## Abstract

The use of complex amplitude correlation to compensate for large in-plane motion in digital speckle pattern interferometry is investigated. The result is compared with experiments where digital speckle photography (DSP) is used for compensation. An advantage of using complex amplitude correlation instead of intensity correlation (as in DSP) is that the phase change describing the deformation is retrieved directly from the correlation peak, and there is no need to compensate for the large movement and then use the interferometric algorithms. A discovered drawback of this method is that the correlation values drop quickly if a phase gradient larger than π is present in the subimages used for cross correlation. This means that, for the complex amplitude correlation to be used, the size of the subimages must be well chosen or a third parämeter in the cross-correlation algorithm that compensates for the phase variation is needed.
Correlation values and wrapped phase maps from the two techniques (intensity and complex amplitude correlation) are presented.

© 2006 Optical Society of America

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### Equations (20)

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(1)
$$I\left(n\right)={I}_{0}+{I}_{M}\text{\hspace{0.17em} cos}\left(\varphi +n\pi /2\right),$$
(2)
$$I\prime \left(n\right)={I}_{0}\prime +{I}_{M}\prime \text{\hspace{0.17em} cos}\left(\varphi \prime +n\pi /2+\Delta \varphi \right),$$
(3)
$$\text{\hspace{1em}}n=0,1,2,3,$$
(4)
$$C=I\left(0\right)-I\left(2\right)=2{I}_{M}\text{\hspace{0.17em} cos \hspace{0.17em}}\varphi ,$$
(5)
$$\mathit{S}=I\left(3\right)-I\left(1\right)=2{I}_{M}\text{\hspace{0.17em} sin \hspace{0.17em}}\varphi \text{,}$$
(6)
$$C\prime =I\prime \left(0\right)-I\prime \left(2\right)=2{I}_{M}\prime \text{\hspace{0.17em} cos}\left(\varphi \prime +\Delta \varphi \right),$$
(7)
$$\mathit{\text{S}}\prime =I\prime \left(3\right)-I\prime \left(1\right)=2{I}_{M}\prime \text{\hspace{0.17em} sin}\left(\varphi \prime +\Delta \varphi \right)\text{,}$$
(8)
$$\Delta {\varphi}_{w}=\text{arctan}\left(\frac{\mathit{\text{S}}\prime \mathit{\text{C}}-\mathit{\text{SC}}\prime}{C\prime C+S\prime S}\right),$$
(9)
$${\text{Ref}}_{\text{int}}={\left({C}^{2}+{S}^{2}\right)}^{1/2},$$
(10)
$${\text{Def}}_{\text{int}}={\left(C{\prime}^{2}+S{\prime}^{2}\right)}^{1/2}.$$
(11)
$${\text{Ref}}_{\text{amp}}=\text{sign}\left(C\right)\sqrt{\left|C\right|}+i\text{\hspace{0.17em} sign}\left(S\right)\sqrt{\left|S\right|},$$
(12)
$${\text{Def}}_{\text{amp}}=\text{sign}\left(C\prime \right)\sqrt{\left|C\prime \right|}+i\text{\hspace{0.17em} sign}\left(S\prime \right)\sqrt{\left|S\prime \right|}.$$
(13)
$${\text{phase}}_{\text{amp}}=\text{arctan}{\left[\frac{\text{Im}\left({c}_{\text{amp}}\right)}{\text{Re}\left({c}_{\text{amp}}\right)}\right]}_{\text{max}},$$
(14)
$$\mu =\frac{2}{\pi}\left[\text{arccos}\left(\frac{\left|{A}_{p}\right|}{D*}\right)-\left(\frac{\left|{A}_{p}\right|}{D*}\right)\sqrt{1-{\left(\frac{\left|{A}_{p}\right|}{D*}\right)}^{2}}\right]=\frac{\Theta -\text{sin \hspace{0.17em}}\Theta}{\pi},$$
(15)
$$\Theta =2\text{\hspace{0.17em} arccos}\left(\frac{\left|{A}_{p}\right|}{D*}\right),$$
(16)
$${\gamma}_{0\text{, int}}={\left(\frac{\Theta -\text{sin \hspace{0.17em}}\Theta}{\pi}\right)}^{2}.$$
(17)
$${\gamma}_{0\text{, amp}}=\left|\frac{\Theta -\text{sin \hspace{0.17em}}\Theta}{\pi}\right|.$$
(18)
$${A}_{x}={a}_{x}\left[1+\frac{L}{{L}_{s}}\left(1-{{l}_{\mathit{sx}}}^{2}\right)\right],$$
(19)
$${A}_{y}={a}_{y}+L{l}_{sx}{\Omega}_{z}$$
(20)
$$\sigma =1.22\lambda \left(1+M\right){f}_{\#},$$