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4 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_{\...
David's user avatar
  • 2,871
4 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 ...
Royi's user avatar
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3 votes
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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} \...
Royi's user avatar
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3 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: ...
kippertoffee's user avatar
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 ...
Peter K.'s user avatar
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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 ...
Laurent Duval's user avatar
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 ...
Cedron Dawg's user avatar
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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?

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 ...
Tom Wenseleers's user avatar
2 votes

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 ...
Jazzmaniac's user avatar
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2 votes

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 ...
Knut Inge's user avatar
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2 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: ...
Royi's user avatar
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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:...
Laurent Duval's user avatar
2 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 {\...
Royi's user avatar
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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 ...
David's user avatar
  • 2,871
1 vote

Is there a formal definition of what it means for a signal to be sparse?

This page gives the following definition: A signal is said to be sparse if it can be represented in a basis or frame (e.g Fourier, Wavelets, Curvelets, etc.) in which the curve obtained by plotting ...
Peter K.'s user avatar
  • 25.9k
1 vote

Improving Main Lobe Width of FFT

Do you know the exact duration in samples of the sleep time? If so, then you can take the un-windowed fft of the 2nd block of N samples, modify the phase of each bin to “undo” the effect of the sleep ...
Bob's user avatar
  • 588
1 vote

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 $...
Royi's user avatar
  • 19.8k
1 vote

Denoising by DCT and hard thresholding

DCTs are very useful at energy compaction, so simply put after a DCT of an image is resolved to a weighted some of some basis functions. After a DCT, the resulting matrix will contains multipliers for ...
hit.at.ro's user avatar
  • 111

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