## Abstract

In recent years, optical interferometry has been applied to the whole-field, noncontact measurement of vibrating or continuously deforming objects. In many cases, a high resolution measurement of kinematic (displacement, velocity, and acceleration, etc.) and deformation parameters (strain,
curvature, and twist, etc.) can give useful information on the dynamic response of the objects concerned. Different signal processing algorithms are applied to two types of interferogram sequences, which were captured by a high-speed camera using different interferometric setups: (1) a speckle or fringe pattern sequence with a temporal carrier and (2) a wrapped phase map sequence. These algorithms include Fourier transform, windowed Fourier transform, wavelet transform, and even a combination of two of these techniques. We will compare these algorithms using the example of a 1D temporal evaluation of interferogram sequences and extend these algorithms to 2D and 3D processing,
so that accurate kinematic and deformation parameters of moving objects can be evaluated with different types of optical interferometry.

© 2007 Optical Society of America

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### Equations (19)

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(1)
$${f}_{\mathrm{I}}\left(x,y;\text{\hspace{0.17em}}t\right)={I}_{b}\left(x,y;\text{\hspace{0.17em}}t\right)+A\left(x,y;\text{\hspace{0.17em}}t\right)\mathrm{cos}\left(\phi \left(x,y;\text{\hspace{0.17em}}t\right)\right)\text{,}$$
(2)
$${f}_{\text{II}}\left(x,y;\text{\hspace{0.17em}}t\right)=\mathrm{exp}\left(j\varphi \left(x,y;\text{\hspace{0.17em}}t\right)\right)\text{,}$$
(3)
$$\widehat{f}\left(\xi \right)={\displaystyle \underset{-\infty}{\overset{+\infty}{\int}}f\left(t\right)\mathrm{exp}}\left(-j\xi t\right)\mathrm{d}t\text{,}$$
(4)
$$f\left(t\right)=\frac{1}{2\pi}{\displaystyle \underset{-\infty}{\overset{+\infty}{\int}}\widehat{f}\left(\xi \right)\mathrm{exp}}\left(j\xi t\right)\mathrm{d}\xi \text{,}$$
(5)
$$\widehat{f}\left(\xi \right)=DC\left(\xi \right)+\widehat{C}\left(\xi \right)+\widehat{C}*\left(\xi \right)\text{,}$$
(6)
$$\phi \left(t\right)=\mathrm{arctan}\text{\hspace{0.17em}}\frac{\mathrm{Im}\left(C\left(t\right)\right)}{\mathrm{Re}\left(C\left(t\right)\right)}\text{,}$$
(7)
$$Sf\left(u,\xi \right)={\displaystyle \underset{-\infty}{\overset{+\infty}{\int}}f\left(t\right)}{{\displaystyle g}}_{u\text{,}\xi}*\left(t\right)\mathrm{d}t\text{,}$$
(8)
$$f\left(t\right)=\frac{1}{2\pi}{\displaystyle \underset{-\infty}{\overset{+\infty}{\int}}{\displaystyle \underset{-\infty}{\overset{+\infty}{\int}}Sf\left(u,\xi \right)}}{g}_{u\text{,}\xi}\left(t\right)\mathrm{d}\xi \mathrm{d}u\text{,}$$
(9)
$${g}_{u\text{,}\xi}\left(t\right)=g\left(t-u\right)\mathrm{exp}\left(j\xi t\right)\text{.}$$
(10)
$$g\left(t\right)=\mathrm{exp}\left(-{t}^{2}/2{\sigma}^{2}\right)\text{,}$$
(11)
$$S{f}_{\mathrm{I}\mathrm{I}}\left(u,\xi \right)=\frac{\sqrt{s}}{2}\text{\hspace{0.17em}}A\left(u\right)\mathrm{exp}\left(j\left[\phi \left(u\right)-\xi u\right]\right)\left(\widehat{g}\left(s\left[\xi -\phi \prime \left(u\right)\right]\right)+\epsilon \left(u,\xi \right)\right)\text{,}$$
(12)
$$\xi \left(u\right)=\phi \prime \left(u\right)\text{,}$$
(13)
$$Wf(a,b)=\frac{1}{\sqrt{a}}{\displaystyle \underset{-\infty}{\overset{+\infty}{\int}}f\left(t\right)}\mathrm{\Psi}*\left(\frac{t-b}{a}\right)\mathrm{d}t=\frac{1}{\sqrt{a}}{\displaystyle \underset{-\infty}{\overset{+\infty}{\int}}f\left(t\right)}{\mathrm{\Psi}}_{ab}*\left(t\right)\mathrm{d}t\text{,}$$
(14)
$$\mathrm{\Psi}\left(t\right)=g\left(t\right)\mathrm{exp}\left(i{\omega}_{0}t\right)\text{,}$$
(15)
$${\mathrm{\Psi}}_{ab}\left(t\right)=\mathrm{\Psi}\left(\frac{t-b}{a}\right)=\mathrm{exp}(-\frac{{\left(t-b\right)}^{2}}{2{a}^{2}})\mathrm{exp}\left(i\text{\hspace{0.17em}}\frac{{\omega}_{0}}{a}\left(t-b\right)\right)\text{.}$$
(16)
$$\xi =\frac{{\omega}_{0}}{a}=\frac{2\pi}{a}\text{.}$$
(17)
$$Wf\left(a,b\right)=\frac{\sqrt{a}}{2}\text{\hspace{0.17em}}A\left(b\right)\left\{\widehat{g}\left[a\left(\xi -\phi \prime \left(b\right)\right)\right]+\epsilon \left(b,\xi \right)\right\}\mathrm{exp}\left[i\phi \left(b\right)\right]\text{,}$$
(18)
$$\phi \prime \left(b\right)={\xi}_{rb}=\frac{{\omega}_{0}}{{a}_{rb}}\text{,}$$
(19)
$${f}_{\text{I I}}\left(t\right)=U\left(x,y;\text{\hspace{0.17em}}{t}_{n}\right)U*\left(x,y;\text{\hspace{0.17em}}{t}_{n-1}\right)=\mathrm{exp}\left(j\phi \left(x,y;\text{\hspace{0.17em}}{t}_{n}\right)\right)\mathrm{exp}\left(-j\phi \left(x,y;\text{\hspace{0.17em}}{t}_{n-1}\right)\right)\text{,}$$