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# 计算机代写|计算机视觉代写Computer Vision代考|CS231N Total Variation (TV)

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## 计算机代写|计算机视觉代写Computer Vision代考|The Rudin-Osher-Fatemi (ROF) Model

There exist implementations of regularization techniques being different from Tikhonov’s approach. More specifically, they differ in the calculation of the regularization term $E_{\mathrm{s}}$. For example, if we take the integral of the absolute gradients instead of the squared magnitudes, we talk about total variation regularization, which was first used in an image processing context by Rudin et al. [22] for noise suppression. When we use total variation as a regularizer, we make use of the observation that noise introduces additional gradient strength, too. In contrast to Tikhonov regularization, however, total variation takes the absolute values of the gradient strength as regularization term, which is $|\nabla \hat{R}(x, y)|=\sqrt{(\partial R \hat{R} / \partial x)^2+(\partial \hat{R} / \partial y)^2}$. Consequently, the energy functional of Rudin et al. can be written as

$$E_{T V}=\frac{1}{2 \lambda} \cdot \iint(\hat{R}(x, y)-I(x, y))^2 \mathrm{~d} x \mathrm{~d} y+\iint|\nabla \hat{R}(x, y)| \mathrm{d} x \mathrm{~d} y$$
According to first letters of the names of the authors of [22], Rudin, Osher, and Fatemi, this energy functional is also known as the ROF model in literature.

The smoothness term differs from Tikhonov regularization, where the L2 norm of the gradient strength is used. A disadvantage of the L2 norm is that it tends to oversmooth the reconstructed image because it penalizes strong gradients too much. However, sharp discontinuities producing strong gradients actually do occur in real images, typically at the boundary between two objects or object and background. In contrast to the L2 norm, the absolute gradient strength (or L1 norm of the gradient strength) has the desirable property that it has no bias in favor of smooth edges. The shift from L2 to L1 norm seems only a slight modification, but in practice it turns out that the quality of the results can be improved considerably, because the bias to oversmoothed reconstructions is removed efficiently.

## 计算机代写|计算机视觉代写Computer Vision代考|Numerical Solution of the ROF Model

A technique for solving (4.18) numerically, which leads to a quite simple update procedure, was suggested by Chambolle [3]. The derivation is rather complicated;

therefore, only an outline will be given here. The interested reader is referred to $[3,4,20]$ for details.

In order to obtain a solution, Chambolle transforms the original problem into a so-called primal-dual formulation. The primal-dual formulation of the problem involves the usage of a 2-dimensional vector field $\mathbf{p}(x, y)=\left[p_1(x, y), p_2(x, y)\right]$. The vector field $\mathbf{p}$ is introduced as an auxiliary variable (also termed dual variable) and helps to convert the regularization term into a differentiable expression. With the help of $\mathbf{p}$, the absolute value $|\mathbf{v}|$ of a 2-dimensional vector $\mathbf{v}$ can be rewritten as
$$|\mathbf{v}|=\sup _{|\mathbf{p}| \leq 1}\langle\mathbf{v}, \mathbf{p}\rangle$$
where $\langle\cdot\rangle$ denotes the dot product and can be written as $\langle\mathbf{v}, \mathbf{p}\rangle=|\mathbf{v}| \cdot|\mathbf{p}| \cdot \cos \theta$, i.e., the product of the lengths of the two vectors $\mathbf{v}$ and $\mathbf{p}$ with the angle $\theta$ in between these two vectors. If $|\mathbf{p}|=1$ and, furthermore, $\mathbf{v}$ and $\mathbf{p}$ point in the same direction (i.e., $\theta=0),\langle\mathbf{v}, \mathbf{p}\rangle$ exactly equals $|\mathbf{v}|$. Therefore, $|\mathbf{v}|$ is the supremum of the dot product for all vectors $\mathbf{p}$ which are constrained to lie within the unit circle, i.e., $|\mathbf{p}| \leq 1$.

## 计算机代写|计算机视觉代写ComputerVision代考|The Rudin-OsherFatemi (ROF) Model

$|\nabla \hat{R}(x, y)|=\sqrt{(\partial R \hat{R} / \partial x)^2+(\partial \hat{R} / \partial y)^2}$. 因此，Rudin 等人的能量函数。可以写成
$$E_{T V}=\frac{1}{2 \lambda} \cdot \iint(\hat{R}(x, y)-I(x, y))^2 \mathrm{~d} x \mathrm{~d} y+\iint|\nabla \hat{R}(x, y)| \mathrm{d} x \mathrm{~d} y$$

## 计算机代写|计算机视觉代写ComputerVision代考|Numerical Solution of the ROF Model

Chambolle [3] 提出了一种数值求解 (4.18) 的技术，这导致了一个非常简单的更新过程。推导比较复杂；

$$|\mathbf{v}|=\sup _{|\mathbf{p}| \leq 1}\langle\mathbf{v}, \mathbf{p}\rangle$$

## MATLAB代写

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