Image Restoration in MATLAB: Deblurring Motion-Blurred Images

# Image Restoration Fundamentals

## 1.1 Overview of Image Restoration

Image restoration aims to recover images degraded by various factors such as motion blur, noise, and distortion. By analyzing the image degradation process and utilizing mathematical models and algorithms, it processes images to restore their original or near-original state. Image restoration has a wide range of applications in medical imaging, remote sensing, video processing, and more.

## 2. Theory of Deblurring Motion Blur Images

Deblurring motion blur images is a significant topic in the field of image restoration, aiming to eliminate the blur caused by camera or object movement and restore the clarity of the images. This chapter will introduce two classic theories of motion blur image deblurring: Wiener filtering and regularization filtering.

### 2.1 Wiener Filtering

#### 2.1.1 Principle of Wiener Filtering

Wiener filtering is a linear filter based on the Minimum Mean Square Error (MSE) criterion, which estimates the optimal image restoration result given a noise model. The principle of Wiener filtering is as follows:

Let f(x, y) be the blurred image, g(x, y) be the original clear image, and n(x, y) be the noise, then the relationship between the blurred image and the original image is:

f(x, y) = g(x, y) * h(x, y) + n(x, y)

Where h(x, y) is the motion blur kernel.

The frequency response of the Wiener filter H(u, v) is:

H(u, v) = G(u, v) * H*(u, v) / (|H(u, v)|^2 * P_n(u, v) + P_g(u, v))

Where G(u, v) and H(u, v) are the Fourier transforms of the original image and the motion blur kernel, respectively, and P_n(u, v) and P_g(u, v) are the noise power spectrum and the original image power spectrum, respectively.

#### 2.1.2 Implementation of Wiener Filtering

The steps to implement Wiener filtering are as follows:

1. Perform Fourier transform on the blurred image.
2. Calculate the Fourier transform of the motion blur kernel.
3. Estimate the noise power spectrum and the original image power spectrum based on the noise model.
4. Calculate the frequency response of the Wiener filter according to the formula.
5. Perform inverse Fourier transform on the Wiener filter to obtain the Wiener filter in the spatial domain.
6. Convolve the Wiener filter with the blurred image to obtain the deblurred image.

### 2.2 Regularization Filtering

#### 2.2.1 Principle of Regularization Filtering

Regularization filtering is a nonlinear filter based on regularization terms, aiming to suppress noise while deblurring the image. The principle of regularization filtering is as follows:

min_u ||f - g * u||^2_2 + λ * R(u)

Where f is the blurred image, g is the motion blur kernel, u is the deblurred image, R(u) is the regularization term, and λ ***

***mon regularization terms include:

***L1 norm regularization:** ||u||_1
***L2 norm regularization:** ||u||_2^2
***Total variation regularization:** ||∇u||_1

#### 2.2.2 Implementation of Regularization Filtering

The steps to implement regularization filtering are as follows:

1. Perform Fourier transform on the blurred image.
2. Calculate the Fourier transform of the motion blur kernel.
3. Choose an appropriate regularization term.
4. Solve for the deblurred image using an optimization algorithm based on the regularization term and the Fourier transform of the blurred image.
5. Perform inverse Fourier transform on the deblurred image to obtain the deblurred image in the spatial domain.

## 3. Practice of Deblurring Motion Blur Images

### 3.1 Deblurring Based on Wiener Filtering

#### 3.1.1 MATLAB Implementation of Wiener Filtering

% Read the motion blur image
image = imread('motion_blur.jpg');

% Convert the image to grayscale
grayImage = rgb2gray(image);

% Create a Wiener filter
h = fspecial('gaussian', [5 5], 1);

% Apply Wiener filtering
deblurredImage = deconvwnr(grayImage, h, 0.001);

% Display the deblurred image
figure;
imshow(deblurredImage);
title('Deblurring Result Based on Wiener Filtering');

#### 3.1.2 Analysis of Wiener Filtering Deblurring Effects

The deblurring effect of Wiener filtering is mainly influenced by the following parameters:

***Filter size:** The