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## EECS207A ﬁnal Project. Image restoration in the frequency domain Report submitted for fulﬁllment of the Requirements for EECS 207A, UCI

December 2, 2004. University of California, Irvine   Compiled on May 22, 2020 at 5:31pm

## Contents

1.1 Introduction
1.2 Theory

## Chapter 1Project Proposal

### 1.1 Introduction

This is an outline of my ﬁnal project for EECS 207A.

The project will be to develop an application which will accept as input a degraded image ﬁle, and will perform image restoration in the frequency domain and will generate the restored image and save it to the disk.

### 1.2 Theory

Assume that the original, undistorted object is $$f$$, Let the Point spread function (PSF) of the camera be $$h$$, and let the degraded resulting image of $$f$$ produced by the camera be $$g.$$

The goal is that, given $$g$$, and assuming an $$h$$, we can restore $$g$$ back as close as possible to $$f$$ , call this restored image $$\tilde{f}\ .$$ To do this, we will use the following model of image degradation

$g=h\circledast \tilde{f}+\eta$

Where $$\eta$$ is the noise and $$\circledast$$ is the convolution operator.

In this project I will not consider noise and will assume it to be zero.

Converting the above equation to the frequency domain, we obtain

$G=H\ \tilde{F}$

Where $$G,H,$$ and $$F$$ are the 2 dimensional Fourier transforms of $$g,h,$$ and $$\tilde{f}$$

Hence the transform of the restored image will be

$\tilde{F}=\frac{G}{H}$

Where in the above, the division will be done term by term between the two matrices. If a value of $$H$$ contain a zero, then this term will not be used.

Now we need to apply the 2D inverse Fourier transform to obtain the spatial restored image $$\tilde{f}$$ from $$\tilde{F}$$.

The PSF used will be the 2D Gaussian, with a certain standard deviation (Will try 5 or 10 pixels, and experiment with these values to ﬁnd the best value to get the best restoration) and with zero mean.

The degraded images will be generated by blurring using either a standard $$3\times 3$$ or $$5\times 5$$ averaging ﬁlter or a Gaussian ﬁlter. The restored image will be visually compared to the original image, and a conclusion as to the eﬀectiveness of this method of restoration will be given.