## Abstract

A novel method is proposed for the direct and simultaneous estimation of multiple phase derivatives corresponding to strain and slope fields from a single moiré fringe pattern in digital holographic moiré. The interference field in a given row/column is a multicomponent complex exponential signal and is represented as a spatially-varying autoregressive (SVAR) process. The spatially-varying coefficients of the SVAR model are computed by approximating them as the linear combination of linearly independent basis functions. Further, the spatially varying poles of the transfer function corresponding to the SVAR model are computed which provide the accurate estimation of the multiple phase derivatives. The simulation and experimental results are provided to substantiate the effectiveness of the proposed method.

© 2014 Optical Society of America

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### Equations (20)

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(1)
$$I[n,m]={e}^{j{\psi}_{1}[n,m]}+{e}^{j{\psi}_{2}[n,m]}+\varsigma [n,m],$$
(2)
$$I[n,m]={e}^{j{\psi}_{1}[n,m]}+{e}^{j{\psi}_{2}^{\prime}[n,m]}+\varsigma [n,m],$$
(3)
$$I[n]={e}^{j{\psi}_{1}[n]}+{e}^{j{\psi}_{2}^{\prime}[n]}+\varsigma [n].$$
(4)
$$\begin{array}{l}{\dot{\psi}}_{1}[n]=\frac{\partial {\psi}_{1}[n]}{\partial n}\\ {\dot{\psi}}_{2}^{\prime}[n]=\frac{\partial {\psi}_{2}^{\prime}[n]}{\partial n}.\end{array}$$
(4)
$$I[n]=\sum _{p=1}^{P}{a}_{p}[n]I[n-p]+\varsigma [n],$$
(5)
$$H(z;n)=\frac{1}{1-{\sum}_{p=1}^{P}{a}_{p}[n]{z}^{-p}}.$$
(6)
$${a}_{p}[n]=\sum _{k=0}^{K}{a}_{pk}{\beta}_{k}[n],$$
(7)
$$I[n]=\sum _{p=1}^{P}\sum _{k=0}^{K}{a}_{pk}{\beta}_{k}[n]I[n-p]+\varsigma [n].$$
(8)
$$\widehat{I}[n]=\sum _{p=1}^{P}\sum _{k=0}^{K}{a}_{pk}{\beta}_{k}[n]I[n-p].$$
(9)
$$\varsigma [n]=I[n]-\widehat{I}[n].$$
(10)
$${\Vert \varsigma \Vert}^{2}={\left|\varsigma [1]\right|}^{2}+{\left|\varsigma [2]\right|}^{2}+\dots +{\left|\varsigma [N]\right|}^{2}.$$
(12)
$$(x,y)=\sum _{n=1}^{N}x[n]{y}^{*}[n],$$
(11)
$${\Vert \varsigma \Vert}^{2}=(\varsigma ,\varsigma ).$$
(12)
$$\begin{array}{ll}{\Vert \varsigma \Vert}^{2}\hfill & =(\varsigma ,\varsigma )\hfill \\ \hfill & ={\Vert I\Vert}^{2}-2(I,\widehat{I})+{\Vert \widehat{I}\Vert}^{2}.\hfill \end{array}$$
(15)
$$\begin{array}{ccc}\hfill {w}_{kp}& =\hfill & \left\{{\beta}_{k}[n]I[n-p]\right\}\hfill \\ \hfill {s}_{kl}(p,q)& =\hfill & ({w}_{kp}-{w}_{lq})\hfill \\ \hfill {s}_{0l}& =\hfill & {\left({s}_{0l}(0,1),\dots ,{s}_{0l}(0,P)\right)}^{\top},\hfill \end{array}$$
(13)
$${\Vert \varsigma \Vert}^{2}={\Vert I\Vert}^{2}-2{A}^{\top}{s}_{0}+{A}^{\top}SA.$$
(18)
$${\beta}_{k}[n]={\left(\frac{n}{N}\right)}^{k}$$
(19)
$${\beta}_{k}[n]=\{\begin{array}{ll}\text{cos}\frac{k\pi n}{2N}\hfill & k\hspace{0.17em}\text{even}\hfill \\ \text{sin}\frac{(k+1)\pi n}{2N}\hfill & k\hspace{0.17em}\text{odd}\hfill \end{array}$$
(20)
$${\dot{\psi}}_{2}[n]=\frac{\partial {\psi}_{2}^{\prime}[n]}{\partial n}-{\omega}_{n}.$$