## Abstract

Optical impulse-response characterization of diffusive media can be of importance in various applications, among them optical imaging in the security and medical fields. We present results of an experimental technique that we developed for acquiring the impulse response, based upon the Kramers–Kronig algorithm, and have been applied for optical imaging of objects hidden behind clothing. We demonstrate three-dimensional imaging with $5\text{}\mathrm{mm}$ depth resolution between diffusive layers.

© 2010 Optical Society of America

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### Equations (8)

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(1)
$${\phi}_{\mathrm{KK}}(\omega )=-\frac{1}{\pi}P{\int}_{{\omega}_{1}}^{{\omega}_{2}}\mathrm{d}{\omega}^{\prime}\frac{\mathrm{ln}|H({\omega}^{\prime})|}{{\omega}^{\prime}-\omega},$$
(2)
$${H}_{\mathrm{KK}}(\omega )=|H(\omega )|\mathrm{exp}[i{\phi}_{\mathrm{KK}}(\omega )]\mathrm{.}$$
(3)
$$h(n)={h}_{e}(n)u(n),$$
(4)
$$u(n)=\{\begin{array}{cc}1& n=0,N/2\\ 2& n=1,2,\dots (N/2)-1\\ 0& (N/2)+1,\dots N-1\end{array}\mathrm{.}$$
(5)
$${h}_{e}(n)=\frac{1}{N}\sum _{k=0}^{N-1}\Re H(k)\mathrm{exp}(2\pi ikn/N)=\mathrm{IFFT}\{\Re H(k)\}\mathrm{.}$$
(6)
$$[\mathrm{IFFT}\{\mathrm{ln}|H({\omega}_{k})|\}u(n)]=\mathrm{IFFT}\{\mathrm{ln}H({\omega}_{k})\}\mathrm{.}$$
(7)
$$H({\omega}_{k})=\mathrm{exp}(\mathrm{FFT}[\mathrm{IFFT}\{\mathrm{ln}|H({\omega}_{k})|\}u(n)]),$$
(8)
$$h({t}_{n})=\mathrm{IFFT}\{H({\omega}_{k})\}=\mathrm{IFFT}\{\mathrm{exp}(\mathrm{FFT}[\mathrm{IFFT}\{\mathrm{ln}|H({\omega}_{k})|\}u(n)])\}\mathrm{.}$$