6
votes
Accepted
The Meaning of the Terms Isotropic and Anisotropic in the Total Variation Framework
In the Total Variation framework we define 2 flavors:
$$ \text{Isotropic TV} \; {TV}_{ {L}_{2} } \left( X \right) = \sum_{ij} \sqrt{ { \left( {D}_{h} X \right) }_{ij}^{2} + { \left( {D}_{v} X \right) ...
- 20.5k
5
votes
Accepted
Deconvolution of a 1D Time Domain Wave Signal Convolved with Series of Rect Signals
Solving a deconvolution isn't easy even in simulated environment not to mention in practice.
The main trick to solve it is using the proper model / prior for the problem and very good measurements (...
- 20.5k
4
votes
Accepted
How to Solve Non Blind Image Deblurring with Total Variation Prior Using ADMM?
Formulation of the Problem
I am solving the problem under the following assumptions:
The blurring operator is Linear and Spatially Invariant (Hence applied by convolution).
The blurring operator is ...
- 20.5k
3
votes
Solve Efficiently the 1D Total Variation Regularized Least Squares Problem (Denoising / Deblurring)
I will answer Total Variation Regularization:
$$ \arg \min_{\boldsymbol{x}} f \left( \boldsymbol{x} \right) = \arg \min_{\boldsymbol{x}} \frac{1}{2} {\left\| A \boldsymbol{x} - \boldsymbol{y} \right\|}...
- 20.5k
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 ...
- 71
3
votes
Accepted
How to Solve Image Denoising with Total Variation Prior Using ADMM?
Formulation of the Denoising Problem
The problem is given by:
$$ \arg \min_{x} \frac{1}{2} {\left\| x - y \right\|}_{2}^{2} + \lambda \operatorname{TV} \left( x \right) = \arg \min_{x} \frac{1}{2} {\...
- 20.5k
3
votes
Accepted
Is the Bilateral Filter a Solution of Some Variational Method?
Yes indeed.
You may have a look on work (Paper) by Michael Elad which is called On the Origin of the Bilateral Filter and Ways to Improve It or Analysis of the Bilateral Filter.
You may have a look on ...
- 20.5k
3
votes
Accepted
Intuitive Meaning of Regularization in Imaging Inverse Problems
One general form of Inverse Problem in Imaging which assumes Linear Operator is given by:
$$ \arg \min_{x} \frac{1}{2} {\left\| A x - b \right\|}_{2}^{2} + \lambda R \left( x \right) $$
Where $ R \...
- 20.5k
3
votes
Accepted
Total Variation of a Signal - Is It Proportional to Signal Energy?
No it is not.
Total Variation is like the amount of changes in the signal.
Though changes require energy it doesn't mean they are proportional.
For instance, imagine that during a Window we see a ...
- 20.5k
3
votes
Using Total Variation Denoising to Clean Accelerometer Data
If your data model is Piece Wise Smooth Signal then you should use Total Variation as regularization.
Let's try comparing 2 methods for Denoising with 2 different regularization (Both works on the ...
- 20.5k
2
votes
Accepted
What Does the Total Variation Norm Mean in the Context of Image Processing
The Total Variation of an image $ I $ can be calculated in one of 3 methods (See The Meaning of the Terms Isotropic and Anisotropic in the Total Variation Framework):
Anisotropic TV - $ \operatorname{...
- 20.5k
2
votes
Gradient of Total Variation (TV) Norm in Total Variation Denoising
To obtain the Gradient of the TV norm, you should refer to the calculus of variations. By examining the TV minimization with Euler-Lagrange equation, e.g,, Eq. (2.5a) in [1], you would see the ...
- 21
2
votes
How to Solve Image Denoising with Total Variation Prior Using ADMM?
I have done a bit of this myself and you'd need to adapt.
There is a Douglas Rachford self implemented and a primal dual approach here implemented in Recovery of Fusion Frame Structured Signal via ...
- 121
1
vote
Accepted
Matrix-vector multiplication representation of Total Variation function
I just found out on my own. Assuming that $X \in \mathbb{R}^{m \times n}$, we define the difference matrices $\Delta_I X = [ X_{i,j} - X_{i-1,j} ]$ and $\Delta_J X = [X_{i,j} - X_{i,j-1}]$, where ...
- 95
1
vote
How to Solve Blind Image Deblurring with Total Variation (TV) Prior Using ADMM?
First, let's analyze the problem by formulating it. The model is given by:
$$ \boldsymbol{y} = H \boldsymbol{x} + \boldsymbol{n} $$
Where $ \boldsymbol{y} $ is the given image, $ H $ is an unknown ...
- 20.5k
1
vote
Accepted
How to Solve an Image Deblurring Problem by Variational Methods Using ADMM?
Remark: This is adapted from How to Solve Image Deblurring with Total Variation Prior Using ADMM?
Formulation of the Problem
I am solving the problem under the following assumptions:
The blurring ...
- 20.5k
1
vote
Accepted
Why Does the Rudin Osher Fatemi (ROF) Method Use Variational Methods for Image Denoising When Denoising Problems Are Not Boundary Value Problems?
This turned out to be easier than I thought. Image processing is a boundary value problem, and the boundary is the set of pixels along the edge of the image. The common notation for this boundary in ...
- 257
1
vote
How Can I Use MATLAB to Solve a Total Variation Denoising / Deblurring Problem?
I will solve this for 1D but it could easily generalize into 2D.
The nice thing about the TV Norm that it can be re formulated by the $ {L}_{1} $ norm of the Derivative Operator:
$$ \operatorname{TV} \...
- 20.5k
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