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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 Fidelity Term This is basically measurements term with the model of AWGN with IID noise. The Regularization Term This is a sparse promoting model by using the Laplace ...


6

Given a set of values $ {\left\{ {s}_{i} \right\}}_{i = 1}^{N} $, we're basically after: $$ \arg \min_{x} \sum_{i = 1}^{N} \left| {s}_{i} - x \right| $$ One should notice that $ \frac{\mathrm{d} \left | x \right | }{\mathrm{d} x} = \operatorname{sign} \left( x \right) $ (Being more rigorous would say it is a Sub Gradient of the non smooth $ {L}_{1} $ Norm ...


6

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 known. There is a measurement noise. So the model is: $$ \boldsymbol{y} = H \boldsymbol{x} + \boldsymbol{n} $$ Where $ H $ is the matrix form of the blurring ...


5

The way to build the matrix is playing with indices of the signal data and the convolution kernel. For example: function [ mK ] = CreateConvMtx1D( vK, numElements, convShape ) % ----------------------------------------------------------------------------------------------- % % [ mK ] = CreateConvMtx1D( vK, numElements, convShape ) % Generates a Convolution ...


5

Formulation of the Denoising Problem The problem is given by: $$ \arg \min_{x} \frac{1}{2} {\left\| x - y \right\|}_{2}^{2} + \lambda \operatorname{TV} \left( x \right) = \arg \min_{x} \frac{1}{2} {\left\| x - y \right\|}_{2}^{2} + \lambda {\left\| D x \right\|}_{1} $$ Where $ D $ is the column stacked derivative operator. In the above I used the Anisotropic ...


5

You can employ Compressed Sensing / Sparse Representation for Super Resolution in Frequency Domain. One way to do so is solving the problem: $$ \arg \min_{\boldsymbol{x}} \frac{1}{2} {\left\| F \boldsymbol{x} - \boldsymbol{y} \right\|}_{2}^{2} + \lambda {\left\| \boldsymbol{x} \right\|}_{1} $$ Where the $ {L}_{1} $ norm is sparsity inducing regularization ...


5

Usually Tikhonov Regularization is applied in the following form: $$ \arg \min_{\boldsymbol{x}} \frac{1}{2} {\left\| A \boldsymbol{x} - \boldsymbol{y} \right\|}_{2}^{2} + \frac{\lambda}{2} {\left\| \boldsymbol{x} \right\|}_{2}^{2} $$ This formulation can be seen as: MAP Estimator with the prior of $ \boldsymbol{x} \sim N \left( 0, {\sigma}_{x}^{2} \right) $....


4

The function is given by: $$ f \left( \boldsymbol{x} \right) = \frac{1}{2} {\left\| \boldsymbol{x} - \boldsymbol{g} \right\|}_{2}^{2} - { \boldsymbol{y} }^{T} \left( \boldsymbol{r} - \boldsymbol{w} \odot \left( D \boldsymbol{x} \right) \right) + \frac{\rho}{2} {\left\| \boldsymbol{r} - \boldsymbol{w} \odot \left( D \boldsymbol{x} \right) \right\|}_{2}^{2} $$ ...


4

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 linear shift invariant blur operator, $ \boldsymbol{x} $ is the image we're after and $ \boldsymbol{n} $ is the added noise. We'll assume it is a White Noise (...


4

Could it be that you are indeed looking for the closest orthogonal matrix $Y$? Then, there is a solution which involves computing the square root of $ D^TD$ . If $E=(D^TD)^{1/2}$ were invertible, the solution would be its inverse. Yet, it is not invertible here. Then, there is a trick. If I remember well, you have to perform an eigenvalue/eigenvector ...


3

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 operator is Linear and Spatially Invariant (Hence applied by convolution). The blurring operator is known. There is a measurement noise. So the model is: $$ \...


3

I will answer Total Variation Regularization: $$ \arg \min_{\boldsymbol{x}} f \left( \boldsymbol{x} \right) = \arg \min_{\boldsymbol{x}} \frac{1}{2} {\left\| A \boldsymbol{x} - \boldsymbol{y} \right\|}_{2}^{2} + \lambda {\left\| D \boldsymbol{x} \right\|}_{1} $$ Now, we can use the method of Majorization Minimization to solve this: $$ \forall t \in \mathbb{R}...


3

Assuming we know how to solve: $$ \arg \min_{\boldsymbol{x}} \frac{1}{2} {\left\| C \boldsymbol{x} - \boldsymbol{b} \right\|}_{2}^{2} + {\left\| E \boldsymbol{x} \right\|}_{1} $$ For any matrix $ E $ one could see that: $$ \boldsymbol{w} \circ D \boldsymbol{x} = \operatorname{Diag} \left( \boldsymbol{w} \right) D \boldsymbol{x} = E \boldsymbol{x} $$ Where $ \...


3

The solution is very similar to what I had in Solve Efficiently the 1D Total Variation Regularized Least Squares Problem (Denoising / Deblurring). The only difference in the MM is setting $ D = I $. This means there is no reason to use the Matrix Inversion Lemma. Hence one need to prevent zeros in the values of $ {\Lambda}_{k} $. The comparison yields: In ...


2

If D is not singular, i.e. $\operatorname*{det}(D) \ne 0$ than simply $C = D^{-1}$ will do the trick, since $D \cdot C = I$, which is obviously unitary. If D is singular, i.e. $\operatorname*{det}(D) = 0$ than the product is also singular, i.e. $\operatorname*{det}(D \cdot C)=0$ which means the product cannot be unitary for all possible matrices $C$.


2

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 Compressed Sensing. Note that Clarice Poon (Bath University) had some nice tutorials on it. Another source is the Numerical Tours from Gabriel Peyre. See Denoising ...


1

As you already now a 1st order CIC filter is identical to a moving average filter. Lets consider the decimation factor to be 2 and having the following time series: input = 10 11 12 13 14 15 Lets have a look at the convolution with the fir filter h=[1 1] 0 10 11 12 13 14 15 1 1 0 0 0 0 0 The first output sample will be 10 0 10 11 12 13 14 15 0 1 1 0 ...


1

That's a pretty tortured way of defining the Fourier Transform. Are the optimizations $\max_{\phi \in [0 , 1)}$ and $\text{argmax}_{\phi \in [0 , 1)}$ included in the Fourier transform? Yes. The optimization steps are completely unnecessary. Instead of "finding the phase that maximizes the integral", we can just simply calculate that phase ...


1

A convolution matrix is really just a diagonal band-structure matrix, where every row is all zeros but for the elements around the diagonal, which are identical (but shifted) for every row: the elements of the kernel.


1

You can't solve the problem in the way you intend to, because $x=H\theta$ is not a constraint in the usual sense. Note that $\theta$ is unknown and must be chosen such that $||y-x||$ is minimized. A constraint is given as $Ax=b$ with $A$ and $b$ fixed and known. If $\theta$ were known then there wouldn't be any minimization problem; the solution would just ...


1

You specify that $T$ is known and invertible, so you know $X_1, X_2$, and then it's really trivial: $y[3]=[X_1,X_2]*[h_1, h_2]$, and $y[9]=[X_7, X_8] *[h_1,h_2]$; write that down as matrix system $$\begin{pmatrix}y[3]\\y[9]\end{pmatrix}=\begin{pmatrix}X_1&X_2\\X_7&X_8\end{pmatrix}\begin{pmatrix}h_2\\h_1\end{pmatrix}$$ and solve it.


1

Though this is a quite old question, my two cents might help others in the future. The UKF is a very nice algorithm, but you need to take care before using it. My concerns are the following: You want to estimate matrices with constraints, but the UKF is for Euclidean manifolds. The problem here is that when you impose a constraint, you are not longer on a ...


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