8 votes
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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 ...
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8 votes
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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 \...
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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} \...
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6 votes
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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: ...
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6 votes
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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 {\...
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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 ...
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4 votes
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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 ...
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4 votes
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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) ...
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4 votes
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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 ...
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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 ...
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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 ...
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  • 101
2 votes
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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_{\...
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  • 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:...
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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 ...
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  • 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: ...
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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 ...
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  • 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 ...
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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 ...
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  • 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 ...
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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 ...
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  • 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 ...
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