12 votes
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Accessing Maximum Value from a Singular Value Decomposed Matrix

The SVD Decompose the image into the (One way to look at it) many matrices. For instance, given an Image $ I $ its SVD is given by: $$ I = U S {V}^{T} = \sum_{i=1}^{\textrm{rank}(I)} {s}_{i} {u}_{i} {...
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10 votes
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How to Check Separability of 2D Filter / Signal / Matrix

Nilesh Padhi, Welcome to the DSP Community. The classic definition of separable means the data (2D) given by $ X \in \mathbb{R}^{m \times n} $ can be written as: $$ X = \sigma u {v}^{T} $$ Where $ \...
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8 votes
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Group delay of $H(\omega)= 1- re^{j \theta}e^{ - j \omega} $

The book's formula is right. Let $$H(w) = 1 - r e^{j(\theta - w)} = [1-r \cos(\theta - w)] + j [-r \sin(\theta - w)]$$ Since the group delay $\tau$ is the negative of the derivative of the phase of $...
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7 votes

Generate the Matrix Form of 2D Convolution Kernel

I created a function to create a Matrix for Image Filtering (Similar ideas to MATLAB's imfilter()): ...
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7 votes
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Sequential Form of the Least Squares Estimator for Linear Least Squares Model

Slope from all samples obtained To summarize the question's problem, you want to calculate the slope based on all samples obtained thus far, and as new samples are obtained, update the slope without ...
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7 votes
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Resources on Solving Convex Optimization Problems in the Compressed Sensing Field

There are few options: Stephen Boyd, Lieven Vandenberghe - Convex Optimization. This is the classic in this field. Very well written book. Also have a look on other papers of Boyd on similar subjects ...
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7 votes
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Super Resolution in Frequency Domain Using Compressed Sensing

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 \...
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7 votes
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Tikhonov Regularization for Complex Matrices

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\| \...
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6 votes
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Least Angle Regression (LARS) without Matrix Inversion

If you want to solve for single value of $ \lambda $ in the model: $$ \arg \min_{x} \frac{1}{2} {\left\| A x - b \right\|}_{2}^{2} + \lambda {\left\| x \right\|}_{1} $$ Then you can use Coordinate ...
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6 votes

Looking for integral-free DSP textbooks

I would like to recommend a great book to you (see below), but before that I would like to point out that there is some confusion in your question. If you see a convolution integral in a textbook, ...
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6 votes
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Ideas on Matrix Factorization / Transformations for $ {L}_{1} $ Minimization

Since $ \epsilon $ is a parameter you need to set, why not trade it with another parameter you need to set to create an easily solvable problem (Relaxation of the Problem)? You can transform the ...
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6 votes
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Proof of complex conjugate symmetry property of DFT

Hint: According to Euler's formula we have $$e^{-j2\pi k}=\cos(2\pi k)-j\sin(2\pi k)=\ldots$$
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Group delay of $H(\omega)= 1- re^{j \theta}e^{ - j \omega} $

This is a slightly tedious but nevertheless straightforward exercise in computing the derivative of a function: $$\begin{align}\tau(\omega)&=-\frac{d\phi(\omega)}{d\omega}=-\frac{d}{d\omega}\...
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6 votes

Sequential Form of the Least Squares Estimator for Linear Least Squares Model

There are really great answers. I will try to give the Sequential Least Squares approach which generalizes to any Linear Model. Sequential Least Squares Model We're after solving the Linear Least ...
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6 votes
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The Gradient / Derivative of Least Squares of 2D Image Convolution

The easiest approach would be writing each case using Matrix Form of the convolution. In this answer we assume the discrete convolution is applied only on valid support (Matching MATLAB's ...
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6 votes
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Orthogonal Basis for a 2D Signals (Compressive Sensing)

If you have a function $ f \left[ m, n \right] \in \mathbb{R}^{M \times N} $ then the DFT of those functions is an orthogonal basis of functions in $ \mathbb{C}^{M \times N} $. So if we have: $$ F \...
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6 votes
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Circular Convolution as Cyclic Shift Operator

You may solve it by 3 steps: Show yourself that a Cyclic Convolution with a vector $ \boldsymbol{e}_{i}^{N} $ is a Cyclic Shift Operator $ {T}_{i - 1} \left( \cdot \right) $. Where $ \boldsymbol{e}_{...
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6 votes
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Generate the Matrix Form of 1D Convolution Kernel

The way to build the matrix is playing with indices of the signal data and the convolution kernel. For example: ...
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6 votes
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Decomposing Sobel Filter

The decomposition of a separable filter is not unique, since $u v' = (u a) (v a^{-1})'$ The solution you are expecting is found by using a $-\sqrt{2}$ factor, namely ...
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5 votes
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Deriving the Matrix Inversion Lemma for RLS Equations vs the Woodbury Derivation

It is not clear what are you asking but I will try answer both things. Deriving the Matrix Inversion Lemma The Matrix Inversion Lemma goes as: $$ {\left( A + U C V \right)}^{-1} = {A}^{-1} - {A}^{-...
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5 votes
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Weighted Nuclear Norm Minimization for Image Denoising

Most of the Denoisers in Image Processing make a simple assumption - The data has small number of freedom degrees while noise has high number. Hence if we try to represent the given noisy data with ...
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5 votes

Proof of complex conjugate symmetry property of DFT

Remember that $e^z$ has a very different meaning than $e^x$ (taking $z\in\mathbb{C}$ and $x\in\mathbb{R}$). If the exponent was real, then, as you state in your question: $$e^x = 1 \iff x=0$$ ...
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5 votes

Proof of complex conjugate symmetry property of DFT

Did you ever wonder about where $\pi $ came from? Watch out... Let us first draw this weird function complex exponential $e^{-2j\pi t}$ for several discrete values of $t\in[0,10]$ (the little blue ...
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5 votes

The Least Norm Solution of Under Determined Linear System

Least Squares solution is always well defined for Linear System of Equations. In your case, which is under determined it means there are many solutions to the Linear Equations. The Least Squares ...
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5 votes
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How Can PCA Be Used in Image Analysis

Imagine you have a set of 10,000 images (32 x 32) of faces. An intuitive way is to think they have a lot in common. One step farther would be that if you take one of the faces you could generate it ...
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5 votes
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Minimize the Cost Function of Values of Vectors Based on Their Amplitude

Since there is no prior at the Vector level this is basically element wise problem. Moreover, if we assume the noise to be White Noise with zero mean then the answer can be very simple. Since the ...
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5 votes

Image Matrix Vector Representation for the Degradation Model

Indeed since both expressions are scalars then they are equal to each other since the transpose of a scalar is the same scalar. See in MATLAB as an example (Calculating $ {x}^{T} H y $ and $ {y}^{T} {...
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5 votes
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How to Solve the Image Dehazing Problem Using ADMM?

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} \...
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5 votes
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Converting Hadamard Product into Matrix Multiplication in Image Deconvolution with Total Variation (TV) Using ADMM

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 $ ...
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5 votes
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The Matrix Form of a 2D Circular Convolution

Yes, indeed. You may represent the convolution in a Matrix Form. Pay attention that this form assumes the image is column / row stacked into a vector. If you're after a circular convolution, you may ...
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