9 votes
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Deconvolution of a 1D Time Domain Wave Signal Convolved with Series of Rect Signals

Solving a deconvolution isn't easy even in simulated environment not to mention in practice. The main trick to solve it is using the proper model / prior for the problem and very good measurements (...
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8 votes
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The Meaning of the Terms Isotropic and Anisotropic in the Total Variation Framework

In the Total Variation framework we define 2 flavors: $$ \text{Isotropic TV} \; {TV}_{ {L}_{2} } \left( X \right) = \sum_{ij} \sqrt{ { \left( {D}_{h} X \right) }_{ij}^{2} + { \left( {D}_{v} X \right) ...
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7 votes
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Is the Bilateral Filter a Solution of Some Variational Method?

Yes indeed. You may have a look on work (Paper) by Michael Elad which is called On the Origin of the Bilateral Filter and Ways to Improve It or Analysis of the Bilateral Filter. You may have a look on ...
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7 votes
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Intuitive Meaning of Regularization in Imaging Inverse Problems

One general form of Inverse Problem in Imaging which assumes Linear Operator is given by: $$ \arg \min_{x} \frac{1}{2} {\left\| A x - b \right\|}_{2}^{2} + \lambda R \left( x \right) $$ Where $ R \...
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7 votes
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Total Variation of a Signal - Is It Proportional to Signal Energy?

No it is not. Total Variation is like the amount of changes in the signal. Though changes require energy it doesn't mean they are proportional. For instance, imagine that during a Window we see a ...
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7 votes

Using Total Variation Denoising to Clean Accelerometer Data

If your data model is Piece Wise Smooth Signal then you should use Total Variation as regularization. Let's try comparing 2 methods for Denoising with 2 different regularization (Both works on the ...
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7 votes
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How to Solve Non Blind Image Deblurring with Total Variation Prior Using ADMM?

Formulation of the Problem I am solving the problem under the following assumptions: The blurring operator is Linear and Spatially Invariant (Hence applied by convolution). The blurring operator is ...
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7 votes
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How to Solve Image Denoising with Total Variation Prior Using ADMM?

Formulation of the Denoising Problem The problem is given by: $$ \arg \min_{x} \frac{1}{2} {\left\| x - y \right\|}_{2}^{2} + \lambda \operatorname{TV} \left( x \right) = \arg \min_{x} \frac{1}{2} {\...
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6 votes
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Solve Efficiently the 1D Total Variation Regularized Least Squares Problem (Denoising / Deblurring)

I will answer Total Variation Regularization: $$ \arg \min_{\boldsymbol{x}} f \left( \boldsymbol{x} \right) = \arg \min_{\boldsymbol{x}} \frac{1}{2} {\left\| A \boldsymbol{x} - \boldsymbol{y} \right\|}...
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6 votes
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What Does the Total Variation Norm Mean in the Context of Image Processing

The Total Variation of an image $ I $ can be calculated in one of 3 methods (See The Meaning of the Terms Isotropic and Anisotropic in the Total Variation Framework): Anisotropic TV - $ \operatorname{...
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5 votes

How Can I Use MATLAB to Solve a Total Variation Denoising / Deblurring Problem?

I will solve this for 1D but it could easily generalize into 2D. The nice thing about the TV Norm that it can be re formulated by the $ {L}_{1} $ norm of the Derivative Operator: $$ \operatorname{TV} \...
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5 votes
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How to Solve Blind Image Deblurring with Total Variation (TV) Prior Using ADMM?

First, let's analyze the problem by formulating it. The model is given by: $$ \boldsymbol{y} = H \boldsymbol{x} + \boldsymbol{n} $$ Where $ \boldsymbol{y} $ is the given image, $ H $ is an unknown ...
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5 votes
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How to Solve an Image Deblurring Problem by Variational Methods Using ADMM?

Remark: This is adapted from How to Solve Image Deblurring with Total Variation Prior Using ADMM? Formulation of the Problem I am solving the problem under the following assumptions: The blurring ...
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4 votes

How to solve ADMM for TV Minimization Problem For Different Sizes $A$ and $x$ in $Ax=b$

The Error in the Model The problem is in the dimensions of the Linear Operator $ A $ in your model compared to the Data Matrix $ X $. The number of columns of the matrix $ A $ must match the number ...
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4 votes
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What Is the Difference Between MRF and Total Variation in Noise Removal?

These are two different concepts that you talk about. First, MRF gives you a framework to do discrete optimization of problems, which respect the Markovian property, that is a pixel is conditioned ...
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3 votes

Deconvolution of a 1D Time Domain Wave Signal Convolved with Series of Rect Signals

Actually I have a similar problem like yours. But in mine, the objective function is not like rectangular pulse but just spikes as shown below. I work in ultrasoinc testing field. So, this example is ...
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3 votes
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Gradient of Total Variation (TV) Norm in Total Variation Denoising

I am by no means an expert on total variation, however I think you should check out this Wikipedia page. It doesn't directly answer your question, but I believe the lemma below illustrates the ...
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2 votes

Gradient of Total Variation (TV) Norm in Total Variation Denoising

To obtain the Gradient of the TV norm, you should refer to the calculus of variations. By examining the TV minimization with Euler-Lagrange equation, e.g,, Eq. (2.5a) in [1], you would see the ...
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  • 21
2 votes

How to Solve Image Denoising with Total Variation Prior Using ADMM?

I have done a bit of this myself and you'd need to adapt. There is a Douglas Rachford self implemented and a primal dual approach here implemented in Recovery of Fusion Frame Structured Signal via ...
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1 vote
Accepted

Why Does the Rudin Osher Fatemi (ROF) Method Use Variational Methods for Image Denoising When Denoising Problems Are Not Boundary Value Problems?

This turned out to be easier than I thought. Image processing is a boundary value problem, and the boundary is the set of pixels along the edge of the image. The common notation for this boundary in ...
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