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Image Restoration by Inverse Filtering

The purpose of image restoration is to estimate or recover the scene without image degradation or distortion caused by non-ideal image system (e.g. the optics of the camera system). Inverse filtering is one of the techniques used for image restoration to obtain a recovered image $f'(x,y)$ from the image data $g(x,y)$ so that $f'(x,y)=f(x,y)$ in the ideal situation $n(x,y)=0$ and $h(x,y)*h'(x,y)=\delta(x,y)$ or $H(\omega_x,\omega_y)H'(\omega_x,\omega_y)=1$.

imagerestoration.gif

The camera system for image acquisition can be modeled mathematically by

\begin{displaymath}
g(x,y)=\int_0^T \int \int_{-\infty}^{\infty} h(x,y,x',y',t) f(x',y',t)dx'dy'dt
+n(x,y)
\end{displaymath}

where $T$ is the exposure time, $n(x,y)$ is some additive noise, and $h(x,y,x',y',t)$ is a function characterizing the distortion introduced by the imaging system, caused by, for example, limited aperture, out of focus, random atmospheric turbulence, and/or relative motion. If the system is ideal, spatial and time invariant, and noise-free, i.e.,

\begin{displaymath}h(x,y,x',y',t)=\delta(x-x',y-y') \end{displaymath}

then the imaging process becomes

\begin{displaymath}g(x,y)=\int_0^T f(x,y,t) dt \end{displaymath}

If the signal is also time invariant (a stationary scene), i.e., $f(x,y,t)=f(x,y)$, the image obtained is simply

\begin{displaymath}g(x,y)=T\;f(x,y) \end{displaymath}

We consider the correction of some possible distortions that may occur during image acquisition.



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next up previous
Next: Relative motion Up: Image_Processing Previous: Image Enhancement by Filtering
Ruye Wang 2007-11-15