8

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 (High SNR). So basically, for deconvolution we're after: $$ \hat{\boldsymbol{x}} = \arg \min_{\boldsymbol{x}} \frac{1}{2} {\left\| H \boldsymbol{x} - \boldsymbol{y}...


7

In the Total Variation framework we define 2 flavors: $$ \text{Isotropic TV} \; {TV}_{ {L}_{2} } \left( X \right) = \sum_{ij} \sqrt{ { \left( {D}_{h} X \right) }_{ij}^{2} + { \left( {D}_{v} X \right) }_{ij}^{2} } $$ $$ \text{Anisotropic TV} \; {TV}_{ {L}_{1} } \left( X \right) = \sum_{ij} \sqrt{ { \left( {D}_{h} X \right) }_{ij}^{2} } + \sqrt{{ \left( {D}_{...


6

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 known. There is a measurement noise. So the model is: $$ \boldsymbol{y} = H \boldsymbol{x} + \boldsymbol{n} $$ Where $ H $ is the matrix form of the blurring ...


6

Formulation of the Denoising Problem The problem is given by: $$ \arg \min_{x} \frac{1}{2} {\left\| x - y \right\|}_{2}^{2} + \lambda \operatorname{TV} \left( x \right) = \arg \min_{x} \frac{1}{2} {\left\| x - y \right\|}_{2}^{2} + \lambda {\left\| D x \right\|}_{1} $$ Where $ D $ is the column stacked derivative operator. In the above I used the Anisotropic ...


6

The Total Variation of an image $I$ can be computed using two formulas: $TV(I) = \sum_{x} \| \nabla I (x) \|_1 $ (anisotropic TV); $TV(I) = \sum_{x} \sqrt{ \| \nabla I (x) \|_2^2 }$ (isotropic TV). In practice, both formulas yield almost the same result. Using matlab, what you propose implements the first formula, while the second one can be obtained by ...


6

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 the way he derives the Bilateral Filter from a Least Squares problem with weighted regularization on the Gradient of the image in the form of: $$ \arg \min_{x} ...


6

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 \left( x \right) $ is the regularization function. One way, which is pretty intuitive in my opinion, is to treat the above as a solution to Maximum a Posteriori ...


6

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 constant signal of high value. Clearly this high energy signal (Unless energy for you is the Variance, usually it is the 2nd moment) yet its Total Variation is zero....


5

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{TV} \left( I \right) = \sum_{x} \| \nabla I (x) \|_1 $. Isotropic TV - $ \operatorname{TV} \left( I \right) = \sum_{x} \| \nabla I (x) \|_2 $. Isotropic ...


5

I will answer Total Variation Regularization: $$ \arg \min_{\boldsymbol{x}} f \left( \boldsymbol{x} \right) = \arg \min_{\boldsymbol{x}} \frac{1}{2} {\left\| A \boldsymbol{x} - \boldsymbol{y} \right\|}_{2}^{2} + \lambda {\left\| D \boldsymbol{x} \right\|}_{1} $$ Now, we can use the method of Majorization Minimization to solve this: $$ \forall t \in \mathbb{R}...


4

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 linear shift invariant blur operator, $ \boldsymbol{x} $ is the image we're after and $ \boldsymbol{n} $ is the added noise. We'll assume it is a White Noise (...


4

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 operator is Linear and Spatially Invariant (Hence applied by convolution). The blurring operator is known. There is a measurement noise. So the model is: $$ \...


4

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: $$ \operatorname{TV} \left( x \right) = \sum_{i = 1}^{N - 1} \left| {x}_{i + 1} - {x}_{i} \right| = {\left\| D x \right\|}_{1} $$ Where $ D $ is the matrix form of the Derivative ...


4

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 Derivative of the Signal): $$ \text{Toal Variation:} \quad \arg \min_{z} \frac{1}{2} {\left\| z - b \right\|}_{2}^{2} + \lambda {\left\| D z \right\|}_{1} $$ $$ ...


4

Apart from Total Variation Denoising you could try a first much simpler approach: a median-filter. You just move a window along your data and replace the current input value by the median of all data in the window. You just have to optimize the window length (by experimenting). By the way, the equations you copied into your question are for 2-dimensional ...


4

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 only on the neighboring ones (roughly stated). Typical applications include binary or multi-class labeling problems. Total variation on the other hand, is ...


3

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 of pixels in each column of $ X $ (Each image). While in your case it matches the number of images. Matrix Form of 2D Linear Operator Let's try to build the ...


3

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 relationship between total variation and divergence. There, it gives a lemma that follows from the Gauss-Ostrogradsky theorem and provides a proof for it, $\int_{\...


3

Well, unless it is a more programming question (how to translate from MATLAB script to C code), you might find interesting the following implementation: click, proposed in this article: A direct algorithm for 1D total variation denoising. Good luck!


2

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 answer. [1] Nonlinear total variation based noise removal algorithms, 1992.


2

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 Compressed Sensing. Note that Clarice Poon (Bath University) had some nice tutorials on it. Another source is the Numerical Tours from Gabriel Peyre. See Denoising ...


1

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 the Vese Guyader book is $\partial(\Omega)$, so that is not the most obvious notation for an image. Of course $\partial$ is often the boundary operator in ...


1

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 kind of like an ultrasoinc signal. But, first I tried with the code from Royi using the ADMM solver from the: answer. For me, I find his solution was painfully ...


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