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14

L1 norm minimization (compressed sensing) can do a relative better job than conventional Fourier denoising in terms of preserving edges. The procedure is to minimize an objective function $$|x-y|^2 + b|f(y)|$$ where $x$ is the noisy signal, $y$ is the denoised signal, $b$ is the regularziation parameter, and $|f(y)|$ is some L1 norm penalty. ...

7

Sounds like you want to denoise and preserve edges. Have you considered nonlocal means? There's some GPL'd C++ code along with a brief writeup of the algorithm by the original authors here: http://www.ipol.im/pub/algo/bcm_non_local_means_denoising/ One caveat, nonlocal means is very slow and the output can be sensitive to the implementation you have. You ...

7

You can consider anisotropic diffusion. There are many methods based on this technique. Generally spoken, it is for images. It is an adaptive denoising method which aims to smooth non-edge parts of an image, and preserve edges. Also, for Total variation minimization, you can use this tutorial. Authors provide MATLAB code also. They recognize the problem as ...

6

Chaohuang has a good answer, but I will also add that one other method that you can use would be via the Haar Wavelet Transform, followed by wavelet co-efficient shrinkage, and an Inverse Haar Transform back to the time-domain. The Haar wavelet transform decomposes your signal into co-efficients of square and difference functions, albeit at different ...

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 ...

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 ...

4

The problem with $\left|f\right|$ is that since is not analytic the standard definition of complex derivative does not apply. A solution is to use Wirtinger derivatives: http://en.wikipedia.org/wiki/Wirtinger_derivatives A detailed account of Wirtinger calculus for signal processing problems is http://arxiv.org/abs/0906.4835 Another (probably simpler) ...

4

A simple method that often works is to apply a median filter.

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 ...

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

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_{\... 2 In the Total Variation framework we define 2 flavors: $$\text{Isotropic TV} \; {TV}_{ {L}_{2} } \left( X \right) = \sum_{ij} \sqrt{ { \left( {D}_{h} X \right) }_{ij}^{2} + { \left( {D}_{v} X \right) }_{ij}^{2} }$$ $$\text{Anisotropic TV} \; {TV}_{ {L}_{1} } \left( X \right) = \sum_{ij} \sqrt{ { \left( {D}_{h} X \right) }_{ij}^{2} } + \sqrt{{ \left( {D}_{... 2 Yes indeed. You mayt have a look on work by Michael Elad which is called On the Origin of the Bilateral Filter and Ways to Improve It or Analysis of the Bilateral Filter. You may have a look on the way he derives the Bilateral Filter from a Least Squares problem with weighted regularization on the Gradient of the image in the form of:$$ \arg \min_{x} \... 1 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) $is the regularization function. One way, which is pretty intuitive in my opinion, is to treat the above as a solution to Maximum a Posteriori ... 1 No it is not. Total Variation is like the amount of changes in the signal. Though changes require energy it doesn't mean they are proportional. For instance, imagine that during a Window we see a constant signal of high value. Clearly this high energy signal (Unless energy for you is the Variance, usually it is the 2nd moment) yet its Total Variation is ... 1 I will solve this for 1D but it could easily generalize into 2D. The nice thing about the TV Norm that it can be re formulated by the$ {L}_{1} $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 ... 1 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 , you would see the answer.  Nonlinear total variation based noise removal algorithms, 1992. 1 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): $$\text{Toal Variation:} \quad \arg \min_{z} \frac{1}{2} {\left\| z - b \right\|}_{2}^{2} + \lambda {\left\| D z \right\|}_{1}$$$\$ ...

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