaaaaa@5965@ .

Then N1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGobGaeyOeI0IaaGymaaaa@391F@  features c 1 ,, c N 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadogal8aadaWgaaqaaKqzadWdbiaaigdaaSWdaeqaaKqz GeWdbiaacYcacqGHMacVcaGGSaGaam4yaSWdamaaBaaabaqcLbmape GaamOtaaWcpaqabaWaaSbaaeaajugWa8qacqGHsislcaaIXaaal8aa beaaaaa@43D7@ are parameterized by α 1 ,, α N 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHXo qylmaaBaaabaqcLbmaqaaaaaaaaaWdbiaaigdaaSWdaeqaaKqzGeWd biaacYcacqGHMacVcaGGSaGaeqySde2cpaWaaSbaaeaajugWa8qaca WGobaal8aabeaadaWgaaqaaKqzadWdbiabgkHiTiaaigdaaSWdaeqa aaaa@4526@ between two nodes and the last feature (for example y) is interpolated via formula (1). Of course there can be calculated x( c ) or z( c ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG4bWdamaabmaabaWdbiaadogaa8aacaGLOaGaayzkaaWd biaabccacaWGVbGaamOCaiaabccacaWG6bWdamaabmaabaWdbiaado gaa8aacaGLOaGaayzkaaaaaa@411F@  using (1). Two examples of h (when N=2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad6eacqGH9aqpcaaIYaaaaa@393A@ ) computed for MHR method26 with good features because of orthogonal rows and columns at Hurwitz-Radon family of matrices:

h( p 1 , p 2 )= y 1 x 1 x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGOb GaaiikaiaadchalmaaBaaabaqcLbmacaaIXaaaleqaaKqzGeGaaiil aiaadchalmaaBaaabaqcLbmacaaIYaaaleqaaKqzGeGaaiykaiabg2 da9KqbaoaalaaakeaajugibiaadMhalmaaBaaabaqcLbmacaaIXaaa leqaaaGcbaqcLbsacaWG4bWcdaWgaaqaaKqzadGaaGymaaWcbeaaaa qcLbsacaWG4bWcdaWgaaqaaKqzadGaaGOmaaWcbeaaaaa@4D82@ + y 2 x 2 x 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHRa WkjuaGdaWcaaGcbaqcLbsacaWG5bWcdaWgaaqaaKqzadGaaGOmaaWc beaaaOqaaKqzGeGaamiEaSWaaSbaaeaajugWaiaaikdaaSqabaaaaK qzGeGaamiEaSWaaSbaaeaajugWaiaaigdaaSqabaaaaa@4320@  (2)

or
h( p 1 , p 2 , p 3 , p 4 )= 1 x 1 2 + x 3 2 ( x 1 x 2 y 1 + x 2 x 3 y 3 + x 3 x 4 y 1 x 1 x 4 y 3 )+ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGOb GaaiikaiaadchajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGa aiilaiaadchajuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqzGeGaai ilaiaadchalmaaBaaabaqcLbmacaaIZaaaleqaaKqzGeGaaiilaiaa dchajuaGdaWgaaWcbaqcLbmacaaI0aaaleqaaKqzGeGaaiykaiabg2 da9KqbaoaalaaakeaajugibiaaigdaaOqaaKqzGeGaamiEaSWaaSba aeaajugWaiaaigdaaSqabaqcfa4aaWbaaSqabeaajugWaiaaikdaaa qcLbsacqGHRaWkcaWG4bqcfa4aaSbaaSqaaKqzadGaaG4maaWcbeaa juaGdaahaaWcbeqaaKqzadGaaGOmaaaaaaqcLbsacaGGOaGaamiEaS WaaSbaaeaajugWaiaaigdaaSqabaqcLbsacaWG4bWcdaWgaaqaaKqz adGaaGOmaaWcbeaajugibiaadMhalmaaBaaabaqcLbmacaaIXaaale qaaKqzGeGaey4kaSIaamiEaKqbaoaaBaaaleaajugWaiaaikdaaSqa baqcLbsacaWG4bWcdaWgaaqaaKqzadGaaG4maaWcbeaajugibiaadM halmaaBaaabaqcLbmacaaIZaaaleqaaKqzGeGaey4kaSIaamiEaSWa aSbaaeaajugWaiaaiodaaSqabaqcLbsacaWG4bWcdaWgaaqaaKqzad GaaGinaaWcbeaajugibiaadMhalmaaBaaabaqcLbmacaaIXaaaleqa aKqzGeGaeyOeI0IaamiEaSWaaSbaaeaajugWaiaaigdaaSqabaqcLb sacaWG4bWcdaWgaaqaaKqzadGaaGinaaWcbeaajugibiaadMhalmaa BaaabaqcLbmacaaIZaaaleqaaKqzGeGaaiykaiabgUcaRaaa@8E5E@ 1 x 2 2 + x 4 2 ( x 1 x 2 y 2 + x 1 x 4 y 4 + x 3 x 4 y 2 x 2 x 3 y 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaaGymaaGcbaqcLbsacaWG4bWcdaWgaaqaaKqzadGaaGOm aaWcbeaadaahaaqabeaajugWaiaaikdaaaqcLbsacqGHRaWkcaWG4b WcdaWgaaqaaKqzadGaaGinaaWcbeaadaahaaqabeaajugWaiaaikda aaaaaKqzGeGaaiikaiaadIhalmaaBaaabaqcLbmacaaIXaaaleqaaK qzGeGaamiEaSWaaSbaaeaajugWaiaaikdaaSqabaqcLbsacaWG5bWc daWgaaqaaKqzadGaaGOmaaWcbeaajugibiabgUcaRiaadIhalmaaBa aabaqcLbmacaaIXaaaleqaaKqzGeGaamiEaSWaaSbaaeaajugWaiaa isdaaSqabaqcLbsacaWG5bqcfa4aaSbaaSqaaKqzadGaaGinaaWcbe aajugibiabgUcaRiaadIhalmaaBaaabaqcLbmacaaIZaaaleqaaKqz GeGaamiEaSWaaSbaaeaajugWaiaaisdaaSqabaqcLbsacaWG5bWcda WgaaqaaKqzadGaaGOmaaWcbeaajugibiabgkHiTiaadIhalmaaBaaa baqcLbmacaaIYaaaleqaaKqzGeGaamiEaSWaaSbaaeaajugWaiaaio daaSqabaqcLbsacaWG5bWcdaWgaaqaaKqzadGaaGinaaWcbeaajugi biaacMcaaaa@7597@ .

The simplest nodes combination is

h( p 1 , p 2 ,..., p m )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGOb GaaiikaiaadchalmaaBaaabaqcLbmacaaIXaaaleqaaKqzGeGaaiil aiaadchajuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqzGeGaaiilai aac6cacaGGUaGaaiOlaiaacYcacaWGWbqcfa4aaSbaaSqaaKqzadGa amyBaaWcbeaajugibiaacMcacqGH9aqpcaaIWaaaaa@4AF1@  (3)

and then there is a formula of interpolation:

y(c)=γ y i +(1γ) y i+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG5b GaaiikaiaadogacaGGPaGaeyypa0Jaeq4SdCMaeyyXICTaamyEaSWa aSbaaeaajugWaiaadMgaaSqabaqcLbsacqGHRaWkcaGGOaGaaGymai abgkHiTiabeo7aNjaacMcacaWG5bWcdaWgaaqaaKqzadGaamyAaiab gUcaRiaaigdaaSqabaaaaa@4D13@ .

Formula (1) gives the infinite number of calculations for unknown feature (determined by choice of F and h) as there is the infinite number of objects to recognize or the infinite number of images to rerieve. Nodes combination is the individual feature of each modeled data. Coefficient γ=F( α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNjabg2da9iaadAeajuaGpaWaaeWaaOqaaKqzGeWd biabeg7aHbGcpaGaayjkaiaawMcaaaaa@3EA4@ and nodes combination h are key factors in PFC data interpolation and object modeling.

N-dimensional probability distribution functions in PFC modeling

Unknown values of features, settled between the nodes, are computed using PFC method as in (1). Key question is dealing with coefficient γ. The simplest way of PFC calculation means h=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadIgacqGH9aqpcaaIWaaaaa@3952@ and γ i = α i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNTWdamaaBaaabaqcLbmapeGaamyAaaWcpaqabaGa eyypa0tcLbsapeGaeqySde2cpaWaaSbaaeaajugWa8qacaWGPbaal8 aabeaaaaa@4092@ (basic probability distribution for each random variable α i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHXo qylmaaBaaabaqcLbmaqaaaaaaaaaWdbiaadMgaaSWdaeqaaaaa@3AA6@ ). Then PFC represents a linear interpolation. Figure 1 is the example of curve (data) modeling when the formula is known: y= 2 x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadMhacqGH9aqpcaaIYaWcpaWaaWbaaeqabaqcLbmapeGa amiEaaaaaaa@3BDC@ . MHR method26 is the example of PFC modeling for feature vector of dimension N=2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad6eacqGH9aqpcaaIYaaaaa@393A@ . Each interpolation requires specific distributions of random variables α i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHXo qylmaaBaaabaqcLbmaqaaaaaaaaaWdbiaadMgaaSWdaeqaaaaa@3AA6@ and γ in (1) depends on parameters α i ε[ 0;1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeg7aHTWdamaaBaaabaqcLbmapeGaamyAaaWcpaqabaqc LbsacqaH1oqzjuaGdaWadaGcbaqcLbsapeGaaGimaiaacUdacaaIXa aak8aacaGLBbGaayzxaaaaaa@4271@ :

γ =F( α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNjaabccacqGH9aqpcaWGgbqcfa4damaabmaakeaa jugib8qacqaHXoqyaOWdaiaawIcacaGLPaaaaaa@3F47@ , F: [ 0;1 ] N 1 [ 0;1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadAeacaGG6aqcfa4damaadmaakeaajugib8qacaaIWaGa ai4oaiaaigdaaOWdaiaawUfacaGLDbaalmaaCaaabeqaaKqzadWdbi aad6eaaaWcpaWaaWbaaeqabaqcLbmapeGaeyOeI0IaaGymaaaajugi biabgkziUMqba+aadaWadaGcbaqcLbsapeGaaGimaiaacUdacaaIXa aak8aacaGLBbGaayzxaaaaaa@4B14@ , F( 0,,0 ) = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadAeajuaGpaWaaeWaaOqaaKqzGeWdbiaaicdacaGGSaGa eyOjGWRaaiilaiaaicdaaOWdaiaawIcacaGLPaaajugib8qacaqGGa Gaeyypa0Jaaeiiaiaaicdaaaa@425F@ , F( 1,,1 ) = 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadAeajuaGpaWaaeWaaOqaaKqzGeWdbiaaigdacaGGSaGa eyOjGWRaaiilaiaaigdaaOWdaiaawIcacaGLPaaajugib8qacaqGGa Gaeyypa0Jaaeiiaiaaigdaaaa@4262@

and F is strictly monotonic for each random variable α i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeg7aHTWdamaaBaaabaqcLbmapeGaamyAaaWcpaqabaaa aa@3AC5@ separately. Coefficient γ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNTWdamaaBaaabaqcLbmapeGaamyAaaWcpaqabaaa aa@3ACD@ are calculated using appropriate function and choice of function is connected with initial requirements and data specifications. Different values of coefficients γ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNTWdamaaBaaabaqcLbmapeGaamyAaaWcpaqabaaa aa@3ACD@ are connected with applied functions F i ( α i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadAeal8aadaWgaaqaaKqzadWdbiaadMgaaSWdaeqaaKqb aoaabmaakeaajugibiabeg7aHLqbaoaaBaaaleaajugWa8qacaWGPb aal8aabeaaaOGaayjkaiaawMcaaaaa@414A@ . These functions γ i = F i ( α i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNTWdamaaBaaabaqcLbmapeGaamyAaaWcpaqabaqc LbsapeGaeyypa0JaamOraSWdamaaBaaabaqcLbmapeGaamyAaaWcpa qabaqcfa4aaeWaaOqaaKqzGeGaeqySde2cdaWgaaqaaKqzadWdbiaa dMgaaSWdaeqaaaGccaGLOaGaayzkaaaaaa@4689@ represent the examples of probability distribution functions for random variable α i ε[ 0;1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHXo qylmaaBaaabaqcLbmaqaaaaaaaaaWdbiaadMgaaSWdaeqaaKqzGeGa eqyTduwcfa4aamWaaOqaaKqzGeWdbiaaicdacaGG7aGaaGymaaGcpa Gaay5waiaaw2faaaaa@4252@ and real number s>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadohacqGH+aGpcaaIWaaaaa@395F@ , i= 1,2,N1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadMgacqGH9aqpcaqGGaGaaGymaiaacYcacaaIYaGaaiil aiabgAci8kaad6eacqGHsislcaaIXaaaaa@401C@ :
γ i = α i s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNTWdamaaBaaabaqcLbmapeGaamyAaaWcpaqabaqc LbsapeGaeyypa0JaeqySde2cpaWaaSbaaeaajugWa8qacaWGPbaal8 aabeaadaahaaqabeaajugWa8qacaWGZbaaaaaa@42EA@ , γ i =sin( α i s ·π/2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNTWdamaaBaaabaqcLbmapeGaamyAaaWcpaqabaqc LbsapeGaeyypa0Jaam4CaiaadMgacaWGUbqcfa4damaabmaakeaaju gibiabeg7aHTWaaSbaaeaajugWa8qacaWGPbaal8aabeaadaahaaqa beaajugWa8qacaWGZbaaaKqzGeGaai4Taiabec8aWjaac+cacaaIYa aak8aacaGLOaGaayzkaaaaaa@4D82@ , γ i =si n s ( α i ·π/2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNTWdamaaBaaabaqcLbmapeGaamyAaaWcpaqabaqc LbsapeGaeyypa0Jaam4CaiaadMgacaWGUbWcpaWaaWbaaeqabaqcLb mapeGaam4CaaaajuaGpaWaaeWaaOqaaKqzGeGaeqySde2cdaWgaaqa aKqzadWdbiaadMgaaSWdaeqaaKqzGeWdbiaacElacqaHapaCcaGGVa GaaGOmaaGcpaGaayjkaiaawMcaaaaa@4DAC@ , γ i =1cos( α i s ·π/2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNTWdamaaBaaabaqcLbmapeGaamyAaaWcpaqabaqc LbsapeGaeyypa0JaaGymaiabgkHiTiaadogacaWGVbGaam4CaKqba+ aadaqadaGcbaqcLbsapeGaeqySde2cpaWaaSbaaeaajugWa8qacaWG Pbaal8aabeaadaahaaqabeaajugWa8qacaWGZbaaaKqzGeGaai4Tai abec8aWjaac+cacaaIYaaak8aacaGLOaGaayzkaaaaaa@4F44@ , γ i =1co s s ( α i ·π/2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNTWdamaaBaaabaqcLbmapeGaamyAaaWcpaqabaqc LbsapeGaeyypa0JaaGymaiabgkHiTiaadogacaWGVbGaam4CaSWdam aaCaaabeqaaKqzadWdbiaadohaaaqcfa4damaabmaakeaajugib8qa cqaHXoqyl8aadaWgaaqaaKqzadWdbiaadMgaaSWdaeqaaKqzGeWdbi aacElacqaHapaCcaGGVaGaaGOmaaGcpaGaayjkaiaawMcaaaaa@4F6E@ , γ i =tan( α i s ·π/4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNTWdamaaBaaabaqcLbmapeGaamyAaaWcpaqabaqc LbsapeGaeyypa0JaamiDaiaadggacaWGUbqcfa4damaabmaakeaaju gib8qacqaHXoqyl8aadaWgaaqaaKqzadWdbiaadMgaaSWdaeqaamaa CaaabeqaaKqzadWdbiaadohaaaqcLbsacaGG3cGaeqiWdaNaai4lai aaisdaaOWdaiaawIcacaGLPaaaaaa@4D9C@ , γ i =ta n s ( α i ·π/4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNTWdamaaBaaabaqcLbmapeGaamyAaaWcpaqabaqc LbsapeGaeyypa0JaamiDaiaadggacaWGUbqcfa4damaaCaaaleqaba qcLbmapeGaam4CaaaajuaGpaWaaeWaaOqaaKqzGeWdbiabeg7aHTWd amaaBaaabaqcLbmapeGaamyAaaWcpaqabaqcLbsapeGaai4Taiabec 8aWjaac+cacaaI0aaak8aacaGLOaGaayzkaaaaaa@4E54@ , γ i =lo g 2 ( α i s +1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNTWdamaaBaaabaqcLbmapeGaamyAaaWcpaqabaqc LbsapeGaeyypa0JaamiBaiaad+gacaWGNbWcpaWaaSbaaeaajugWa8 qacaaIYaaal8aabeaajuaGdaqadaGcbaqcLbsapeGaeqySde2cpaWa aSbaaeaajugWa8qacaWGPbaal8aabeaadaahaaqabeaajugWa8qaca WGZbaaaKqzGeGaey4kaSIaaGymaaGcpaGaayjkaiaawMcaaaaa@4D0F@ γ i =lo g 2 s ( α i +1 ),  γ i = ( 2 α 1 ) s ,  γ i =2/π·arcsin( α i s ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNTWdamaaBaaabaqcLbmapeGaamyAaaWcpaqabaqc LbsapeGaeyypa0JaamiBaiaad+gacaWGNbWcpaWaaSbaaeaajugWa8 qacaaIYaaal8aabeaadaahaaqabeaajugWa8qacaWGZbaaaKqba+aa daqadaGcbaqcLbsacqaHXoqylmaaBaaabaqcLbmapeGaamyAaaWcpa qabaqcLbsapeGaey4kaSIaaGymaaGcpaGaayjkaiaawMcaaKqzGeWd biaacYcacaqGGaGaeq4SdC2cpaWaaSbaaeaajugWa8qacaWGPbaal8 aabeaajugib8qacqGH9aqpjuaGpaWaaeWaaOqaaKqzGeWdbiaaikda lmaaCaaajuaGbeqaaKqzadGaeqySdegaaKqzGeGaai4eGiaaigdaaO WdaiaawIcacaGLPaaalmaaCaaabeqaaKqzadWdbiaadohaaaqcLbsa caGGSaGaaeiiaiabeo7aNTWdamaaBaaabaqcLbmapeGaamyAaaWcpa qabaqcLbsapeGaeyypa0JaaGOmaiaac+cacqaHapaCcaGG3cGaamyy aiaadkhacaWGJbGaam4CaiaadMgacaWGUbqcfa4damaabmaakeaaju gibiabeg7aHTWaaSbaaeaajugWa8qacaWGPbaal8aabeaadaahaaqa beaajugWa8qacaWGZbaaaaGcpaGaayjkaiaawMcaaaaa@7B53@ , γ i = ( 2/π·arcsin α i ) s γ i =12/π·arccos( α i s ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNTWdamaaBaaabaqcLbmapeGaamyAaaWcpaqabaqc LbsapeGaeyypa0tcfa4damaabmaakeaajugib8qacaaIYaGaai4lai abec8aWjaacElacaWGHbGaamOCaiaadogacaWGZbGaamyAaiaad6ga cqaHXoqyl8aadaWgaaqaaKqzadWdbiaadMgaaSWdaeqaaaGccaGLOa Gaayzkaaqcfa4aaWbaaSqabeaajugWa8qacaWGZbaaaKqba+aacaqG SaqcLbsapeGaaeiiaiabeo7aNTWdamaaBaaabaqcLbmapeGaamyAaa WcpaqabaqcLbsapeGaeyypa0JaaGymaiabgkHiTiaaikdacaGGVaGa eqiWdaNaai4TaiaadggacaWGYbGaam4yaiaadogacaWGVbGaam4CaK qba+aadaqadaGcbaqcLbsapeGaeqySde2cpaWaaSbaaeaajugWa8qa caWGPbaal8aabeaadaahaaqabeaajugWa8qacaWGZbaaaaGcpaGaay jkaiaawMcaaaaa@6DC4@ , γ i =1 ( 2/π·arccos α i ) s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNTWdamaaBaaabaqcLbmapeGaamyAaaWcpaqabaqc LbsapeGaeyypa0JaaGymaiabgkHiTKqba+aadaqadaGcbaqcLbsape GaaGOmaiaac+cacqaHapaCcaGG3cGaamyyaiaadkhacaWGJbGaam4y aiaad+gacaWGZbGaeqySde2cpaWaaSbaaeaajugWa8qacaWGPbaal8 aabeaaaOGaayjkaiaawMcaaSWaaWbaaeqabaqcLbmapeGaam4Caaaa aaa@5176@ , γ i =4/π·arctan( α i s ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNTWdamaaBaaabaqcLbmapeGaamyAaaWcpaqabaqc LbsapeGaeyypa0JaaGinaiaac+cacqaHapaCcaGG3cGaamyyaiaadk hacaWGJbGaamiDaiaadggacaWGUbqcfa4damaabmaakeaajugib8qa cqaHXoqyl8aadaWgaaqaaKqzadWdbiaadMgaaSWdaeqaamaaCaaabe qaaKqzadWdbiaadohaaaaak8aacaGLOaGaayzkaaaaaa@4FD2@ , γ i = ( 4/π·arctan α i ) s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNTWdamaaBaaabaqcLbmapeGaamyAaaWcpaqabaqc LbsapeGaeyypa0tcfa4damaabmaakeaajugib8qacaaI0aGaai4lai abec8aWjaacElacaWGHbGaamOCaiaadogacaWG0bGaamyyaiaad6ga cqaHXoqyl8aadaWgaaqaaKqzadWdbiaadMgaaSWdaeqaaaGccaGLOa GaayzkaaWcdaahaaqabeaajugWa8qacaWGZbaaaaaa@4FCE@ , γ i =ctg( π/2 α i s ·π/4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNLqba+aadaWgaaWcbaqcLbmapeGaamyAaaWcpaqa baqcLbsapeGaeyypa0Jaam4yaiaadshacaWGNbqcfa4damaabmaake aajugib8qacqaHapaCcaGGVaGaaGOmaiaacobicqaHXoqyl8aadaWg aaqaaKqzadWdbiaadMgaaSWdaeqaamaaCaaabeqaaKqzadWdbiaado haaaqcLbsacaGG3cGaeqiWdaNaai4laiaaisdaaOWdaiaawIcacaGL Paaaaaa@5208@ ,
γ i =ct g s ( π/2 α i .π/4 ),  γ i =24/π·arcctg( α i s ),  γ i = ( 24/π·arcctg α i ) s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNTWdamaaBaaabaqcLbmapeGaamyAaaWcpaqabaqc LbsapeGaeyypa0Jaam4yaiaadshacaWGNbqcfa4damaaCaaaleqaba qcLbmapeGaam4CaaaajuaGpaWaaeWaaOqaaKqzGeWdbiabec8aWjaa c+cacaaIYaGaeyOeI0IaeqySde2cpaWaaSbaaeaajugWa8qacaWGPb aal8aabeaajugib8qacaGGUaGaeqiWdaNaai4laiaaisdaaOWdaiaa wIcacaGLPaaajugib8qacaGGSaGaaeiiaiabeo7aNTWdamaaBaaaba qcLbmapeGaamyAaaWcpaqabaqcLbsapeGaeyypa0JaaGOmaiabgkHi TiaaisdacaGGVaGaeqiWdaNaai4TaiaadggacaWGYbGaam4yaiaado gacaWG0bGaam4zaKqba+aadaqadaGcbaqcLbsapeGaeqySdewcfa4d amaaBaaaleaajugib8qacaWGPbaal8aabeaajuaGdaahaaWcbeqaaK qzGeWdbiaadohaaaaak8aacaGLOaGaayzkaaqcLbsapeGaaiilaiaa bccacqaHZoWzl8aadaWgaaqaaKqzadWdbiaadMgaaSWdaeqaaKqzGe Wdbiabg2da9Kqba+aadaqadaGcbaqcLbsapeGaaGOmaiabgkHiTiaa isdacaGGVaGaeqiWdaNaai4TaiaadggacaWGYbGaam4yaiaadogaca WG0bGaam4zaiabeg7aHTWdamaaBaaabaqcLbmapeGaamyAaaWcpaqa baaakiaawIcacaGLPaaalmaaCaaabeqaaKqzadWdbiaadohaaaaaaa@8B4A@

Figure 1 PFC linear 2D modeling of function with seven nodes (in left window) and options in right window (modeling functions γ and nodes combination h).

or any strictly monotonic function between points ( 0;0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeaeaaaaaaaaa8qacaaIWaGaai4oaiaaicdaaOWdaiaawIca caGLPaaaaaa@3B12@ and ( 1;1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeaeaaaaaaaaa8qacaaIXaGaai4oaiaaigdaaOWdaiaawIca caGLPaaaaaa@3B14@ – for example combinations of these functions. Interpolations of function y= 2 x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadMhacqGH9aqpcaaIYaWcpaWaaWbaaeqabaqcLbmapeGa amiEaaaaaaa@3BDC@ for N= 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad6eacqGH9aqpcaqGGaGaaGOmaaaa@39DD@ , h = 0 and γ= α s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNjabg2da9iabeg7aHTWdamaaCaaabeqaaKqzadWd biaadohaaaaaaa@3D63@ with s= 0.8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadohacqGH9aqpcaqGGaGaaGimaiaac6cacaaI4aaaaa@3B74@ (Figure 2) or γ=lo g 2 ( α+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNjabg2da9iaadYgacaWGVbGaam4zaKqba+aadaWg aaWcbaqcLbmapeGaaGOmaaWcpaqabaqcfa4aaeWaaOqaaKqzGeWdbi abeg7aHjabgUcaRiaaigdaaOWdaiaawIcacaGLPaaaaaa@4515@ (Figure 3) are quite better then linear interpolation (Figure 1).

Figure 2 PFC two-dimensional modeling of function y= 2 x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadMhacqGH9aqpcaaIYaWcpaWaaWbaaeqabaqcLbmapeGa amiEaaaaaaa@3BDC@ with seven nodes as Figure 1 and h=0, γ= α 0.8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadIgacqGH9aqpcaaIWaGaaiilaiaabccacqaHZoWzcqGH 9aqpcqaHXoqyjuaGpaWaaWbaaSqabeaajugWa8qacaaIWaGaaiOlai aaiIdaaaaaaa@4327@ .
Figure 3 PFC two-dimensional reconstruction of function y= 2 x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadMhacqGH9aqpcaaIYaWcpaWaaWbaaeqabaqcLbmapeGa amiEaaaaaaa@3BDC@ with seven nodes as Figure 1 and h=0, γ =lo g 2 ( α+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadIgacqGH9aqpcaaIWaGaaiilaiaabccacqaHZoWzcaqG GaGaeyypa0JaamiBaiaad+gacaWGNbWcpaWaaSbaaeaajugWa8qaca aIYaaal8aabeaajuaGdaqadaGcbaqcLbsapeGaeqySdeMaey4kaSIa aGymaaGcpaGaayjkaiaawMcaaaaa@492A@ .

Functions γ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNTWdamaaBaaabaqcLbmapeGaamyAaaWcpaqabaaa aa@3ACD@ are strictly monotonic for each random variable α i ε[ 0;1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeg7aHLqba+aadaWgaaWcbaqcLbsapeGaamyAaaWcpaqa baqcLbsacqaH1oqzjuaGdaWadaGcbaqcLbsapeGaaGimaiaacUdaca aIXaaak8aacaGLBbGaayzxaaaaaa@4260@ as γ=F( α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNjabg2da9iaadAeajuaGpaWaaeWaaOqaaKqzGeWd biabeg7aHbGcpaGaayjkaiaawMcaaaaa@3EA4@ is N-dimensional probability distribution function, for example:

γ= 1 N1 i1 N1 γ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHZo WzcqGH9aqpjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaad6ea cqGHsislcaaIXaaaaKqbaoaaqahakeaajugibiabeo7aNLqbaoaaBa aaleaajugWaiaadMgaaSqabaaabaqcLbmacaWGPbGaeyOeI0IaaGym aaWcbaqcLbmacaWGobGaeyOeI0IaaGymaaqcLbsacqGHris5aaaa@4E18@ , γ= i=1 N1 γ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHZo WzcqGH9aqpjuaGdaqeWbGcbaqcLbsacqaHZoWzjuaGdaWgaaWcbaqc LbmacaWGPbaaleqaaaqaaKqzadGaamyAaiabg2da9iaaigdaaSqaaK qzadGaamOtaiabgkHiTiaaigdaaKqzGeGaey4dIunaaaa@491A@

and every monotonic combination of γi such as

γ=F( α ), F: [ 0;1 ] N 1 [ 0;1 ], F( 0,,0 ) = 0, F( 1,,1 )1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNjabg2da9iaadAeajuaGpaWaaeWaaOqaaKqzGeWd biabeg7aHbGcpaGaayjkaiaawMcaaKqzGeWdbiaacYcacaGGGcGaam OraiaacQdajuaGpaWaamWaaOqaaKqzGeWdbiaaicdacaGG7aGaaGym aaGcpaGaay5waiaaw2faaSWaaWbaaeqabaqcLbmapeGaamOtaaaal8 aadaahaaqabeaajugWa8qacqGHsislcaaIXaaaaKqzGeGaeyOKH4Ac fa4damaadmaakeaajugib8qacaaIWaGaai4oaiaaigdaaOWdaiaawU facaGLDbaajugib8qacaGGSaGaaiiOaiaadAeajuaGpaWaaeWaaOqa aKqzGeWdbiaaicdacaGGSaGaeyOjGWRaaiilaiaaicdaaOWdaiaawI cacaGLPaaajugib8qacaqGGaGaaeypaiaabccacaaIWaGaaiilaiaa cckacaWGgbqcfa4damaabmaakeaajugib8qacaaIXaGaaiilaiabgA ci8kaacYcacaaIXaaak8aacaGLOaGaayzkaaqcfaOaaeypaKqzGeWd biaabccacaaIXaaaaa@70A3@ .

For example when N= 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad6eacqGH9aqpcaqGGaGaaG4maaaa@39DE@  there is a bilinear interpolation:

γ 1 = α 1 , γ 2 = α 2  ,γ= ½( α 1 +  α 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNTWdamaaBaaabaqcLbmapeGaaGymaaWcpaqabaqc LbsapeGaeyypa0JaeqySde2cpaWaaSbaaeaajugWa8qacaaIXaaal8 aabeaajugib8qacaGGSaGaeq4SdCwcfa4damaaBaaaleaajugWa8qa caaIYaaal8aabeaajugib8qacqGH9aqpcqaHXoqyl8aadaWgaaqaaK qzadWdbiaaikdaaSWdaeqaaKqzGeWdbiaacckacaGGSaGaeq4SdCMa eyypa0Jaaeiiaiaac2lapaGaaiika8qacqaHXoqyl8aadaWgaaqaaK qzadWdbiaaigdaaSWdaeqaaKqzGeWdbiabgUcaRiaabccacqaHXoqy l8aadaWgaaqaaKqzadWdbiaaikdaaSWdaeqaaKqzGeGaaiykaaaa@5E9C@ (4)

or a bi-quadratic interpolation:

γ 1 = α 1 2 , γ 2 = α 2 2  ,γ= ½( α 1 2 +  α 2 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNTWdamaaBaaabaqcLbmapeGaaGymaaWcpaqabaqc LbsapeGaeyypa0JaeqySde2cpaWaaSbaaeaajugWa8qacaaIXaaal8 aabeaadaahaaqabeaajugWa8qacaaIYaaaaKqzGeGaaiilaiabeo7a NTWdamaaBaaabaqcLbmapeGaaGOmaaWcpaqabaqcLbsapeGaeyypa0 JaeqySde2cpaWaaSbaaeaajugWa8qacaaIYaaal8aabeaadaahaaqa beaajugWa8qacaaIYaaaaKqzGeGaaiiOaiaacYcacqaHZoWzcqGH9a qpcaqGGaGaaiyVa8aacaGGOaWdbiabeg7aHTWdamaaBaaabaqcLbma peGaaGymaaWcpaqabaWaaWbaaeqabaqcLbmapeGaaGOmaaaajugibi abgUcaRiaabccacqaHXoqyl8aadaWgaaqaaKqzadWdbiaaikdaaSWd aeqaamaaCaaabeqaaKqzadWdbiaaikdaaaqcLbsapaGaaiykaaaa@665D@  (5)

or a bi-cubic interpolation:

γ 1 = α 1 3 , γ 2 = α 2 3  ,γ= ½( α 1 3 +  α 2 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNTWdamaaBaaabaqcLbmapeGaaGymaaWcpaqabaqc LbsapeGaeyypa0JaeqySde2cpaWaaSbaaeaajugWa8qacaaIXaaal8 aabeaadaahaaqabeaajugWa8qacaaIZaaaaKqzGeGaaiilaiabeo7a NLqba+aadaWgaaWcbaqcLbmapeGaaGOmaaWcpaqabaqcLbsapeGaey ypa0JaeqySde2cpaWaaSbaaeaajugWa8qacaaIYaaal8aabeaadaah aaqabeaajugWa8qacaaIZaaaaKqzGeGaaiiOaiaacYcacqaHZoWzcq GH9aqpcaqGGaGaaiyVa8aacaGGOaWdbiabeg7aHTWdamaaBaaabaqc LbmapeGaaGymaaWcpaqabaWaaWbaaeqabaqcLbmapeGaaG4maaaaju gibiabgUcaRiaabccacqaHXoqyl8aadaWgaaqaaKqzadWdbiaaikda aSWdaeqaamaaCaaabeqaaKqzadWdbiaaiodaaaqcLbsapaGaaiykaa aa@66EF@  (6)

or others modeling functions γ. Choice of functions γ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNTWdamaaBaaabaqcLbmapeGaamyAaaWcpaqabaaa aa@3ACD@ and value s depends on the specifications of feature vectors and individual requirements. What is very important in PFC method: two data sets (for example a handwritten letter or signature) may have the same set of nodes (feature vectors: pixel coordinates, pressure, speed, angles) but different h or γ results in different interpolations (Figure 4-6). Here are three examples of PFC reconstruction (Figure 4-6) for N=2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad6eacqGH9aqpcaaIYaaaaa@393A@ and four nodes: (-1.5;-1), (1.25;3.15), (4.4;6.8) and (8;7). Formula of the curve is not given. Algorithm of PFC retrieval, interpolation and modeling consists of five steps: first choice of nodes pi (feature vectors), then choice of nodes combination h( p 1 , p 2 ,, p m ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadIgajuaGpaWaaeWaaOqaaKqzGeWdbiaadchal8aadaWg aaqaaKqzadWdbiaaigdaaSWdaeqaaKqzGeWdbiaacYcacaWGWbqcfa 4damaaBaaaleaajugWa8qacaaIYaaal8aabeaajugib8qacaGGSaGa eyOjGWRaaiilaiaadchal8aadaWgaaqaaKqzadWdbiaad2gaaSWdae qaaaGccaGLOaGaayzkaaaaaa@49D6@ , choice of distribution (modeling function) γ=F( α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNjabg2da9iaadAeajuaGpaWaaeWaaOqaaKqzGeWd biabeg7aHbGcpaGaayjkaiaawMcaaaaa@3EA4@ , determining values of α i ε[ 0;1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeg7aHLqba+aadaWgaaWcbaqcLbsapeGaamyAaaWcpaqa baqcLbsacqaH1oqzjuaGdaWadaGcbaqcLbsapeGaaGimaiaacUdaca aIXaaak8aacaGLBbGaayzxaaaaaa@4260@ and finally the computations (1).

Figure 4 PFC 2D modeling for γ= α 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNjabg2da9iabeg7aHTWdamaaCaaabeqaaKqzadWd biaaikdaaaaaaa@3D27@ and h= 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadIgacqGH9aqpcaqGGaGaaGimaaaa@39F5@ .
Figure 5 PFC 2D reconstruction for γ=sin( α 2 ·π/2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNjabg2da9iaadohacaWGPbGaamOBaKqba+aadaqa daGcbaqcLbsapeGaeqySdewcfa4damaaCaaaleqabaqcLbmapeGaaG OmaaaajugibiaacElacqaHapaCcaGGVaGaaGOmaaGcpaGaayjkaiaa wMcaaaaa@486C@ and h in (2).
Figure 6 PFC 2D interpolation for γ=tan( α 2 ·π/4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNjabg2da9iaadshacaWGHbGaamOBaKqba+aadaqa daGcbaqcLbsacqaHXoqylmaaCaaabeqaaKqzadWdbiaaikdaaaqcLb sacaGG3cGaeqiWdaNaai4laiaaisdaaOWdaiaawIcacaGLPaaaaaa@47BA@ and h= ( x 2 / x 1 )+ ( y 2 / y 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadIgacqGH9aqpcaqGGaqcfa4damaabmaakeaajugib8qa caWG4bWcpaWaaSbaaeaajugWa8qacaaIYaaal8aabeaajugib8qaca GGVaGaamiEaSWdamaaBaaabaqcLbmapeGaaGymaaWcpaqabaaakiaa wIcacaGLPaaajugib8qacqGHRaWkcaqGGaqcfa4damaabmaakeaaju gib8qacaWG5bWcpaWaaSbaaeaajugWa8qacaaIYaaal8aabeaajugi b8qacaGGVaGaamyEaSWdamaaBaaabaqcLbmapeGaaGymaaWcpaqaba aakiaawIcacaGLPaaaaaa@50E5@ .

Image retrieval via PFC high-dimensional feature reconstruction

After the process of image segmentation and during the next steps of retrieval, recognition or identification, there is a huge number of features included in N-dimensional feature vector. These vectors can be treated as “points” in N-dimensional feature space. For example in artificial intelligence there is a high-dimensional search space (the set of states that can be reached in a search problem) or hypothesis space (the set of hypothesis that can be generated by a machine learning algorithm). This paper is dealing with multidimensional feature spaces that are used in computer vision, image processing and machine learning. Having monochromatic (binary) image which consists of some objects, there is only 2-dimensional feature space ( x i , y i )  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeaeaaaaaaaaa8qacaWG4bWcpaWaaSbaaeaajugWa8qacaWG Pbaal8aabeaajugib8qacaGGSaGaamyEaSWdamaaBaaabaqcLbmape GaamyAaaWcpaqabaaakiaawIcacaGLPaaajugib8qacaGGGcaaaa@42DF@ – coordinates of black pixels or coordinates of white pixels. No other parameters are needed. Thus any object can be described by a contour (closed binary curve). Binary images are attractive in processing (fast and easy) but don’t include important information. If the image has grey shades, there is 3-dimensional feature space ( x i , y i , z i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeaeaaaaaaaaa8qacaWG4bWcpaWaaSbaaeaajugWa8qacaWG Pbaal8aabeaajugib8qacaGGSaGaamyEaSWdamaaBaaabaqcLbmape GaamyAaaWcpaqabaqcLbsapeGaaiilaiaadQhajuaGpaWaaSbaaSqa aKqzadWdbiaadMgaaSWdaeqaaaGccaGLOaGaayzkaaaaaa@4679@ with grey shade. For example most of medical images are written in grey shades to get quite fast processing. But when there are color images (three parameters for RGB or other color systems) with textures or medical data or some parameters, then it is N-dimensional feature space. Dealing with the problem of classification learning for high-dimensional feature spaces in artificial intelligence and machine learning (for example text classification and recognition), there are some methods: decision trees, k-nearest neighbors, perceptrons, naïve Bayes or neural networks methods. All of these methods are struggling with the curse of dimensionality: the problem of having too many features. And there are many approaches to get less number of features and to reduce the dimension of feature space for faster and less expensive calculations. This paper aims at inverse problem to the curse of dimensionality: dimension N of feature space (i.e. number of features) is unchanged, but number of feature vectors (i.e. “points” in N-dimensional feature space) is reduced into the set of nodes. So the main problem is as follows: how to fix the set of feature vectors for the image and how to retrieve the features between the “nodes”? This paper aims in giving the answer of this question.

Grey scale image retrieval using PFC 3D method

Binary images are just the case of 2D points ( x,y ): 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeaeaaaaaaaaa8qacaWG4bGaaiilaiaadMhaaOWdaiaawIca caGLPaaajugib8qacaGG6aGaaeiiaiaaicdaaaa@3E44@  or 1, black or white, so retrieval of monochromatic images is done for the closed curves (first and last node are the same) as the contours of the objects for N= 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad6eacqGH9aqpcaqGGaGaaGOmaaaa@39DD@ and examples as (Figure 1-6). Grey scale images are the case of 3D points ( x,y,s ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeaeaaaaaaaaa8qacaWG4bGaaiilaiaadMhacaGGSaGaam4C aaGcpaGaayjkaiaawMcaaaaa@3D32@ with s as the shade of grey. So the grey scale between the nodes p 1 =( x 1 , y 1 , s 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadchal8aadaWgaaqaaKqzadWdbiaaigdaaSWdaeqaaKqz GeWdbiabg2da9Kqba+aadaqadaGcbaqcLbsapeGaamiEaSWdamaaBa aabaqcLbmapeGaaGymaaWcpaqabaqcLbsapeGaaiilaiaadMhal8aa daWgaaqaaKqzadWdbiaaigdaaSWdaeqaaKqzGeWdbiaacYcacaWGZb WcpaWaaSbaaeaajugWa8qacaaIXaaal8aabeaaaOGaayjkaiaawMca aaaa@4AE1@ and p 2 =( x 2 , y 2 , s 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadchal8aadaWgaaqaaKqzadWdbiaaikdaaSWdaeqaaKqz GeWdbiabg2da9Kqba+aadaqadaGcbaqcLbsapeGaamiEaSWdamaaBa aabaqcLbmapeGaaGOmaaWcpaqabaqcLbsapeGaaiilaiaadMhal8aa daWgaaqaaKqzadWdbiaaikdaaSWdaeqaaKqzGeWdbiaacYcacaWGZb qcfa4damaaBaaaleaajugWa8qacaaIYaaal8aabeaaaOGaayjkaiaa wMcaaaaa@4B73@ is computed with γ =F( α ) =F( α 1 , α 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNjaabccacqGH9aqpcaWGgbqcfa4damaabmaakeaa jugibiabeg7aHbGccaGLOaGaayzkaaqcLbsapeGaaeiiaiabg2da9i aadAeapaGaaiikaiabeg7aHTWaaSbaaeaajugWa8qacaaIXaaal8aa beaajugib8qacaGGSaGaeqySde2cpaWaaSbaaeaajugWa8qacaaIYa aal8aabeaajugibiaacMcaaaa@4D4D@ as (1) and for example (4)-(6) or others modeling functions γ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNTWdamaaBaaabaqcLbmapeGaamyAaaWcpaqabaaa aa@3ACD@ . As the simple example two successive nodes of the image are: left upper corner with coordinates p 1 =( x 1 , y 1 ,2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadchal8aadaWgaaqaaKqzadWdbiaaigdaaSWdaeqaaKqz GeWdbiabg2da9Kqba+aadaqadaGcbaqcLbsapeGaamiEaSWdamaaBa aabaqcLbmapeGaaGymaaWcpaqabaqcLbmapeGaaiilaKqzGeGaamyE aSWdamaaBaaabaqcLbmapeGaaGymaaWcpaqabaqcLbsapeGaaiilai aaikdaaOWdaiaawIcacaGLPaaaaaa@4994@ and right down corner p 2 =( x 2 , y 2 ,10 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadchajuaGpaWaaSbaaSqaaKqzadWdbiaaikdaaSWdaeqa aKqzGeWdbiabg2da9Kqba+aadaqadaGcbaqcLbsapeGaamiEaSWdam aaBaaabaqcLbmapeGaaGOmaaWcpaqabaqcLbsapeGaaiilaiaadMha juaGpaWaaSbaaSqaaKqzadWdbiaaikdaaSWdaeqaaKqzGeWdbiaacY cacaaIXaGaaGimaaGcpaGaayjkaiaawMcaaaaa@4A3E@ . The image retrieval with the grey scales 2-10 between p 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadchal8aadaWgaaqaaKqzadWdbiaaigdaaSWdaeqaaaaa @39E8@ and p 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadchal8aadaWgaaqaaKqzadWdbiaaikdaaSWdaeqaaaaa @39E9@ looks as follows for a bilinear interpolation (4) (Figure 7) (Figure 8).

Figure 7 Reconstructed grey scale numbered at each pixel.
Figure 8 Grey scale image with shades of grey retrieved at each pixel.
The feature vector of dimension N= 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad6eacqGH9aqpcaqGGaGaaG4maaaa@39DE@ is called a voxel.

Color image retrieval via PFC method

Color images in for example RGB color system (r,g,b) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa aeaaaaaaaaa8qacaWGYbGaaiilaiaadEgacaGGSaGaamOya8aacaGG Paaaaa@3C37@ are the set of points (x,y,r,g,b) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa aeaaaaaaaaa8qacaWG4bGaaiilaiaadMhacaGGSaGaamOCaiaacYca caWGNbGaaiilaiaadkgapaGaaiykaaaa@3F92@ in a feature space of dimension N=5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad6eacqGH9aqpcaaI1aaaaa@393D@ . There can be more features, for example texture t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadshaaaa@379E@ , and then one pixel (x,y,r,g,b,t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa aeaaaaaaaaa8qacaWG4bGaaiilaiaadMhacaGGSaGaamOCaiaacYca caWGNbGaaiilaiaadkgacaGGSaGaamiDa8aacaGGPaaaaa@413B@ exists in a feature space of dimension N=6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad6eacqGH9aqpcaaI2aaaaa@393E@ . But there are the sub-spaces of a feature space of dimension N 1 <N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad6eajuaGpaWaaSbaaSqaaKqzadWdbiaaigdaaSWdaeqa aKqzGeWdbiabgYda8iaad6eaaaa@3CCA@ , for example, (x,y,g) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa aeaaaaaaaaa8qacaWG4bGaaiilaiaadMhacaGGSaGaam4za8aacaGG Paaaaa@3C54@ , (x,y,b) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa aeaaaaaaaaa8qacaWG4bGaaiilaiaadMhacaGGSaGaamOya8aacaGG Paaaaa@3C4F@ or (x,y,t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa aeaaaaaaaaa8qacaWG4bGaaiilaiaadMhacaGGSaGaamiDa8aacaGG Paaaaa@3C61@ are points in a feature sub-space of dimension N 1 =3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaqFbjugiba baaaaaaaaapeGaamOtaSWdamaaBaaabaqcLbmapeGaaGymaaWcpaqa baGaeyypa0tcLbsapeGaaG4maaaa@3D24@ . Reconstruction and interpolation of color coordinates or texture parameters is done like in chapter 3.1 for dimension N=3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaqFbjugiba baaaaaaaaapeGaamOtaiabg2da9iaaiodaaaa@3A37@ . Appropriate combination of α 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaqFbjugibi abeg7aHTWaaSbaaeaajugWaabaaaaaaaaapeGaaGymaaWcpaqabaaa aa@3B6F@ and α 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaqFbjugibi abeg7aHTWaaSbaaeaajugWaabaaaaaaaaapeGaaGOmaaWcpaqabaaa aa@3B70@ leads to modeling of color r,g,b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaqFbjugiba baaaaaaaaapeGaamOCaiaacYcacaWGNbGaaiilaiaadkgaaaa@3BCB@ or texture t or another feature between the nodes. And for example (x,y,r,t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaqFbjugibi aacIcaqaaaaaaaaaWdbiaadIhacaGGSaGaamyEaiaacYcacaWGYbGa aiilaiaadshapaGaaiykaaaa@3F04@ , (x,y,g,t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa aeaaaaaaaaa8qacaWG4bGaaiilaiaadMhacaGGSaGaam4zaiaacYca caWG0bWdaiaacMcaaaa@3DFD@ , (x,y,b,t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa aeaaaaaaaaa8qacaWG4bGaaiilaiaadMhacaGGSaGaamOyaiaacYca caWG0bWdaiaacMcaaaa@3DF8@ ) are points in a feature sub-space of dimension N 1 =4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad6eal8aadaWgaaqaaKqzadWdbiaaigdaaSWdaeqaaKqz GeWdbiabg2da9iaaisdaaaa@3C29@ called doxels. Appropriate combination of α1, a2 and a3 leads to modeling of texture t or another feature between the nodes. For example color image, given as the set of doxels (x,y,r,t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa aeaaaaaaaaa8qacaWG4bGaaiilaiaadMhacaGGSaGaamOCaiaacYca caWG0bWdaiaacMcaaaa@3E08@ , is described for coordinates (x,y) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa aeaaaaaaaaa8qacaWG4bGaaiilaiaadMhapaGaaiykaaaa@3AB8@ via pairs (r,t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa aeaaaaaaaaa8qacaWGYbGaaiilaiaadshapaGaaiykaaaa@3AAD@ interpolated between nodes ( x 1 , y 1 ,2,1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa aeaaaaaaaaa8qacaWG4bqcfa4damaaBaaaleaajugWa8qacaaIXaaa l8aabeaajugib8qacaGGSaGaamyEaKqba+aadaWgaaWcbaqcLbmape GaaGymaaWcpaqabaqcLbsapeGaaiilaiaaikdacaGGSaGaaGyma8aa caGGPaaaaa@4485@  and ( x 2 , y 2 ,10,9) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa aeaaaaaaaaa8qacaWG4bWcpaWaaSbaaeaajugWa8qacaaIYaaal8aa beaajugib8qacaGGSaGaamyEaSWdamaaBaaabaqcLbmapeGaaGOmaa WcpaqabaqcLbsapeGaaiilaiaaigdacaaIWaGaaiilaiaaiMdapaGa aiykaaaa@442C@ as follows Figure 9 So dealing with feature space of dimension N and using PFC method there is no problem called “the curse of dimensionality” and no problem called “feature selection” because each feature is important. There is no need to reduce the dimension N and no need to establish which feature is “more important” or “less important”. Every feature that depends from N 1 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad6eal8aadaWgaaqaaKqzadWdbiaaigdaaSWdaeqaaKqz GeWdbiabgkHiTiaaigdaaaa@3C0D@ other features can be interpolated (reconstructed) in the feature sub-space of dimension N 1 <N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad6eal8aadaWgaaqaaKqzadWdbiaaigdaaSWdaeqaaKqz GeWdbiabgYda8iaad6eaaaa@3C3C@ via PFC method. But having a feature space of dimension N and using PFC method there is another problem: how to reduce the number of feature vectors and how to interpolate (retrieve) the features between the known vectors (called nodes).

Figure 9 Color image with color and texture parameters (r,t) interpolated at each pixel.

Difference between two given approaches (the curse of dimensionality with feature selection and PFC interpolation) can be illustrated as follows. There is a feature matrix of dimension means the number of features (dimension of feature space) and M is the number of feature vectors (interpolation nodes)–columns are feature vectors of dimension N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad6eaaaa@3778@ . One approach (Figure 10): the curse of dimensionality with feature selection wants to eliminate some rows from the feature matrix and to reduce dimension N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad6eaaaa@3778@ to N 1 <N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad6eal8aadaWgaaqaaKqzadWdbiaaigdaaSWdaeqaaKqz GeWdbiabgYda8iaad6eaaaa@3C3C@ . Second approach (Figure 11) for PFC method wants to eliminate some columns from the feature matrix and to reduce dimension M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad2eaaaa@3777@ to M 1 <M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad2eal8aadaWgaaqaaKqzadWdbiaaigdaaSWdaeqaaKqz GeWdbiabgYda8iaad2eaaaa@3C3A@ . So after feature selection (Figure 10) there are nine feature vectors (columns): M= 9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad2eacqGH9aqpcaqGGaGaaGyoaaaa@39E3@ in a feature sub-space of dimension N 1 = 6 <N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad6eajuaGpaWaaSbaaSqaaKqzadWdbiaaigdaaSWdaeqa aKqzGeWdbiabg2da9iaabccacaaI2aGaaeiiaiabgYda8iaad6eaaa a@3FD6@ (three features are fixed as less important and reduced). But feature elimination is a very unclear matter. And what to do if every feature is denoted as meaningful and then no feature are to be reduced? For PFC method (Figure 11) there are seven feature vectors (columns): M 1 = 7 <M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad2eal8aadaWgaaqaaKqzadWdbiaaigdaaSWdaeqaaKqz GeWdbiabg2da9iaabccacaaI3aGaaeiiaiabgYda8iaad2eaaaa@3F47@ in a feature space of dimension N=9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad6eacqGH9aqpcaaI5aaaaa@3941@ . Then no feature is eliminated and the main problem is dealing with interpolation or extrapolation of feature values, like for example image retrieval (Figure 7-9).

Figure 10 The curse of dimensionality with feature selection wants to eliminate some rows from the feature matrix and to reduce dimension N.
Figure 11 PFC method wants to eliminate some columns from the feature matrix and to reduce dimension M.

Recognition tasks via high-dimensional feature vectors’ interpolation

The process of biometric recognition and identification consists of three parts: pre-processing, image segmentation with feature extraction and recognition or verification. Pre-processing is a common stage for all methods with binarization, thinning, size standardization. Proposed online approach is based on 2D curve modeling and multi-dimensional feature vectors’ interpolation. Feature extraction gives the key points (nodes as N-dimensional feature vectors) that are used in PFC curve reconstruction and identification. PFC method enables signature and handwriting recognition, which is used for biometric purposes, because human signature or handwriting consists of non-typical curves and irregular shapes (for example (Figure 4-6). The language does not matter because each symbol is treated as a curve. This process of recognition consists of three parts:

  1. Before recognition: continual and never-ending building the data basis: patterns’ modeling – choice of nodes combination, probabilistic distribution function (1) and values of features (pen pressure, speed, pen angle etc.) appearing in high dimensional feature vectors for known signature or handwritten letters of some persons in the basis;
  2. Feature extraction: unknown author – fixing the values in feature vectors for unknown signature or handwritten words: N-dimensional feature vectors (x,y,p,s,a,t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa aeaaaaaaaaa8qacaWG4bGaaiilaiaadMhacaGGSaGaamiCaiaacYca caWGZbGaaiilaiaadggacaGGSaGaamiDa8aacaGGPaaaaa@4144@ with x, y-points’ coordinates, p-pen pressure, s-speed of writing, a- pen angle or any other features t;
  3. The result: recognition or identification - comparing the results of PFC interpolation for known patterns from the data basis with features of unknown object.

Signature modeling and multidimensional recognition

Human signature or handwriting consists mainly of non-typical curves and irregular shapes. So how to model two-dimensional handwritten characters via PFC method? Each model has to be described (1) by the set of nodes, nodes combination h and a function γ=F( α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNjabg2da9iaadAeajuaGpaWaaeWaaOqaaKqzGeGa eqySdegakiaawIcacaGLPaaaaaa@3E85@ for each letter. Other features in multi-dimensional feature space are not visible but used in recognition process (for example p-pen pressure, s-speed of writing, a- pen angle). Less complicated models can take h( p 1 , p 2 ,, p m )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadIgajuaGpaWaaeWaaOqaaKqzGeWdbiaadchal8aadaWg aaqaaKqzadWdbiaaigdaaSWdaeqaaKqzGeWdbiaacYcacaWGWbqcfa 4damaaBaaaleaajugWa8qacaaIYaaal8aabeaajugib8qacaGGSaGa eyOjGWRaaiilaiaadchajuaGpaWaaSbaaSqaaKqzadWdbiaad2gaaS WdaeqaaaGccaGLOaGaayzkaaqcfaOaeyypa0tcLbsapeGaaGimaaaa @4D51@ and then the formula of interpolation (1) looks as follows:
y(c)=γ y i +(1γ) y i+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG5b GaaiikaiaadogacaGGPaGaeyypa0Jaeq4SdCMaeyyXICTaamyEaSWa aSbaaeaajugWaiaadMgaaSqabaqcLbsacqGHRaWkcaGGOaGaaGymai abgkHiTiabeo7aNjaacMcacaWG5bqcfa4aaSbaaSqaaKqzadGaamyA aiabgUcaRiaaigdaaSqabaaaaa@4DA1@ . (7)

Formula (7) represents the simplest linear interpolation for basic probability distribution if γ = α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNjaabccacqGH9aqpcaqGGaGaeqySdegaaa@3C37@ . Here are some examples of non-typical curves and irregular shapes as the whole signature or a part of signature, reconstructed via PFC method for seven nodes (x,y) (Figure 1-3).

Figure 12-18 are two-dimensional subspace of N-dimensional feature space, for example (x,y,p,s,a,t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa aeaaaaaaaaa8qacaWG4bGaaiilaiaadMhacaGGSaGaamiCaiaacYca caWGZbGaaiilaiaadggacaGGSaGaamiDa8aacaGGPaaaaa@4144@ when N= 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad6eacqGH9aqpcaqGGaGaaGOnaaaa@39E1@ . If the recognition process is working “offline” and features p-pen pressure, s-speed of writing, a- pen angle or another feature t are not given, the only information before recognition is situated in x,y-points’ coordinates. After pre-processing (binarization, thinning, size standarization), feature extraction is second part of biometric identification. Choice of characteristic points (nodes) for unknown letter or handwritten symbol is a crucial factor in object recognition. The range of coefficients x has to be the same like the x range in the basis of patterns. When the nodes are fixed, each coordinate of every chosen point on the curve ( x 0 c , y 0 c ), ( x 1 c , y 1 c ),, ( x M c , y M c ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeaeaaaaaaaaa8qacaWG4bWcpaWaaSbaaeaajugWa8qacaaI Waaal8aabeaadaahaaqabeaajugWa8qacaWGJbaaaKqzGeGaaiilai aadMhal8aadaWgaaqaaKqzadWdbiaaicdaaSWdaeqaamaaCaaabeqa aKqzadWdbiaadogaaaaak8aacaGLOaGaayzkaaqcLbsapeGaaiilai aabccajuaGpaWaaeWaaOqaaKqzGeWdbiaadIhal8aadaWgaaqaaKqz adWdbiaaigdaaSWdaeqaamaaCaaabeqaaKqzadWdbiaadogaaaqcLb sacaGGSaGaamyEaSWdamaaBaaabaqcLbmapeGaaGymaaWcpaqabaWa aWbaaeqabaqcLbmapeGaam4yaaaaaOWdaiaawIcacaGLPaaajugib8 qacaGGSaGaeyOjGWRaaiilaiaabccajuaGpaWaaeWaaOqaaKqzGeWd biaadIhal8aadaWgaaqaaKqzadWdbiaad2eaaSWdaeqaamaaCaaabe qaaKqzadWdbiaadogaaaqcLbsacaGGSaGaamyEaSWdamaaBaaabaqc LbmapeGaamytaaWcpaqabaWaaWbaaeqabaqcLbmapeGaam4yaaaaaO WdaiaawIcacaGLPaaaaaa@6A2F@ is accessible to be used for comparing with the models. Then probability distribution function γ=F( α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNjabg2da9iaadAeajuaGpaWaaeWaaOqaaKqzGeGa eqySdegakiaawIcacaGLPaaaaaa@3E85@ and nodes combination h have to be taken from the basis of modeled letters to calculate appropriate second coordinates y i ( j ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadMhal8aadaWgaaqaaKqzadWdbiaadMgaaSWdaeqaamaa CaaabeqaamaabmaabaqcLbmapeGaamOAaaWcpaGaayjkaiaawMcaaa aaaaa@3E16@ of the pattern S j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadofajuaGpaWaaSbaaSqaaKqzadWdbiaadQgaaSWdaeqa aaaa@3A8D@ for first coordinates x i c ,i= 0,1,,M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadIhal8aadaWgaaqaaKqzadWdbiaadMgaaSWdaeqaamaa CaaabeqaaKqzadWdbiaadogaaaqcLbsacaGGSaGaamyAaiabg2da9i aabccacaaIWaGaaiilaiaaigdacaGGSaGaeyOjGWRaaiilaiaad2ea aaa@4626@ . After interpolation it is possible to compare given handwritten symbol with a letter in the basis of patterns. Comparing the results of PFC interpolation for required second coordinates of a model in the basis of patterns with points on the curve ( x 0 c , y 0 c ), ( x 1 c , y 1 c ),, ( x M c , y M c ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeaeaaaaaaaaa8qacaWG4bWcpaWaaSbaaeaajugWa8qacaaI Waaal8aabeaadaahaaqabeaajugWa8qacaWGJbaaaKqzGeGaaiilai aadMhal8aadaWgaaqaaKqzadWdbiaaicdaaSWdaeqaamaaCaaabeqa aKqzadWdbiaadogaaaaak8aacaGLOaGaayzkaaqcLbsapeGaaiilai aabccajuaGpaWaaeWaaOqaaKqzGeWdbiaadIhal8aadaWgaaqaaKqz adWdbiaaigdaaSWdaeqaamaaCaaabeqaaKqzadWdbiaadogaaaqcLb sacaGGSaGaamyEaSWdamaaBaaabaqcLbmapeGaaGymaaWcpaqabaWa aWbaaeqabaqcLbmapeGaam4yaaaaaOWdaiaawIcacaGLPaaajugib8 qacaGGSaGaeyOjGWRaaiilaiaabccajuaGpaWaaeWaaOqaaKqzGeWd biaadIhal8aadaWgaaqaaKqzadWdbiaad2eaaSWdaeqaamaaCaaabe qaaKqzadWdbiaadogaaaqcLbsacaGGSaGaamyEaSWdamaaBaaabaqc LbmapeGaamytaaWcpaqabaWaaWbaaeqabaqcLbmapeGaam4yaaaaaO WdaiaawIcacaGLPaaaaaa@6A2F@ , one can say if the letter or symbol is written by person P1, P2 or another. The comparison and decision of recognition26 is done via minimal distance criterion. Curve points of unknown handwritten symbol are: ( x 0 c , y 0 c ), ( x 1 c , y 1 c ),, ( x M c , y M c ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeaeaaaaaaaaa8qacaWG4bWcpaWaaSbaaeaajugWa8qacaaI Waaal8aabeaadaahaaqabeaajugWa8qacaWGJbaaaKqzGeGaaiilai aadMhal8aadaWgaaqaaKqzadWdbiaaicdaaSWdaeqaamaaCaaabeqa aKqzadWdbiaadogaaaaak8aacaGLOaGaayzkaaqcLbsapeGaaiilai aabccajuaGpaWaaeWaaOqaaKqzGeWdbiaadIhal8aadaWgaaqaaKqz adWdbiaaigdaaSWdaeqaamaaCaaabeqaaKqzadWdbiaadogaaaqcLb sacaGGSaGaamyEaSWdamaaBaaabaqcLbmapeGaaGymaaWcpaqabaWa aWbaaeqabaqcLbmapeGaam4yaaaaaOWdaiaawIcacaGLPaaajugib8 qacaGGSaGaeyOjGWRaaiilaiaabccajuaGpaWaaeWaaOqaaKqzGeWd biaadIhal8aadaWgaaqaaKqzadWdbiaad2eaaSWdaeqaamaaCaaabe qaaKqzadWdbiaadogaaaqcLbsacaGGSaGaamyEaSWdamaaBaaabaqc LbmapeGaamytaaWcpaqabaWaaWbaaeqabaqcLbmapeGaam4yaaaaaO WdaiaawIcacaGLPaaaaaa@6A2F@ .The criterion of recognition for models S j = { ( x 0 c , y 0 (j) ), ( x 1 c , y 1 (j) ),, ( x M c , y M (j) ),j=0,1,2,3K } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadofal8aadaWgaaqaaKqzadWdbiaadQgaaSWdaeqaaKqz GeWdbiabg2da9iaabccajuaGpaWaaiWaaOqaaKqbaoaabmaakeaaju gib8qacaWG4bWcpaWaaSbaaeaajugWa8qacaaIWaaal8aabeaadaah aaqabeaajugWa8qacaWGJbaaaKqzGeGaaiilaiaadMhal8aadaWgaa qaaKqzadWdbiaaicdaaSWdaeqaamaaCaaabeqaaKqzadGaaiika8qa caWGQbWdaiaacMcaaaaakiaawIcacaGLPaaajugib8qacaGGSaGaae iiaKqba+aadaqadaGcbaqcLbsapeGaamiEaSWdamaaBaaabaqcLbma peGaaGymaaWcpaqabaWaaWbaaeqabaqcLbmapeGaam4yaaaajugibi aacYcacaWG5bWcpaWaaSbaaeaajugWa8qacaaIXaaal8aabeaadaah aaqabeaajugWaiaacIcapeGaamOAa8aacaGGPaaaaaGccaGLOaGaay zkaaqcLbsapeGaaiilaiabgAci8kaacYcacaqGGaqcfa4damaabmaa keaajugib8qacaWG4bWcpaWaaSbaaeaajugWa8qacaWGnbaal8aabe aadaahaaqabeaajugWa8qacaWGJbaaaKqzGeGaaiilaiaadMhal8aa daWgaaqaaKqzadWdbiaad2eaaSWdaeqaamaaCaaabeqaaKqzadGaai ika8qacaWGQbWdaiaacMcaaaaakiaawIcacaGLPaaajugib8qacaGG SaGaamOAaiabg2da9iaaicdacaGGSaGaaGymaiaacYcacaaIYaGaai ilaiaaiodacqGHMacVcaWGlbaak8aacaGL7bGaayzFaaaaaa@8221@ is given as:

i=0 M | y i c y i (j) | min MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaabCaO qaaKqbaoaaemaakeaajugibiaadMhalmaaBaaabaqcLbmacaWGPbaa leqaamaaCaaabeqaaKqzadGaam4yaaaajugibiabgkHiTiaadMhalm aaBaaabaqcLbmacaWGPbaaleqaamaaCaaabeqaaKqzadGaaiikaiaa dQgacaGGPaaaaaGccaGLhWUaayjcSdaaleaajugWaiaadMgacqGH9a qpcaaIWaaaleaajugWaiaad2eaaKqzGeGaeyyeIuoacqGHsgIRciGG TbGaaiyAaiaac6gaaaa@5639@  Or i=0 M | y i c y i (j) | 2 min MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaaO qaaKqbaoaaqahakeaajuaGdaabdaGcbaqcLbsacaWG5bWcdaWgaaqa aKqzadGaamyAaaWcbeaadaahaaqabeaajugWaiaadogaaaqcLbsacq GHsislcaWG5bWcdaWgaaqaaKqzadGaamyAaaWcbeaadaahaaqabeaa jugWaiaacIcacaWGQbGaaiykaaaaaOGaay5bSlaawIa7aSWaaWbaae qabaqcLbmacaaIYaaaaaWcbaqcLbmacaWGPbGaeyypa0JaaGimaaWc baqcLbmacaWGnbaajugibiabggHiLdaaleqaaKqzGeGaeyOKH4Qaci yBaiaacMgacaGGUbaaaa@5992@ . (8)

Minimal distance criterion helps us to fix a candidate for unknown writer as a person from the model S j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadofal8aadaWgaaqaaKqzadWdbiaadQgaaSWdaeqaaaaa @39FF@ in the basis. If the recognition process is “online” and features p-pen pressure, s-speed of writing, a- pen angle or some feature t are given, then there is more information in the process of author recognition, identification or verification in a feature space (x,y,p,s,a,t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa aeaaaaaaaaa8qacaWG4bGaaiilaiaadMhacaGGSaGaamiCaiaacYca caWGZbGaaiilaiaadggacaGGSaGaamiDa8aacaGGPaaaaa@4144@ of dimension N= 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad6eacqGH9aqpcaqGGaGaaGOnaaaa@39E1@ or others. Some person may know how the signature of another man looks like for example (Figures 4-6) or (Figures 12-18), but other extremely important features p,s,a,t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadchacaGGSaGaam4CaiaacYcacaWGHbGaaiilaiaadsha aaa@3C81@  are not visible. Dimension N of a feature space may be very high, but this is no problem. As it is illustrated (Figure 10) (Figure 11) the problem connected with the curse of dimensionality with feature selection does not matter. There is no need to fix which feature is less important and can be eliminated. Every feature is very important and each of them can be interpolated between the nodes using PFC high-dimensional interpolation. For example pressure of the pen p differs during the signature writing and p is changing for particular letters or fragments of the signature. Then feature vector (x,y,p) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa aeaaaaaaaaa8qacaWG4bGaaiilaiaadMhacaGGSaGaamiCa8aacaGG Paaaaa@3C5D@ of dimension N 1 =3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad6eal8aadaWgaaqaaKqzadWdbiaaigdaaSWdaeqaaKqz GeWdbiabg2da9iaaiodaaaa@3C28@ is dealing with p interpolation at the point ( x,y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeaeaaaaaaaaa8qacaWG4bGaaiilaiaacMhaaOWdaiaawIca caGLPaaaaaa@3B89@ via modeling functions (4)-(6) or others. If angle of the pen a differs during the signature writing and a is changing for particular letters or fragments of the signature, then feature vector (x,y,a) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa aeaaaaaaaaa8qacaWG4bGaaiilaiaadMhacaGGSaGaamyya8aacaGG Paaaaa@3C4E@ of dimension N 1 =3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad6eal8aadaWgaaqaaKqzadWdbiaaigdaaSWdaeqaaKqz GeWdbiabg2da9iaaiodaaaa@3C28@ is dealing with a interpolation at the point ( x,y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeaeaaaaaaaaa8qacaWG4bGaaiilaiaadMhaaOWdaiaawIca caGLPaaaaaa@3B8A@ via modeling functions (4)-(6) or others. If speed of the writing s differs during the signature writing and s is changing for particular letters or fragments of the signature, then feature vector (x,y,s) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa aeaaaaaaaaa8qacaWG4bGaaiilaiaadMhacaGGSaGaam4Ca8aacaGG Paaaaa@3C60@ of dimension N 1 =3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad6eal8aadaWgaaqaaKqzadWdbiaaigdaaSWdaeqaaKqz GeWdbiabg2da9iaaiodaaaa@3C28@ is dealing with s interpolation at the point ( x,y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeaeaaaaaaaaa8qacaWG4bGaaiilaiaadMhaaOWdaiaawIca caGLPaaaaaa@3B8A@ via modeling functions (4)-(6) or others. This PFC 3D interpolation is the same like in chapter 3.1 grey scale image retrieval but for selected pairs ( α 1 , α 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa GaeqySde2cdaWgaaqaaKqzadaeaaaaaaaaa8qacaaIXaaal8aabeaa jugib8qacaGGSaGaeqySdewcfa4damaaBaaaleaajugWa8qacaaIYa aal8aabeaajugibiaacMcaaaa@4226@ – only for the points of signature between ( x 1 , y 1 ,2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeaeaaaaaaaaa8qacaWG4bWcpaWaaSbaaeaajugWa8qacaaI Xaaal8aabeaajugib8qacaGGSaGaamyEaSWdamaaBaaabaqcLbmape GaaGymaaWcpaqabaqcLbsapeGaaiilaiaaikdaaOWdaiaawIcacaGL Paaaaaa@42D0@ and ( x 2 , y 2 ,10 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeaeaaaaaaaaa8qacaWG4bqcfa4damaaBaaaleaajugWa8qa caaIYaaal8aabeaajugib8qacaGGSaGaamyEaSWdamaaBaaabaqcLb mapeGaaGOmaaWcpaqabaqcLbsapeGaaiilaiaaigdacaaIWaaak8aa caGLOaGaayzkaaaaaa@4419@ : If a feature sub-space is dimension N 1 =4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad6eal8aadaWgaaqaaKqzadWdbiaaigdaaSWdaeqaaKqz GeWdbiabg2da9iaaisdaaaa@3C29@ and feature vector is for example (x,y,p,s) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa aeaaaaaaaaa8qacaWG4bGaaiilaiaadMhacaGGSaGaamiCaiaacYca caWGZbWdaiaacMcaaaa@3E05@ , then PFC 4D interpolation is the same like in chapter 3.2 color image retrieval but for selected pairs (α1,a2) – only for the points of signature between ( x 1 , y 1 ,2,1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeaeaaaaaaaaa8qacaWG4bqcfa4damaaBaaaleaajugWa8qa caaIXaaal8aabeaajugib8qacaGGSaGaamyEaSWdamaaBaaabaqcLb mapeGaaGymaaWcpaqabaqcLbsapeGaaiilaiaaikdacaGGSaGaaGym aaGcpaGaayjkaiaawMcaaaaa@44C9@ and ( x 2 , y 2 ,10,9 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeaeaaaaaaaaa8qacaWG4bqcfa4damaaBaaaleaajugWa8qa caaIYaaal8aabeaajugib8qacaGGSaGaamyEaKqba+aadaWgaaWcba qcLbmapeGaaGOmaaWcpaqabaqcLbsapeGaaiilaiaaigdacaaIWaGa aiilaiaaiMdaaOWdaiaawIcacaGLPaaaaaa@461A@ : If a feature sub-space is dimension N 1 =5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad6eal8aadaWgaaqaaKqzadWdbiaaigdaaSWdaeqaaKqz GeWdbiabg2da9iaaiwdaaaa@3C2A@ and feature vector is for example (x,y,p,s,a) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa aeaaaaaaaaa8qacaWG4bGaaiilaiaadMhacaGGSaGaamiCaiaacYca caWGZbGaaiilaiaadggapaGaaiykaaaa@3F9B@ , then PFC 5D interpolation is the same like in chapter 3.2 color image retrieval but for selected pairs ( α 1 , α 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa GaeqySde2cdaWgaaqaaKqzadaeaaaaaaaaa8qacaaIXaaal8aabeaa jugib8qacaGGSaGaeqySdewcfa4damaaBaaaleaajugWa8qacaaIYa aal8aabeaajugibiaacMcaaaa@4226@  – only for the points of signature between ( x 1 , y 1 ,2,1,30 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeaeaaaaaaaaa8qacaWG4bWcpaWaaSbaaeaajugWa8qacaaI Xaaal8aabeaajugib8qacaGGSaGaamyEaSWdamaaBaaabaqcLbmape GaaGymaaWcpaqabaqcLbsapeGaaiilaiaaikdacaGGSaGaaGymaiaa cYcacaaIZaGaaGimaaGcpaGaayjkaiaawMcaaaaa@4662@ and ( x 2 , y 2 ,10,9,60 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeaeaaaaaaaaa8qacaWG4bWcpaWaaSbaaeaajugWa8qacaaI Yaaal8aabeaajugib8qacaGGSaGaamyEaSWdamaaBaaabaqcLbmape GaaGOmaaWcpaqabaqcLbsapeGaaiilaiaaigdacaaIWaGaaiilaiaa iMdacaGGSaGaaGOnaiaaicdaaOWdaiaawIcacaGLPaaaaaa@4728@ : (Figure 19-21) are the examples of denotation for the features that are not visible during the signing or handwriting but very important in the process of “online” recognition, identification or verification.

Figure 12 PFC 2D interpolation for γ= α s ,  s=1,  h= y 1 x 1 x 2 + y 2 x 2 x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNjabg2da9iabeg7aHTWdamaaCaaabeqaaKqzadWd biaadohaaaqcLbsacaGGSaGaaiiOaiaacckacaWGZbGaeyypa0JaaG ymaiaacYcacaGGGcGaaiiOaiaadIgacqGH9aqpjuaGdaWcaaGcpaqa aKqzGeWdbiaadMhal8aadaWgaaqaaKqzadWdbiaaigdaaSWdaeqaaa GcbaqcLbsapeGaamiEaSWdamaaBaaabaqcLbmapeGaaGymaaWcpaqa baaaaKqzGeWdbiaadIhajuaGpaWaaSbaaSqaaKqzadWdbiaaikdaaS WdaeqaaKqzGeWdbiabgUcaRKqbaoaalaaak8aabaqcLbsapeGaamyE aKqba+aadaWgaaWcbaqcLbmapeGaaGOmaaWcpaqabaaakeaajugib8 qacaWG4bqcfa4damaaBaaaleaajugWa8qacaaIYaaal8aabeaaaaqc LbsapeGaamiEaSWdamaaBaaabaqcLbmapeGaaGymaaWcpaqabaaaaa@64BD@ .
Figure 13 PFC 2D modeling for γ= ( 2 α 1 ) s ,  s=1,  h= y 1 x 1 x 2 + y 2 x 2 x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNjabg2da9Kqbaoaabmaak8aabaqcLbsapeGaaGOm aKqba+aadaahaaWcbeqaaKqzadWdbiabeg7aHbaajugibiabgkHiTi aaigdaaOGaayjkaiaawMcaaSWdamaaCaaabeqaaKqzadWdbiaadoha aaqcLbsacaGGSaGaaiiOaiaacckacaWGZbGaeyypa0JaaGymaiaacY cacaGGGcGaaiiOaiaadIgacqGH9aqpjuaGdaWcaaGcpaqaaKqzGeWd biaadMhal8aadaWgaaqaaKqzadWdbiaaigdaaSWdaeqaaaGcbaqcLb sapeGaamiEaSWdamaaBaaabaqcLbmapeGaaGymaaWcpaqabaaaaKqz GeWdbiaadIhal8aadaWgaaqaaKqzadWdbiaaikdaaSWdaeqaaKqzGe WdbiabgUcaRKqbaoaalaaak8aabaqcLbsapeGaamyEaKqba+aadaWg aaWcbaqcLbmapeGaaGOmaaWcpaqabaaakeaajugib8qacaWG4bWcpa WaaSbaaeaajugWa8qacaaIYaaal8aabeaaaaqcLbsapeGaamiEaSWd amaaBaaabaqcLbmapeGaaGymaaWcpaqabaaaaa@6B75@ .
Figure 14 PFC 2D reconstruction for γ= ( 2 α 1 ) s ,  s=1,  h= y 1 x 1 x 2 + y 2 x 2 x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNjabg2da9Kqbaoaabmaak8aabaqcLbsapeGaaGOm aKqba+aadaahaaWcbeqaaKqzadWdbiabeg7aHbaajugibiabgkHiTi aaigdaaOGaayjkaiaawMcaaSWdamaaCaaabeqaaKqzadWdbiaadoha aaqcLbsacaGGSaGaaiiOaiaacckacaWGZbGaeyypa0JaaGymaiaacY cacaGGGcGaaiiOaiaadIgacqGH9aqpjuaGdaWcaaGcpaqaaKqzGeWd biaadMhal8aadaWgaaqaaKqzadWdbiaaigdaaSWdaeqaaaGcbaqcLb sapeGaamiEaSWdamaaBaaabaqcLbmapeGaaGymaaWcpaqabaaaaKqz GeWdbiaadIhal8aadaWgaaqaaKqzadWdbiaaikdaaSWdaeqaaKqzGe WdbiabgUcaRKqbaoaalaaak8aabaqcLbsapeGaamyEaKqba+aadaWg aaWcbaqcLbmapeGaaGOmaaWcpaqabaaakeaajugib8qacaWG4bWcpa WaaSbaaeaajugWa8qacaaIYaaal8aabeaaaaqcLbsapeGaamiEaSWd amaaBaaabaqcLbmapeGaaGymaaWcpaqabaaaaa@6B75@ .
Figure 15 PFC 2D reconstruction for γ=lo g 2 ( α s +1 ),  s=0,8,  h= y 1 x 1 x 2 + y 2 x 2 x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNjabg2da9iaadYgacaWGVbGaam4zaSWdamaaBaaa baqcLbmapeGaaGOmaaWcpaqabaqcfa4dbmaabmaak8aabaqcLbsape GaeqySdewcfa4damaaCaaaleqabaqcLbmapeGaam4Caaaajugibiab gUcaRiaaigdaaOGaayjkaiaawMcaaKqzGeGaaiilaiaacckacaGGGc Gaam4Caiabg2da9iaaicdacaGGSaGaaGioaiaacYcacaGGGcGaaiiO aiaadIgacqGH9aqpjuaGdaWcaaGcpaqaaKqzGeWdbiaadMhajuaGpa WaaSbaaSqaaKqzadWdbiaaigdaaSWdaeqaaaGcbaqcLbsapeGaamiE aSWdamaaBaaabaqcLbmapeGaaGymaaWcpaqabaaaaKqzGeWdbiaadI hajuaGpaWaaSbaaSqaaKqzadWdbiaaikdaaSWdaeqaaKqzGeWdbiab gUcaRKqbaoaalaaak8aabaqcLbsapeGaamyEaSWdamaaBaaabaqcLb mapeGaaGOmaaWcpaqabaaakeaajugib8qacaWG4bqcfa4damaaBaaa leaajugWa8qacaaIYaaal8aabeaaaaqcLbsapeGaamiEaSWdamaaBa aabaqcLbmapeGaaGymaaWcpaqabaaaaa@70F1@ .
Figure 16 PFC 2D interpolation for γ=si n s ( α π 2 ),  s=1,  h= y 1 x 1 x 2 + y 2 x 2 x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNjabg2da9iaadohacaWGPbGaamOBaSWdamaaCaaa beqaaKqzadWdbiaadohaaaqcfa4aaeWaaOWdaeaajugib8qacqaHXo qycqGHflY1juaGdaWcaaGcpaqaaKqzGeWdbiabec8aWbGcpaqaaKqz GeWdbiaaikdaaaaakiaawIcacaGLPaaajugibiaacYcacaGGGcGaai iOaiaadohacqGH9aqpcaaIXaGaaiilaiaacckacaGGGcGaamiAaiab g2da9Kqbaoaalaaak8aabaqcLbsapeGaamyEaSWdamaaBaaabaqcLb mapeGaaGymaaWcpaqabaaakeaajugib8qacaWG4bWcpaWaaSbaaeaa jugWa8qacaaIXaaal8aabeaaaaqcLbsapeGaamiEaKqba+aadaWgaa WcbaqcLbmapeGaaGOmaaWcpaqabaqcLbsapeGaey4kaSscfa4aaSaa aOWdaeaajugib8qacaWG5bqcfa4damaaBaaaleaajugWa8qacaaIYa aal8aabeaaaOqaaKqzGeWdbiaadIhal8aadaWgaaqaaKqzadWdbiaa ikdaaSWdaeqaaaaajugib8qacaWG4bWcpaWaaSbaaeaajugWa8qaca aIXaaal8aabeaaaaa@70B2@ .
Figure 17 PFC 2D modeling for γ=si n s ( α π 2 ),  s=0,8,  h=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNjabg2da9iaadohacaWGPbGaamOBaKqba+aadaah aaWcbeqaaKqzadWdbiaadohaaaqcfa4aaeWaaOWdaeaajugib8qacq aHXoqycqGHflY1juaGdaWcaaGcpaqaaKqzGeWdbiabec8aWbGcpaqa aKqzGeWdbiaaikdaaaaakiaawIcacaGLPaaajugibiaacYcacaGGGc GaaiiOaiaadohacqGH9aqpcaaIWaGaaiilaiaaiIdacaGGSaGaaiiO aiaacckacaWGObGaeyypa0JaaGimaaaa@57CB@ .
Figure 18 PFC 2D modeling for γ=1 2 π arccos( α s ),  s=0,5,  h=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNjabg2da9iaaigdacqGHsisljuaGdaWcaaGcpaqa aKqzGeWdbiaaikdaaOWdaeaajugib8qacqaHapaCaaGaaeyyaiaabk hacaqGJbGaae4yaiaab+gacaqGZbqcfa4aaeWaaOWdaeaajugibiab eg7aHLqbaoaaCaaaleqabaqcLbmapeGaam4CaaaaaOGaayjkaiaawM caaKqzGeGaaiilaiaacckacaGGGcGaam4Caiabg2da9iaaicdacaGG SaGaaGynaiaacYcacaGGGcGaaiiOaiaadIgacqGH9aqpcaaIWaaaaa@59BB@ .
Figure 19 Reconstructed speed of the writing s at the pixels of signature.
Figure 20 Reconstructed pen pressure p and speed of the writing s as (p,s) at the pixels of signature.
Figure 21 Reconstructed pen pressure p, speed of the writing s and angle a as (p,s,a) at the pixels of signature.

Even if from technical reason or other reasons only some points of signature or handwriting (feature nodes) are given in the process of “online” recognition, identification or verification, the values of features between nodes are computed via multidimensional PFC interpolation like for example between ( x 1 , y 1 ,2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeaeaaaaaaaaa8qacaWG4bWcpaWaaSbaaeaajugWa8qacaaI Xaaal8aabeaajugib8qacaGGSaGaamyEaSWdamaaBaaabaqcLbmape GaaGymaaWcpaqabaqcLbsapeGaaiilaiaaikdaaOWdaiaawIcacaGL Paaaaaa@42D0@ and ( x 2 , y 2 ,10 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeaeaaaaaaaaa8qacaWG4bWcpaWaaSbaaeaajugWa8qacaaI Yaaal8aabeaajugib8qacaGGSaGaamyEaSWdamaaBaaabaqcLbmape GaaGOmaaWcpaqabaqcLbsapeGaaiilaiaaigdacaaIWaaak8aacaGL OaGaayzkaaaaaa@438B@ on Figure 19, between ( x 1 , y 1 ,2,1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeaeaaaaaaaaa8qacaWG4bWcpaWaaSbaaeaajugWa8qacaaI Xaaal8aabeaajugib8qacaGGSaGaamyEaSWdamaaBaaabaqcLbmape GaaGymaaWcpaqabaqcLbsapeGaaiilaiaaikdacaGGSaGaaGymaaGc paGaayjkaiaawMcaaaaa@443B@ and ( x 2 , y 2 ,10,9 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeaeaaaaaaaaa8qacaWG4bqcfa4damaaBaaaleaajugWa8qa caaIYaaal8aabeaajugib8qacaGGSaGaamyEaKqba+aadaWgaaWcba qcLbmapeGaaGOmaaWcpaqabaqcLbsapeGaaiilaiaaigdacaaIWaGa aiilaiaaiMdaaOWdaiaawIcacaGLPaaaaaa@461A@ on (Figure 20) or between ( x 1 , y 1 ,2,1,30 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeaeaaaaaaaaa8qacaWG4bWcpaWaaSbaaeaajugWa8qacaaI Xaaal8aabeaajugib8qacaGGSaGaamyEaKqba+aadaWgaaWcbaqcLb mapeGaaGymaaWcpaqabaqcLbsapeGaaiilaiaaikdacaGGSaGaaGym aiaacYcacaaIZaGaaGimaaGcpaGaayjkaiaawMcaaaaa@46F0@ and ( x 2 , y 2 ,10,9,60 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeaeaaaaaaaaa8qacaWG4bWcpaWaaSbaaeaajugWa8qacaaI Yaaal8aabeaajugib8qacaGGSaGaamyEaKqba+aadaWgaaWcbaqcLb mapeGaaGOmaaWcpaqabaqcLbsapeGaaiilaiaaigdacaaIWaGaaiil aiaaiMdacaGGSaGaaGOnaiaaicdaaOWdaiaawIcacaGLPaaaaaa@47B6@ on (Figure 21). Reconstructed features are compared with the features in the basis of patterns like parameter y in (8) and appropriate criterion gives the result. So persons with the parameters of their signatures are allocated in the basis of patterns. The curve does not have to be smooth at the nodes because handwritten symbols are not smooth. The range of coefficients x has to be the same for all models because of comparing appropriate coordinates. Every letter or a part of signature is modeled by PFC via three factors: the set of high-dimensional feature nodes, probability distribution function γ=F( α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNjabg2da9iaadAeajuaGpaWaaeWaaOqaaKqzGeWd biabeg7aHbGcpaGaayjkaiaawMcaaaaa@3EA4@ and nodes combination h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadIgaaaa@3792@ . These three factors are chosen individually for each letter or a part of signature therefore this information about modeled curves seems to be enough for specific PFC multidimensional curve interpolation and handwriting identification. What is very important, PFC N-dimensional modeling is independent of the language or a kind of symbol (letters, numbers, characters or others). One person may have several patterns for one handwritten letter or signature. Summarize: every person has the basis of patterns for each handwritten letter or symbol, described by the set of feature nodes, modeling function γ=F( α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNjabg2da9iaadAeajuaGpaWaaeWaaOqaaKqzGeWd biabeg7aHbGcpaGaayjkaiaawMcaaaaa@3EA4@ and nodes combination h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadIgaaaa@3792@ . Whole basis of patterns consists of models S j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadofal8aadaWgaaqaaKqzadWdbiaadQgaaSWdaeqaaaaa @39FF@ for j= 0,1,2,3K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadQgacqGH9aqpcaqGGaGaaGimaiaacYcacaaIXaGaaiil aiaaikdacaGGSaGaaG4maiabgAci8kaadUeaaaa@4099@ .

Modeling and interpolation of non-typical curves and irregular shapes

PFC two dimensional interpolations and modeling enables to solve classic problem in numerical methods for example (Figure 2) how to parameterize and to model known function. But having the set of nodes there is another problem connected with handwriting and human signing: how to model or to reconstruct the curve which is the part of signature or handwriting but which is non-typical or irregular. Human signature and handwriting consists of non-typical curves and irregular shapes. PFC method (1) is the way of modeling and interpolation for non-typical curves and irregular shapes – contours as closed curves (if first node and last node is the same). Here are some examples of modeled non-typical or irregular curves as a part of signature or handwriting for five nodes:
( 0.9;4.736 ), ( 0.5;0.666 ), ( 0;0 ), ( 0.5;0.666 ), ( 0.9;4.736 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaqadaWdaeaapeGaeyOeI0IaaGimaiaac6cacaaI5aGaai4o aiaaisdacaGGUaGaaG4naiaaiodacaaI2aaacaGLOaGaayzkaaGaai ilaiaacckadaqadaWdaeaapeGaeyOeI0IaaGimaiaac6cacaaI1aGa ai4oaiaaicdacaGGUaGaaGOnaiaaiAdacaaI2aaacaGLOaGaayzkaa GaaiilaiaacckadaqadaWdaeaapeGaaGimaiaacUdacaaIWaaacaGL OaGaayzkaaGaaiilaiaacckadaqadaWdaeaapeGaaGimaiaac6caca aI1aGaai4oaiabgkHiTiaaicdacaGGUaGaaGOnaiaaiAdacaaI2aaa caGLOaGaayzkaaGaaiilaiaacckadaqadaWdaeaapeGaaGimaiaac6 cacaaI5aGaai4oaiabgkHiTiaaisdacaGGUaGaaG4naiaaiodacaaI 2aaacaGLOaGaayzkaaaaaa@6688@

Figures 22-24 are the examples of very specific modeling for non-typical and irregular curves as a signature. PFC interpolation is used for parameterization and reconstruction of curves in the plane.

Figure 22 PFC 2D interpolation for γ=lo g 2 s ( α+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNjabg2da9iaadYgacaWGVbGaam4zaSWdamaaBaaa baqcLbmapeGaaGOmaaWcpaqabaWaaWbaaeqabaqcLbmapeGaam4Caa aajuaGdaqadaGcpaqaaKqzGeWdbiabeg7aHjabgUcaRiaaigdaaOGa ayjkaiaawMcaaaaa@46E0@ , s=0.8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadohacqGH9aqpcaaIWaGaaiOlaiaaiIdaaaa@3AD2@ , h=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadIgacqGH9aqpcaaIWaaaaa@3953@ .
Figure 23 PFC 2D interpolation for γ=sin( α s * π 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNjabg2da9iGacohacaGGPbGaaiOBaKqbaoaabmaa k8aabaqcLbsacqaHXoqylmaaCaaabeqaaKqzadWdbiaadohaaaqcLb sacaGGQaqcfa4aaSaaaOWdaeaajugib8qacqaHapaCaOWdaeaajugi b8qacaaIYaaaaaGccaGLOaGaayzkaaaaaa@48BA@ , s=1.8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadohacqGH9aqpcaaIXaGaaiOlaiaaiIdaaaa@3AD3@ , h=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadIgacqGH9aqpcaaIWaaaaa@3953@ .
Figure 24 PFC 2D interpolation for γ= ( 2 α 1 ) s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeo7aNjabg2da9Kqbaoaabmaak8aabaqcLbsapeGaaGOm aSWaaWbaaeqabaqcLbmacqaHXoqyaaqcLbsacqGHsislcaaIXaaaki aawIcacaGLPaaal8aadaahaaqabeaajugWa8qacaWGZbaaaaaa@448B@ , s=1.2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadohacqGH9aqpcaaIXaGaaiOlaiaaikdaaaa@3ACD@ , h=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadIgacqGH9aqpcaaIWaaaaa@3953@ .

Conclusion

The method of Probabilistic Features Combination (PFC) enables interpolation and modeling of high-dimensional N data using features’ combinations and different coefficients γ: polynomial, sinusoidal, cosinusoidal, tangent, cotangent, logarithmic, exponential, arc sin, arc cos, arc tan, arc cot or power function. Functions for γ calculations are chosen individually at each data modeling and it is treated as N-dimensional probability distribution function: γ depends on initial requirements and features’ specifications. PFC method leads to data interpolation as handwriting or signature identification and image retrieval via discrete set of feature vectors in N- dimensional feature space. So PFC method makes possible the combination of two important problems: interpolation and modeling in a matter of image retrieval or writer identification. Main features of PFC method are: PFC interpolation develops a linear interpolation in multidimensional feature spaces into other functions as N-dimensional probability distribution functions; PFC is a generalization of MHR method and PNC method via different nodes combinations; interpolation of L points is connected with the computational cost of rank O(L) as in MHR and PNC method; nodes combination and coefficients γ are crucial in the process of data probabilistic parameterization and interpolation: they are computed individually for a single feature. Future works are going to applications of PFC method in signature and handwriting biometric recognition: choice and features of nodes combinations h and coefficients γ.

Acknowledgements

None.

Conflict of interest

Author declares that there is no conflict of interest.

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