8
votes
Accepted
Adding Variance \ Weights Information When Solving a Basis Pursuit Denoising Problem (BPDN)
Your formulation:
$$ \arg \min_{\boldsymbol{x}} \frac{1}{2} {\left\| A \boldsymbol{x} - \boldsymbol{y} \right\|}_{2}^{2} + \lambda {\left\| \boldsymbol{x} \right\|}_{1} $$
Has 2 elements:
The ...
- 41.4k
8
votes
Accepted
Why Sparse Priors Like Total Variation Opts to Concentrate Derivatives at a Small Number of Pixels?
I will divide my answer into 3 sections.
The Distribution of the Derivative of Images
Take a real world image, any image.
Apply the derivative operator on it (Namely apply the kernel $ \left[ 1, -1 \...
- 41.4k
7
votes
Accepted
Convex Optimization with $ {L}_{1, 2} $ Regularization Term
The problem is given by:
$$\begin{equation}
\arg \min_{X} \frac{1}{2} \sum_{k} {\left\| {T}_{k} {X}_{:, k} - {Y}_{:, k} \right\|}_{2}^{2} + \lambda {\left\| G X \right\|}_{2, 1} \\ = \arg \min_{X} \...
- 41.4k
6
votes
Accepted
Solving a Weighted Basis Pursuit Denoising Problem (BPDN) with MATLAB / CVX
A MATLAB code which implements the problem as defined and solve it using CVX is given by:
...
- 41.4k
6
votes
Accepted
Solving LASSO (Basis Pursuit Denoising Form) with LARS
There are 2 forms of the Basis Pursuit problem:
$$\begin{align*}
\text{The $ \lambda $ Form:} & \quad && \arg \min_{x} &&\frac{1}{2} {\left\| A x - b \right\|}_{2}^{2} + \lambda {\...
- 41.4k
6
votes
Is There a Sparse Representation for Noise?
Let's think about it in a different way - Generate Noise from a Dictionary.
Let's create a Dictionary $ A \in \mathbb{R}^{m \times n} $ where each of its rows is normalized (Has Euclidean Norm of $ 1 ...
- 41.4k
4
votes
Accepted
Constrained LASSO Problem - $ {L}_{1} $ Regularized Least Squares with Linear Equality Constraints
Indeed you can not solve the problem ignoring the equality constraints and then project the solution onto the set of solution for the constraint. It is easy to build real world example which shows ...
- 41.4k
4
votes
Accepted
Implementation of Block Orthogonal Matching Pursuit (BOMP) Algorithm - Fix Given Code
Answer taken from Implementation of Block Orthogonal Matching Pursuit (BOMP) Algorithm.
The Block Orthogonal Matching Pursuit (BOMP) Algorithm is basically the Orthogonal Matching Pursuit (OMP) ...
- 41.4k
4
votes
Accepted
Implementation of Block Orthogonal Matching Pursuit (BOMP) Algorithm
The Block Orthogonal Matching Pursuit (BOMP) Algorithm is basically the Orthogonal Matching Pursuit (OMP) Algorithm with single major difference - Instead of selecting single index which maximizes the ...
- 41.4k
4
votes
Denoising by DCT and hard thresholding
Have a look on the following optimization problem:
$$ \arg \min_{x} \frac{1}{2} {\left\| A x - y \right\|} + \lambda {\left\| x \right\|}_{0} $$
Where $ {\left\| \cdot \right\|}_{0} $ is counting ...
- 41.4k
2
votes
how do you know if your matrix is sparse after sparsifying transform?
If you can't find anything in the literature about a threshold, you can develop your own with the following procedure:
Generate $$N=\frac{ln(1-M)}{R}$$ random matrices $\boldsymbol{A}$ that you are ...
- 101
2
votes
Accepted
Differences Between Two $ {L}_{1} $ Norm Minimization Schemes
The first equation you have is often called the Quadratic Problem, which through the use of Duality can be shown to be equivalent to the Basis Pursuit De-Noising (BPDN) given as:
$$ \arg \min_{\...
- 2,751
2
votes
Is it common to impose the sparsity on the Fourier coefficient itself?
This question is typically the subject of a paper like Robust Uncertainty Principles: Exact Signal Reconstruction From Highly Incomplete Frequency Information, Emmanuel J. Candès, Justin Romberg, 2006:...
- 30.3k
2
votes
Compressive Sensing and Sparsity
It depends on the application and on what basis you decide to look at. If there are only a few targets and little to no clutter, then a radar image can be considered sparse in the image domain i.e. it ...
- 2,751
2
votes
Ifft through Matrix multiplication
You can alternatively create a DFT matrix in matlab using this code:
exp(-1j*2*pi* ((0:N-1)/N).' * (0:N-1))
And the IDFT matrix thus:
...
- 383
2
votes
Accepted
How to make the impulse response sparse? How does one know that the channel is sparse?
How you parameterize your sparsity will depend on your application. The authors of that paper, in a paragraph on page 231 say:
which is why they clump the coefficients together in $P$ blocks of ...
- 23k
2
votes
Accepted
When can the impulse response become zero?
The notion of sparsity entails that an object, living for instance in an $n$-dimensional space, can be described (in the suitable basis/frame) by a number $k$ of meaningful components (each above a ...
- 30.3k
2
votes
Estimate peak width from a vector that is a superposition of unknown number of identical Gaussian peaks with different heights?
My first comment would be why the heck are you using R if you are concerned with processing speed, or are you just prototyping algorithms?
Anyway, Without getting into how I derived it, here is a ...
- 6,903
2
votes
Estimate peak width from a vector that is a superposition of unknown number of identical Gaussian peaks with different heights?
Ha just figured out a faster and better method just using BIC-optimized selection of optimal peak width, using a banded covariate matrix with shifted Gaussian peak shapes of given width & using ...
- 151
1
vote
custom raw compression
Raw files are (ideally) the raw readout of a sensor. Suitable for research, or if you want to eek out all possible information from a sensor using fancy offline processing. Now or in 10 years. In some ...
- 2,294
1
vote
Is There a Sparse Representation for Noise?
The question of the existence of a sparse basis of noise is closely related to the question of the effective dimensionality of the noise subspace.
First, it is important to realise that noise is a ...
- 4,324
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