## 计算机代写|图像处理代写Image Processing代考|Constrained Least Squares Restoration

The method of Wiener filtering is a statistical method. The optimal criterion it uses is based on the respective correlation matrices of the image and noise, so the result obtained is only optimal in the average sense. The constrained least squares restoration method is another constrained restoration method. It only needs knowledge about the noise mean and variance to get the optimal result for each given image.
In the frequency domain, the formula for constrained least squares recovery is
$$F_e(u, v)=\left[\frac{H^*(u, v)}{|H(u, v)|^2+s|L(u, v)|^2}\right] G(u, v) \quad u, v=0,1, \ldots, M-1$$
In this equation, $L(u, v)$ represents the 2-D Fourier transform corresponding to the function that extends the Laplacian (calculating the sum of the second-order partial derivatives along the $X$ and $Y$ directions) to the image size. The form of Equation (3.41) is somewhat similar to the form of the Wiener filter. The main difference is that there is no need for knowledge of other statistical parameters except for the estimation of noise mean and variance.

Figure $3.8$ shows a comparison example of Wiener filtering and constrained least square filtering in two degraded situations. Figure $3.8 \mathrm{a}$ is the degraded image obtained by blurring the Cameraman image with a filter of a defocus radius of $R=3$. Figure $3.8 \mathrm{~b}$ is the result of restoring Figure $3.8 \mathrm{a}$ with Wiener filtering, Figure $3.8 \mathrm{c}$ is the result of restoring Figure $3.8 \mathrm{a}$ with the constrained least square filter. Figure $3.8 \mathrm{~d}$ is a degraded image with random noise of a variance of 4 superimposed on Figure $3.8 \mathrm{a}$, Figure $3.8 \mathrm{e}$ is the result of restoring Figure $3.8 \mathrm{~d}$ with Wiener filter, Figure $3.8 \mathrm{f}$ is the result of restoring Fig. $3.8 \mathrm{~d}$ with a constrained least square filter. It can be seen from these figures that when there is both blur and noise, the effect of constrained least square filtering is slightly better than Wiener filtering; when there is only blur but no noise, the effects of the two methods are the same.

## 计算机代写|图像处理代写Image Processing代考|Blind De-Convolution Process

The process of implementing blind de-convolution is shown in Figure $3.10$ (Cho and Lee 2009). To refine $\boldsymbol{f}_e$ and $\boldsymbol{h}$ progressively, three steps need to be iteratively calculated: prediction (including bilateral filtering, impulsive filtering, and gradient amplitude thresholding), kernel estimation, and deconvolution. Put the prediction here at the beginning of the loop is to provide the initial $f_e$ value for kernel estimation.

In the prediction step, two gradient maps of $f\left(G_x=\partial f / \partial x\right.$ and $\left.G_y=\partial f / \partial y\right)$ need to be calculated to eliminate the noise in the smooth region and predict the significant edges in $f$. Except at the beginning of the iteration, the input to the prediction step is the estimate of $f$ obtained in the deconvolution step in the previous iteration. In the kernel estimation step, the predicted gradient image and the gradient image of $g$ need to be used. In the deconvolution step, $\boldsymbol{h}$ and $g$ need to be used to obtain an estimate of $\boldsymbol{f}$, which will be used in the prediction step of the next iteration.

To make the estimation of $\boldsymbol{h}$ and $\boldsymbol{f}$ more effective and efficient, the scheme from coarse to fine can be adopted. At the roughest level, a down-sampled version of $g$ can be used to initialize the process of the prediction step. After the final estimate of $f$ is roughly obtained, it is up-sampled by bilinear interpolation and then used in the next finer level of prediction step. Such a coarse-to-fine scheme can still achieve better results when the blur is large and when only using image filtering to predict may not be sufficient to capture sharp edges.

In the process of iteratively updating $\boldsymbol{h}$ and $\boldsymbol{f}$ from coarse to fine, grayscale versions of $\boldsymbol{g}$ and $f$ can be used. After obtaining the final $\boldsymbol{h}$ at the finest scale (input image size), perform final deconvolution with $\boldsymbol{h}$ for each color channel to obtain a de-blurring result.

# 图像处理代考

## 计算机代写|图像处理代写Image Processing代考|Constrained Least Squares Restoration

$$F_e(u, v)=\left[\frac{H^*(u, v)}{|H(u, v)|^2+s|L(u, v)|^2}\right] G(u, v) \quad u, v=0,1, \ldots, M-1$$

## 计算机代写|图像处理代写Image Processing代考|Blind De-Convolution Process

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