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Nanomedicine Research

Mini Review Volume 5 Issue 3

Data Visualization Using Hodge Decomposition - A Short Review

Monika Bahl

Amity University, India

Correspondence: Monika Bahl, Amity Institute of Applied Sciences, Amity University, Noida, India

Received: November 02, 2017 | Published: April 4, 2017

Citation: Bahl M (2017) Data Visualization Using Hodge Decomposition - A Short R eview. J Nanomed Res 5(3): 00115. DOI: 10.15406/jnmr.2017.05.00115

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Abstract

 It has been seen that a vector field decomposition method called the Helmholtz Hodge Decomposition (HHD) can analyze scalar fields present universally in nature. It aids to reveal complex internal flows including energy flows in interference and diffraction optical fields. A gradient field defined in a region R, can be separated into solenoidal and irrotational components. HHD applied onto Magnetic Resonance Elasticity data can also aid to retain the curl field, while revealing the tissue elasticity in such medical measurements.

Keywords: helmholtz hodge decomposition, phase gradient, magnetic resonance, elastography

Abbreviations

HHD: Helmholtz Hodge Decomposition; OAM: Orbital Angular Momentum; MRI: Magnetic Resonance Imaging; MRE: Magnetic Resonance Elastography

Introduction

A vector field decomposition technique namely Helmholtz Hodge Decomposition (HHD) allows the field to be segregated into a solenoidal (divergence–free part) and an irrotational (curl–free part).1–9 Many problems in electromagnetism, MRI.6 and fluid and smoke simulations.7 use this decomposition method to visualize real–time data. HHD aids to represent the homogeneous data explicitly by extracting the critical points like sources, sinks and vortices. It had been applied to polarized vector fields and to reconstruct phase for wavefront distortions. We used HHD on scalar optical fields and studied the Orbital angular momentum (OAM) in diffraction optics that has been reported.10–15

An identity relates the OAM in an optical field to its phase and amplitude distribution.16 The vectorial nature of the fields is disregarded in such cases. We had shown the usefulness of the HHD in analyzing all such fields, including the ones obtained in interference optics where a single state of polarization (SOP) is assumed. We constructed a phase gradient field φ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgEGirl abeA8aQbaa@39BC@  from a scalar field.17,18 by using the relation

φ= Im[ ψ * ψ] I MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgEGirl abeA8aQjabg2da9maalaaabaGaciysaiaac2gacaGGBbGaeqiYdK3a aWbaaeqabaGaaiOkaaaacqGHhis0cqaHipqEcaGGDbaabaGaamysaa aaaaa@4512@   (1)

Where ψ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI8a5b aa@3847@ a scalar is field resulting due to interference or diffraction and I= ψ ψ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMeacq GH9aqpcqaHipqEdaahaaqabeaacqGHxiIkaaGaeqiYdKhaaa@3CFA@  is the intensity distribution.

ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaey4bIe Taeqy1dygaaa@39C8@  that directed in the local propagation vector direction normal to the phase contour surfaces carries all the features of the wave. The phase gradient field in a singular beam has a non zero curl.19–27 Hence, the solenoidal or the curl part that is an explicit component of HHD carries the circulating energy features of the field. The irrotational part reveals solely the spreading of energy, whether diverging or converging.

It is worth noting that different research groups have used different names (Helmholtz, Hodge, Helmholtz–Hodge or Hodge–Helmholtz) for this decomposition. The Helmholtz decomposition theorem suggested the segregation of a vector field, defined on real domains, into the solenoidal and the irrotational components. While the Hodge decomposition talked about a third component that is harmonic and is both solenoidal as well as irrotational. This decomposition theorem was defined for differential forms on Riemannian manifolds. Thus, the Hodge decomposition is the differential form analog of the Helmholtz decomposition in vector analysis. To the best of our knowledge, there is no origin of the Helmholtz–Hodge theorem or some formal merging of the names Helmholtz and Hodge.

The HHD technique that we adopted for decomposition is addressed in the next section. The approach was applied on to random fields in which both positive and negative curvatures were added. Spherical waves with positive and negative divergence, random wave fronts containing vortices with curling phase gradients, a vortex lattice field resulting from the superposition of plane waves were some of the fields that were investigated to test our HHD method. All the results showed clear explicitly segmented data and proved that our method worked really fine with scalar optical fields. We also believe that this method will definitely augment the flow visualization of velocity, pressure and temperature using optical methods. This technique can also be applied to process data fields as in medicine. It was anticipated and later seen that HHD using our least squares approach yields good results when applied to Magnetic Resonance Elastography (MRE) in human brain tissues. There are various methods to measure tissue elasticity that are used worldwide. Majority of them use sophisticated hardware and software and are not really cost effective. But our technique is very simple and does not require any complicated assessments and interpretations.

MRE consists of measurements of mechanical properties of a tissue using the Magnetic Resonance Imaging (MRI). An external actuator vibrates the tissue of interest and MRI data is obtained. This encodes the shear wave propagation into the MRI image. Tissue elasticity is then measured using wave inversion, after isolation of these shears waves. But the problem is ill–posed and extremely sensitive to denoising methods. Some noise solutions to sort out this issue have been used. Retaining the curl field to study tissue elasticity using HHD by our least square method also gives good results. The method has been explained in detail in the next section.

Helmholtz hodge decomposition

The Helmholtz Hodge Decomposition (HHD) is based on the Helmholtz theorem.28–30 which states that a vector field which F is on a bounded domain V in R3, and is twice continuously differentiable, and whose divergence .F=b(r) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgEGirl aac6cacaWGgbGaeyypa0JaamOyaiaacIcacaWGYbGaaiykaaaa@3DB9@  and curl ×F=c(r) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgEGirl abgEna0kaadAeacqGH9aqpcaWGJbGaaiikaiaadkhacaGGPaaaaa@3F1F@ are known, can be segregated into components f 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAgada WgaaqcfasaaiaaigdaaKqbagqaaaaa@38FC@ and f 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAgada Wgaaqcfasaaiaaikdaaeqaaaaa@386F@  determined by

F= f 1 + f 2 ϕ+×A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAeacq GH9aqpcaWGMbWaaSbaaKqbGeaacaaIXaaabeaajuaGcqGHRaWkcaWG MbWaaSbaaKqbGeaacaaIYaaajuaGbeaacqGHshI3cqGHhis0cqaHvp GzcqGHRaWkcqGHhis0cqGHxdaTcaWGbbaaaa@4923@   (2)

Where ϕ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabew9aMn aabmaabaGaamOCaaGaayjkaiaawMcaaaaa@3AC1@  and A( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeada qadaqaaiaadkhaaiaawIcacaGLPaaaaaa@39BF@  are scalar and vector potentials respectively, that can be obtained from the Poisson’s equations

ϕ(r)= 1 4π V b(r) r d v MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabew9aMj aacIcacaWGYbGaaiykaiabg2da9maalaaabaGaaGymaaqaaiaaisda cqaHapaCaaWaa8quaeaadaWcaaqaaiaadkgacaGGOaGaamOCaiaacM caaeaacaWGYbaaaaqaaiaadAfaaeqacqGHRiI8aiaadsgaceWG2bGb auaaaaa@4818@    (3)

A(r)= 1 4π V c(r) r d v MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeaca GGOaGaamOCaiaacMcacqGH9aqpdaWcaaqaaiaaigdaaeaacaaI0aGa eqiWdahaamaapefabaWaaSaaaeaacaWGJbGaaiikaiaadkhacaGGPa aabaGaamOCaaaaaeaacaWGwbaabeGaey4kIipacaWGKbGabmODayaa faaaaa@4717@   (4)

These potentials ϕ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabew9aMn aabmaabaGaamOCaaGaayjkaiaawMcaaaaa@3AC1@  and A( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeada qadaqaaiaadkhaaiaawIcacaGLPaaaaaa@39BF@  allow the field f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAgaaa a@3764@ to be segregated into the curl free and divergence free components.

The boundary conditions imposed in HHD ensure a normal boundary flow on the curl free component and a tangential flow on the divergence free component. Considering n ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqad6gaga qcaaaa@377C@  as the outward normal to the boundary Ω this implies that for a unique decomposition,

  1. The irrotational component f1 is normal to the boundary dΩ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsgacq qHPoWvaaa@38F0@  of Ω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfM6axb aa@3807@ ,e. f 1 × n ^ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadAgaga WcamaaBaaajuaibaGaaGymaaqabaqcfaOaey41aqRabmOBayaajaGa eyypa0JaaGimaaaa@3DE8@  , and
  2. The solenoidal component f2 is parallel to the boundary dΩ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsgacq qHPoWvaaa@38F0@  of Ω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfM6axb aa@3807@ , i.e. f 2 n ^ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadAgaga WcamaaBaaajuaibaGaaGOmaaqabaqcfaOaeyyXICTabmOBayaajaGa eyypa0JaaGimaaaa@3E1C@

The method adopted to solve the HHD problem involves minimizing the errors in the terms which are constructed from the initial guesses. f 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAgada Wgaaqcfasaaiaaigdaaeqaaaaa@386E@  and f 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAgada WgaaqcfasaaiaaikdaaKqbagqaaaaa@38FD@  are considered as the initial guesses for the curl free and the divergence free component fields of a vector field f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAgaaa a@3764@ .

The error terms/ residuals × f 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaafmaaba Gaey4bIeTaey41aqRaamOzamaaBaaajuaibaGaaGymaaqcfayabaaa caGLjWUaayPcSdaaaa@3FC0@ , f 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaafmaaba Gaey4bIeTaeyyXICTaamOzamaaBaaajuaibaGaaGOmaaqcfayabaaa caGLjWUaayPcSdaaaa@3FF4@  and f 1 + f 2 f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaafmaaba GaamOzamaaBaaajuaibaGaaGymaaqabaqcfaOaey4kaSIaamOzamaa BaaajuaibaGaaGOmaaqabaqcfaOaeyOeI0IaamOzaaGaayzcSlaawQ a7aaaa@4161@ are then reduced to a minimum.

In Cartesian coordinate system, the difference operator operating on a scalar function f, is defined as

d f = f x x x ^ + f y y y ^ + f z z z ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsgace WGMbGbaSaacqGH9aqpdaWcaaqaaiabgkGi2kaadAgadaWgaaqaaiaa dIhaaeqaaaqaaiabgkGi2kaadIhaaaGabmiEayaajaGaey4kaSYaaS aaaeaacqGHciITcaWGMbWaaSbaaeaacaWG5baabeaaaeaacqGHciIT caWG5baaaiqadMhagaqcaiabgUcaRmaalaaabaGaeyOaIyRaamOzam aaBaaabaGaamOEaaqabaaabaGaeyOaIyRaamOEaaaaceWG6bGbaKaa aaa@4FFF@   (5)

Using Finite Difference Approximation, the MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgkGi2c aa@37DF@ operator can be written as a matrix given by

= 1 2 [ 0 1 0 0 1 0 1 0 1 0 0 1 0 0 1 0 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgkGi2k abg2da9maalaaabaGaaGymaaqaaiaaikdaaaWaamWaaeaafaqabeqb faaaaaqaaiaaicdaaeaacqGHsislcaaIXaaabaGaaGimaaqaaiabl+ UimbqaaiaaicdaaeaacaaIXaaabaGaaGimaaqaaiabgkHiTiaaigda aeaacqWIXlYtaeaacqWIUlstaeaacaaIWaaabaGaaGymaaqaaiaaic daaeaacqWIXlYtaeaacaaIWaaabaGaeSO7I0eabaGaeSy8I8eabaGa eSy8I8eabaGaeSy8I8eabaGaeyOeI0IaaGymaaqaaiaaicdaaeaacq WIVlctaeaacaaIWaaabaGaaGymaaqaaiaaicdaaaaacaGLBbGaayzx aaaaaa@5C6D@    (6)

This is true for 1D but for 3D, it is expanded as

x,3D = I m I m x x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgkGi2o aaBaaajuaibaGaamiEaiaacYcacaaIZaGaamiraaqabaqcfaOaeyyp a0JaamysamaaBaaajuaibaGaamyBaaqabaqcfaOaey4LIqSaamysam aaBaaajuaibaGaamyBaaqcfayabaGaey4LIqSaeyOaIy7aaSbaaKqb GeaacaWG4baabeaajuaGcqGHijYUcqGHciITdaWgaaqcfasaaiaadI haaeqaaaaa@4DE4@   (7)

where I m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMeada Wgaaqcfasaaiaad2gaaKqbagqaaaaa@3916@ is an m×m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad2gacq GHxdaTcaWGTbaaaa@3A74@ Identity matrix and  is the Kronecker delta product. Similarly,

y,3D = I m y I m y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgkGi2o aaBaaajuaibaGaamyEaiaacYcacaaIZaGaamiraaqabaqcfaOaeyyp a0JaamysamaaBaaajuaibaGaamyBaaqcfayabaGaey4LIqSaeyOaIy 7aaSbaaKqbGeaacaWG5baabeaajuaGcqGHxkcXcaWGjbWaaSbaaKqb GeaacaWGTbaabeaajuaGcqGHijYUcqGHciITdaWgaaqcfasaaiaadM haaeqaaaaa@4DE7@   (8)

z,3D = z I m I m z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgkGi2o aaBaaajuaibaGaamOEaiaacYcacaaIZaGaamiraaqcfayabaGaeyyp a0JaeyOaIy7aaSbaaKqbGeaacaWG6baabeaajuaGcqGHxkcXcaWGjb WaaSbaaKqbGeaacaWGTbaajuaGbeaacqGHxkcXcaWGjbWaaSbaaKqb GeaacaWGTbaabeaajuaGcqGHijYUcqGHciITdaWgaaqcfasaaiaadQ haaeqaaaaa@4DEA@   (9)

The curl, in Cartesian coordinate system, is written using finite difference operator matrix as

×f=[ x ^ y ^ z ^ x y z f x f y f z ]=[ 0 z y z 0 x y x 0 ] [ f x f y f z ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgEGirl abgEna0kaadAgacqGH9aqpdaWadaqaauaabeqadmaaaeaaceWG4bGb aKaaaeaaceWG5bGbaKaaaeaaceWG6bGbaKaaaeaadaWcaaqaaiabgk Gi2cqaaiabgkGi2kaadIhaaaaabaWaaSaaaeaacqGHciITaeaacqGH ciITcaWG5baaaaqaamaalaaabaGaeyOaIylabaGaeyOaIyRaamOEaa aaaeaacaWGMbWaaSbaaeaacaWG4baabeaaaeaacaWGMbWaaSbaaeaa caWG5baabeaaaeaacaWGMbWaaSbaaeaacaWG6baabeaaaaaacaGLBb GaayzxaaGaeyypa0ZaamWaaeaafaqabeWadaaabaGaaGimaaqaaiab gkHiTiabgkGi2oaaBaaabaGaamOEaaqabaaabaGaeyOaIy7aaSbaae aacaWG5baabeaaaeaacqGHciITdaWgaaqaaiaadQhaaeqaaaqaaiaa icdaaeaacqGHsislcqGHciITdaWgaaqaaiaadIhaaeqaaaqaaiabgk HiTiabgkGi2oaaBaaabaGaamyEaaqabaaabaGaeyOaIy7aaSbaaeaa caWG4baabeaaaeaacaaIWaaaaaGaay5waiaaw2faauaabeqabeaaae aaaaWaamWaaeaafaqabeWabaaabaGaamOzamaaBaaabaGaamiEaaqa baaabaGaamOzamaaBaaabaGaamyEaaqabaaabaGaamOzamaaBaaaba GaamOEaaqabaaaaaGaay5waiaaw2faaaaa@7232@   (10)

Similarly, for divergence,

f= f x x + f y y + f z z =[ x y z ] [ f x f y f z ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgEGirl abgwSixlaadAgacqGH9aqpdaWcaaqaaiabgkGi2kaadAgadaWgaaqa aiaadIhaaeqaaaqaaiabgkGi2kaadIhaaaGaey4kaSYaaSaaaeaacq GHciITcaWGMbWaaSbaaeaacaWG5baabeaaaeaacqGHciITcaWG5baa aiabgUcaRmaalaaabaGaeyOaIyRaamOzamaaBaaabaGaamOEaaqaba aabaGaeyOaIyRaamOEaaaacqGH9aqpdaWadaqaauaabeqabmaaaeaa cqGHciITdaWgaaqaaiaadIhaaeqaaaqaaiabgkGi2oaaBaaabaGaam yEaaqabaaabaGaeyOaIy7aaSbaaeaacaWG6baabeaaaaaacaGLBbGa ayzxaaqbaeqabeqaaaqaaaaadaWadaqaauaabeqadeaaaeaacaWGMb WaaSbaaeaacaWG4baabeaaaeaacaWGMbWaaSbaaeaacaWG5baabeaa aeaacaWGMbWaaSbaaeaacaWG6baabeaaaaaacaGLBbGaayzxaaaaaa@626A@

Thus, the HHD can then be summarized as:

[ 0 z y 0 0 0 z 0 x 0 0 0 y x 0 0 0 0 0 0 0 x y z I 0 0 I 0 0 0 I 0 0 I 0 0 0 I 0 0 I ] [ f 1x f 1y f 1z f 2x f 2y f 2z ]=[ 0 0 0 0 f x f y f z ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaaba qbaeqabCGbaaaaaeaacaaIWaaabaGaeyOeI0IaeyOaIy7aaSbaaeaa caWG6baabeaaaeaacqGHciITdaWgaaqaaiaadMhaaeqaaaqaaiaaic daaeaacaaIWaaabaGaaGimaaqaaiabgkGi2oaaBaaabaGaamOEaaqa baaabaGaaGimaaqaaiabgkHiTiabgkGi2oaaBaaabaGaamiEaaqaba aabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaeyOeI0IaeyOaIy7a aSbaaeaacaWG5baabeaaaeaacqGHciITdaWgaaqaaiaadIhaaeqaaa qaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaa baGaaGimaaqaaiaaicdaaeaacqGHciITdaWgaaqaaiaadIhaaeqaaa qaaiabgkGi2oaaBaaabaGaamyEaaqabaaabaGaeyOaIy7aaSbaaeaa caWG6baabeaaaeaacaWGjbaabaGaaGimaaqaaiaaicdaaeaacaWGjb aabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaamysaaqaaiaaicda aeaacaaIWaaabaGaamysaaqaaiaaicdaaeaacaaIWaaabaGaaGimaa qaaiaadMeaaeaacaaIWaaabaGaaGimaaqaaiaadMeaaaaacaGLBbGa ayzxaaqbaeqabeqaaaqaaaaadaWadaqaauaabeqageaaaaqaaiaadA gadaWgaaqaaiaaigdacaWG4baabeaaaeaacaWGMbWaaSbaaeaacaaI XaGaamyEaaqabaaabaGaamOzamaaBaaabaGaaGymaiaadQhaaeqaaa qaaiaadAgadaWgaaqaaiaaikdacaWG4baabeaaaeaacaWGMbWaaSba aeaacaaIYaGaamyEaaqabaaabaGaamOzamaaBaaabaGaaGOmaiaadQ haaeqaaaaaaiaawUfacaGLDbaacqGH9aqpdaWadaqaauaabeqaheaa aaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaWGMb WaaSbaaeaacaWG4baabeaaaeaacaWGMbWaaSbaaeaacaWG5baabeaa aeaacaWGMbWaaSbaaeaacaWG6baabeaaaaaacaGLBbGaayzxaaaaaa@8958@   (12)

The boundary condition considered is that the fields tend to go to zero at infinity. By applying the above mentioned boundary conditions, the system of equations can be efficiently solved. 12 can be solved as an equation Px=Q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadcfaca WG4bGaeyypa0Jaamyuaaaa@3A27@  where P MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadcfaaa a@374E@  represents [ 0 z y 0 0 0 z 0 x 0 0 0 y x 0 0 0 0 0 0 0 x y z I 0 0 I 0 0 0 I 0 0 I 0 0 0 I 0 0 I ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaaba qbaeqabCGbaaaaaeaacaaIWaaabaGaeyOeI0IaeyOaIy7aaSbaaeaa caWG6baabeaaaeaacqGHciITdaWgaaqaaiaadMhaaeqaaaqaaiaaic daaeaacaaIWaaabaGaaGimaaqaaiabgkGi2oaaBaaabaGaamOEaaqa baaabaGaaGimaaqaaiabgkHiTiabgkGi2oaaBaaabaGaamiEaaqaba aabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaeyOeI0IaeyOaIy7a aSbaaeaacaWG5baabeaaaeaacqGHciITdaWgaaqaaiaadIhaaeqaaa qaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaa baGaaGimaaqaaiaaicdaaeaacqGHciITdaWgaaqaaiaadIhaaeqaaa qaaiabgkGi2oaaBaaabaGaamyEaaqabaaabaGaeyOaIy7aaSbaaeaa caWG6baabeaaaeaacaWGjbaabaGaaGimaaqaaiaaicdaaeaacaWGjb aabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaamysaaqaaiaaicda aeaacaaIWaaabaGaamysaaqaaiaaicdaaeaacaaIWaaabaGaaGimaa qaaiaadMeaaeaacaaIWaaabaGaaGimaaqaaiaadMeaaaaacaGLBbGa ayzxaaaaaa@6A90@ , while [ f 1x f 1y f 1z f 2x f 2y f 2z ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaaba qbaeqabyqaaaaabaGaamOzamaaBaaabaGaaGymaiaadIhaaeqaaaqa aiaadAgadaWgaaqaaiaaigdacaWG5baabeaaaeaacaWGMbWaaSbaae aacaaIXaGaamOEaaqabaaabaGaamOzamaaBaaabaGaaGOmaiaadIha aeqaaaqaaiaadAgadaWgaaqaaiaaikdacaWG5baabeaaaeaacaWGMb WaaSbaaeaacaaIYaGaamOEaaqabaaaaaGaay5waiaaw2faaaaa@4921@  and [ 0 0 0 0 f x f y f z ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaaba qbaeqabCqaaaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaGim aaqaaiaadAgadaWgaaqaaiaadIhaaeqaaaqaaiaadAgadaWgaaqaai aadMhaaeqaaaqaaiaadAgadaWgaaqaaiaadQhaaeqaaaaaaiaawUfa caGLDbaaaaa@4188@  are represented by x and Q respectively.

Since the system of equations is not full rank, appropriate weights are applied to the residuals or error terms in order to get a unique solution. The weight parameters α, β and γ are defined for the curl free, divergence free and the sum residual respectively.

Thus, one minimizes the expression W( PxQ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaafmaaba Gaam4vamaabmaabaGaamiuaiaadIhacqGHsislcaWGrbaacaGLOaGa ayzkaaaacaGLjWUaayPcSdaaaa@3F9A@  where W is a diagonal matrix defined as

W=diag( [α α α β γ γ γ ] ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadEfacq GH9aqpcaWGKbGaamyAaiaadggacaWGNbWaaeWaaeaafaqabeqaeaaa aeaafaqabeqadaaabaqbaeqabeGaaaqaauaabeqabeaaaeaaaaGaai 4waiabeg7aHbqaaiabeg7aHbaaaeaacqaHXoqyaeaacqaHYoGyaaaa baGaeq4SdCgabaGaeq4SdCgabaGaeq4SdCgaaiaac2fafaqabeqaba aabaaaaaGaayjkaiaawMcaaaaa@4B03@   (13)

The results obtained with the specific boundary condition are shown in next section. It is to be noted that the fields are assumed to go to zero at infinity. One clearly observes the vanishing of the normal component of the field in the solenoidal part and tangential component in the irrotational part at the boundaries. The boundary conditions imposed ensure a unique and orthogonal decomposition of the original field. The curl free part is the projection of the original field onto the space of solenoidal fields. Similarly, the divergence free part is the projection of the original field onto the space of irrotational fields. This is possible only when proper boundary conditions are satisfied.

HHD has, hitherto, seemed to be a great technique to extract and detect vortices, but we were unable to determine their strength. Vortices of any topological charge, however high it was, appeared similar. Thus, this seemed to be more of a visualization and analyzation technique than a method for measurement of the strength of the singularities.

Decomposition of scalar fields

In this section, the decomposition using HHD method for the some of the scalar fields is demonstrated. The simulation work has been done using Matlab.

The phase of the beam is shown in part (a) in Figure 1 (I) shows a Hodge decomposed beam for a positive spherical beam. The computed phase gradient field is shown in part (b). Part (c) shows the Hodge decomposed divergence free part, while part (d) shows the curl free component. As can be seen, the solenoidal field is zero in the core area far away from the boundary in case of spherical beams. The normal component of the field vanishes near to the boundary and the field lines tend to get parallel here. The irrotational component shows diverging field lines emanating from the center. As expected, the normal component tends to be perpendicular at all points on the boundary. We thus visualize that the Orbital Angular Momentum, which is associated with the circulating phase, is explicitly absent, en masse, in a spherical wave.

Figure 1 The Hodge decomposition applied to (I) a spherical wave with a positive divergence and (II) a vortex lattice field.

Speckle fields have also been decomposed and we envisage that HHD can be used to produce speckle free fields wherever required. An experimental result that was a vortex lattice field was considered next for decomposition and explicitly segregated components obtained, as shown in Figure 1 (II).

One can, on similar lines, decompose any field obtained in interference / diffractive optics, and study its propagation dynamics and other topological features. We envisaged that fields in MRE where one requires the curl waves could also be decomposed using our technique of decomposition. We tried our method on brain acquisition data collected from a lab and obtained results. We saw that the shear waves were more clarified as compared to ones obtained using other methods. It is attributed to the fact that the HHD removes the low–frequency artifact that causes overestimation of wavelengths. This helps to reduce the noise and thus, clarifies shear waves. These shear waves are then inverted to create a mechanical property map that gives an estimation of the brain mechanical stiffness. Figure 2 (I) and (II) show the shear waves obtained using our HHD method and the inverted map to see mechanical property in brain tissues.

Figure 2 The (I) clarified shear waves obtained using HHD and (II) inverted shear waves.

Conclusion

We had established that the Helmholtz Hodge decomposition can be used as a tool to analyze scalar fields and had demonstrated that the HHD can be used to segment the solenoidal and irrotational components in them. This has been solved in the rectangular coordinate system with general boundary conditions. The propagation of optical beams in circular cross–sectional channels is also of interest. The HHD method described above yields important results in the study of propagation of optical beams. The segregated component fields give a lot of insight into the generation and annihilation of optical vortices during propagation. Our HHD method has been applied on to the brain tissues and the clarified shear waves have been studied. These shear waves were more clarified as compared to the ones obtained using other methods. This is due to the fact that the HHD removes the low–frequency artifact that causes an overestimation of wavelengths. This helps to reduce the noise and thus, clarifies shear waves. These shear waves were then inverted to study the brain mechanical stiffness. The state of tissues at any instant of time is thus revealed using our method of wave separation. This can, hence, be used to study the state of tissues in case of treatment of any disease in the human body. An estimation of the state of health of tissues can thus be obtained using the decomposition technique.

Acknowledgments

Special Thanks to Dr. Eric Barnhill, University of Edinburgh for providing the brain acquisition data file.

Conflicts of interest

None.

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