## Abstract

We have developed what we believe is a new technique for obtaining
a whole-field image representing the deforming amounts of a diffuse
object. The object is supposedly continuously deforming and does
not stop deforming during the measurement. This technique uses
arccosine operations to extract the absolute, not signed, value of the
phase. We assume that a right-phase change retains almost the same
value in a small local area. This retention determines the sign of
the phase and consequently the value of the phase change. The
deformation phase during any term of the deforming process is shown as
a map through the temporal-phase unwrapping of the calculated
phase.

© 2001 Optical Society of America

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### Equations (10)

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(1)
$$I\left(\mathbf{r},t\right)={O}_{D}\left(\mathbf{r}\right)+{O}_{\mathrm{ALT}}\left(\mathbf{r}\right)cos\left\{\mathrm{\psi}\left(\mathbf{r},t\right)\right\},$$
(2)
$$|\mathrm{\psi}\left(\mathbf{r},{t}_{k}\right)|={cos}^{-1}\left\{\frac{I\left(\mathbf{r},{t}_{k}\right)-\left[\frac{{I}_{max}\left(\mathbf{r}\right)+{I}_{min}\left(\mathbf{r}\right)}{2}\right]}{\left[\frac{{I}_{max}\left(\mathbf{r}\right)-{I}_{min}\left(\mathbf{r}\right)}{2}\right]}\right\}.$$
(3)
$${\mathrm{\delta}}_{\mathit{kl}}\left(\mathbf{r}\right)=\pm |\mathrm{\psi}\left(\mathbf{r},{t}_{l}\right)|-\left\{\pm |\mathrm{\psi}\left(\mathbf{r},{t}_{k}\right)|\right\}$$
(4)
$$cos\left[\mathrm{\psi}\left(\mathbf{r},{t}_{0}\right)+\mathrm{\Delta}\mathrm{\psi}\left(\mathbf{r},{t}_{j}-{t}_{0}\right)\right]=cos\left[\mathrm{\psi}\left(\mathbf{r},{t}_{0}\right)\right]cos\left[\mathrm{\Delta}\mathrm{\psi}\left(\mathbf{r},{t}_{j}-{t}_{0}\right)\right]-sin\left[\mathrm{\psi}\left(\mathbf{r},{t}_{0}\right)\right]\times sin\left[\mathrm{\Delta}\mathrm{\psi}\left(\mathbf{r},{t}_{j}-{t}_{0}\right)\right].$$
(5)
$$sin\left[\mathrm{\psi}\left(\mathbf{r},{t}_{0}\right)\right]sin\left[\mathrm{\Delta}\mathrm{\psi}\left(\mathbf{r},{t}_{j}-{t}_{0}\right)\right]=cos\left[\mathrm{\psi}\left(\mathbf{r},{t}_{0}\right)\right]cos\left[\mathrm{\Delta}\mathrm{\psi}\left(\mathbf{r},{t}_{j}-{t}_{0}\right)\right]-cos\left[\mathrm{\psi}\left(\mathbf{r},{t}_{0}\right)+\mathrm{\Delta}\mathrm{\psi}\left(\mathbf{r},{t}_{j}-{t}_{0}\right)\right]=cos\left[\mathrm{\psi}\left(\mathbf{r},{t}_{0}\right)\right]cos\left[\mathrm{\Delta}\mathrm{\psi}\left(\mathbf{r},{t}_{j}-{t}_{0}\right)\right]-cos\left[\mathrm{\psi}\left(\mathbf{r},{t}_{j}\right)\right].$$
(6)
$$\mathrm{\psi}\left(\mathbf{r},{t}_{0}\right)=\mathrm{arg}\left\{cos\left[\mathrm{\psi}\left(\mathbf{r},{t}_{0}\right)\right]sin\left[\mathrm{\Delta}\mathrm{\psi}\left(\mathbf{r},{t}_{j}-{t}_{0}\right)\right]+isin\left[\mathrm{\psi}\left(\mathbf{r},{t}_{0}\right)\right]sin\left[\mathrm{\Delta}\mathrm{\psi}\left(\mathbf{r},{t}_{j}-{t}_{0}\right)\right]\right\}.$$
(7)
$$\mathrm{\delta}\prime \left(\mathbf{r},t\right)=\pm |\mathrm{\psi}\left(\mathbf{r},t\right)|-\mathrm{\psi}\left(\mathbf{r},{t}_{0}\right),$$
(8)
$$\mathrm{\Delta}\mathrm{\psi}\prime \left(\mathbf{r},t\right)=\mathrm{arg}\left[{C}_{r}\left(\mathbf{r},t\right)+{\mathit{iC}}_{i}\left(\mathbf{r},t\right)\right].$$
(9)
$$\mathrm{\Delta}\mathrm{\psi}\u2034\left(\mathbf{r},{t}_{n}\right)=\sum _{l=1}^{l=n}\mathrm{arg}\left\{expi\left[\mathrm{\Delta}\mathrm{\psi}\u2033\left(\mathbf{r},{t}_{l}\right)-\mathrm{\Delta}\mathrm{\psi}\u2033\left(\mathbf{r},{t}_{l-1}\right)\right]\right\}.$$
(10)
$$\mathrm{\varphi}\left(\mathbf{r},{t}_{q}-{t}_{p}\right)=\mathrm{\Delta}\mathrm{\psi}\u2034\left(\mathbf{r},{t}_{q}\right)-\mathrm{\Delta}\mathrm{\psi}\u2034\left(\mathbf{r},{t}_{p}\right).$$