# CS代写|图像处理作业代写Image Processing代考|ECE867

## CS代写|图像处理作业代写Image Processing代考|Unconstrained Restoration

The unconstrained restoration method treats the image as only a digital matrix and restores the image from a mathematical point of view without considering the physical constraints that the image should be subjected to after restoration.

Starting from the general image degradation model given in Figure 3.1, it can be obtained from Equation (3.7):
$$\boldsymbol{n}=g-h f$$
Without any prior knowledge of $\boldsymbol{n}$, image restoration can be described as looking for an estimate $\boldsymbol{f}{\varepsilon}$ of the original image $\boldsymbol{f}$ so that $\boldsymbol{h} \boldsymbol{f}_e$ is closest to the degraded image $\boldsymbol{g}$ in the sense of minimum mean square error, that is, the norm of $n$ is to be minimized: $$\mid \boldsymbol{n}\left|^2=\boldsymbol{n}^T \boldsymbol{n}=\right| g-\boldsymbol{h} f_e |^2=\left(\boldsymbol{g}-\boldsymbol{h} f_e\right)^T\left(\boldsymbol{g}-\boldsymbol{h} f_e\right)$$ According to the above equation, the restoration problem can be regarded as to determine the minimum value of the following equation for $f_i$ : $$L\left(f_c\right)=\left|g-\boldsymbol{h} f_c\right|^2$$ Here it is only needed to differentiate $L$ to $\boldsymbol{f}{\varepsilon}$ and set the result to 0 , when $\boldsymbol{h}^{-1}$ exists, then the unconstrained restoration equation can be obtained:
$$\boldsymbol{f}_e=\left(\boldsymbol{h}^T \boldsymbol{h}\right)^{-1} \boldsymbol{h}^T \boldsymbol{g}=\boldsymbol{h}^{-1}\left(\boldsymbol{h}^T\right)^{-1} \boldsymbol{h}^T \boldsymbol{g}=\boldsymbol{h}^{-1} \boldsymbol{g}$$

## CS代写|图像处理作业代写Image Processing代考|Constrained Restoration

Also starting from Equation (3.7), constrained restoration considers selecting a linear operator $\boldsymbol{Q}$ (transformation matrix) of $\boldsymbol{f}e$ so that $\left|\boldsymbol{Q} \boldsymbol{f}{\mathrm{e}}\right|^2$ is the smallest. This problem can be solved by the Lagrangian multiplier method. Let $l$ be the Lagrangian multiplier, find $\boldsymbol{f}_e$ that minimizes the following criterion function:
$$L\left(f_e\right)=\left|Q f_c\right|^2+l\left(\left|g-H f_c\right|^2-|n|^2\right)$$
Similar to solving Equation (3.20), the constrained restoration equation can be obtained (let $s=1 / l)$ :
$$\boldsymbol{f}_e=\left[\boldsymbol{H}^T \boldsymbol{H}+s \boldsymbol{Q}^T \boldsymbol{Q}\right]^{-1} \boldsymbol{H}^T \boldsymbol{g}$$
3.2.2.2 Wiener Filter
The Wiener filter is a minimum mean square error filter. It can be derived directly from Equation (3.37). In the frequency domain, the general representation of the Wiener filter is:
$$F_c(u, v)=H_W(u, v) G(u, v)=\frac{H^4(u, v)}{|H(u, v)|^2+s\left[S_n(u, v) / S_f(u, v)\right]} G(u, v)$$
In the equation, $s$ is a parameter (see below), and $S_f(u, v)$ and $S_n(u, v)$ are the Fourier transform of the correlation matrix elements of the original image and noise, respectively. There are several variants of Equation (3.38):

1. If $s=1, H_{\mathrm{w}}(u, v)$ is the standard Wiener filter.
2. If $s$ is a variable, it is called a parametric Wiener filter.
3. When there is no noise, $S_n(u, v)=0$, the Wiener filter degenerates into the ideal inverse filter of the previous sub-section.

Because $s$ must be adjusted to satisfy Equation (3.37), when $s=1$, the optimal solution that satisfies Equation (3.37) cannot be obtained by using Equation (3.38), but it is optimal in the sense of minimization of $E\left{\left[f(x, y)-f_c(x, y)\right]^2\right}$. Here, both $f(\bullet)$ and $f_e(\bullet)$ are regarded as random variables, and a statistical criterion is thus obtained.

In practice, $S_n(u, v)$ and $S_f(u, v)$ are often unknown. At this time, Equation (3.38) can be approximated by the following equation (where $K$ is a predetermined constant):
$$F_e(u, v) \approx \frac{H^*(u, v)}{|H(u, v)|^2+K} G(u, v)$$

# 图像处理代考

## CS代写|图像处理作业代写Image Processing代考|Unconstrained Restoration

$$\boldsymbol{n}=g-h f$$

$$|\boldsymbol{n}|^2=\boldsymbol{n}^T \boldsymbol{n}=\left|g-\boldsymbol{h} f_e\right|^2=\left(\boldsymbol{g}-\boldsymbol{h} f_e\right)^T\left(\boldsymbol{g}-\boldsymbol{h} f_e\right)$$

$$L\left(f_c\right)=\left|g-\boldsymbol{h} f_c\right|^2$$

$$\boldsymbol{f}_e=\left(\boldsymbol{h}^T \boldsymbol{h}\right)^{-1} \boldsymbol{h}^T \boldsymbol{g}=\boldsymbol{h}^{-1}\left(\boldsymbol{h}^T\right)^{-1} \boldsymbol{h}^T \boldsymbol{g}=\boldsymbol{h}^{-1} \boldsymbol{g}$$

## CS代写|图像处理作业代写Image Processing代考|Constrained Restoration

$$L\left(f_e\right)=\left|Q f_c\right|^2+l\left(\left|g-H f_c\right|^2-|n|^2\right)$$

$$\boldsymbol{f}_e=\left[\boldsymbol{H}^T \boldsymbol{H}+s \boldsymbol{Q}^T \boldsymbol{Q}\right]^{-1} \boldsymbol{H}^T \boldsymbol{g}$$
3.2.2.2 Wiener 滤波器
Wiener 滤波器是最小均方误差滤波器。它可以直接从方程 (3.37) 推导出来。在频域中，维纳滤波器的一 般表示为:
$$F_c(u, v)=H_W(u, v) G(u, v)=\frac{H^4(u, v)}{|H(u, v)|^2+s\left[S_n(u, v) / S_f(u, v)\right]} G(u, v)$$

1. 如果 $s=1, H_{\mathrm{w}}(u, v)$ 是标准的维纳滤波器。
2. 如果 $s$ 是一个变量，它被称为参数维纳滤波器。
3. 没有噪音的时候， $S_n(u, v)=0$ ，维纳滤波器退化为上一小节的理想逆滤波器。
因为 $s$ 必须调整以满足方程 (3.37)，当 $s=1$ ，满足式 (3.37) 的最优解不能通过式 (3.38) 得到，但在 量，从而获得统计标准。
在实践中， $S_n(u, v)$ 和 $S_f(u, v)$ 往往是末知的。此时，方程 (3.38) 可以近似为以下方程（其中 $K$ 是预定 常数) :
$$F_e(u, v) \approx \frac{H^*(u, v)}{|H(u, v)|^2+K} G(u, v)$$

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