Questions tagged [linear-algebra]

is a branch of algebra, concerning linear nature of objects: vector or vector spaces, linear transformations, systems of linear equations, quadratic and bi-linear forms, among the main tools used in linear algebra is the determinants of the matrix pair. The theory of invariants and tensor calculus is usually considered as integral parts of linear algebra.

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28 views

Equation of line in the Homogeneous coordinate system

Given two points P and Q we can convert them to the homogeneous coordinate system, compute their cross product and thus get the ...
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1answer
39 views

What is the relation between eigenvalues and state-space response in control systems?

I understand the mathematics behind it but I want to know what happens physically in a real-life system. How do the eigenvalues come into the picture from a non-mathematical (physical) point of view? ...
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1answer
175 views

Super Resolution in Frequency Domain Using Compressed Sensing

To be noted that I'm very new to this topic, I would like to understand the fundamentals of how to get Super Resolution in Frequency Domain estimation using the Compressed Sensing Model. I am also ...
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1answer
88 views

Linear correlation of two vectors?

We had a lecture on digital watermark detection using linear correlation. Here it was explained that the linear correlation of two vectors is equal to their scalar product divided by the dimension. So,...
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1answer
129 views

Circular Convolution as Cyclic Shift Operator

Given the following signal vectors: $$ γ=[ψ_0,0,ψ_1,0,ψ_2,0,…,ψ_{N-1},0]^T\in \mathbb{R}^{2N}, ϕ=[1,\frac{1}{2},0,…,0,\frac{1}{2}]^T \in \mathbb{R}^{2N}$$ I want to show that the convolution of $γ$ ...
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31 views

Normalizing directed graphs from image space to compute joint differences

My question is how to transform a 'skeleton' (directed graph) from image space so that I can compare it joint-wise against another 'template' or reference skeleton that may be differently sized/...
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1answer
50 views

Relation between the matrix trace and the amplitude of each element

Assume a diagonal matrix $\mathbf X$ whose size $N\times N$ and its diagonal elements are $0.5 + 0.5i$, and the vector $\mathbf p$ of size $N\times 1$ whose elements have similar amplitude. I have ...
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40 views

Taylor series approximation in Harris corner detection

While watching through the computer vision lecture on interest point detection, computing $E(u,v)$ requires computing the quantity $$E(u,v) = \sum_{x,y}(I(x+u,y+v) - I(x,y))^2$$ In the lecture, $I(x+u,...
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31 views

Harris corner detection shape of $E(u,v)$

I am taking a computer vision class and I have just learnt about the Harris corner detection concept. A corner is detected when a small shift in a window function defined around the corner results in ...
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1answer
41 views

Transform a data set by exploting the vectorfield

I am somewhat new in the field of Digital Signal Processing / Image processing. As shown in the figure, I have 4 straight lines $f_i(x)$ with $i = 1,\dots, 4$ that pass through $g(x)$. Similiarly ...
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2answers
34 views

Image Matrix Vector Representation for the Degradation Model

I am trying to understand the the degradation model equation but I have doubt that how come y^t.x.h will be equal to x^t.h^t.y . Aren't they transpose of each other.
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1answer
55 views

Minimize the Cost Function of Values of Vectors Based on Their Amplitude

I have two vectors $X = [x_1,x_2,x_3,x_4]$; and $Y = [y_1,y_2,y_3,y_4]$; I know that $|x_1|$ = $|y_1|$, and $|x_2|$ = $|y_2|$,... so on. it means the difference is only in the sign. it might be ...
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83 views

A hypercomplex encoding to preserve spatial/temporal information? [closed]

I have recently come across the idea of encoding a 1D signal (i.e. a mono audio) as a complex vector instead of as a vector of reals, where the imaginary part is used to encode the cells' positions. ...
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1answer
30 views

What do coeffcients mean from Matlab?

I ask for a brief explanation of each coefficient and their sources of related information. Thank you very much in advance. The coefficients comes from this code below: I've got a robust linear ...
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4answers
66 views

On the simplification using trigonometric functions

Assume I have a matrix $D$ whose its entries are as below : Where $A$ and $B$ can be written using using the trigonometric functions for (1) as: My question, Is it possible to simplify (1) more? ...
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71 views

unitary matrix complexity multiplication

Having a unitary matrix $X$ whose size is $n \times n$ and a vector $z$ whose length is $n$, and let's have: $$y = X^H {\rm diag}(z)X$$ where $X^H$ is the conjugate transpose of $X$. My question,...
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1answer
50 views

Calculate the Derivative of Gradient Field of an Image

I meet a confusing thing in image processing recently.... Assume the image $x \in \mathbb{R}^n$, with its derivative (difference) matrix: $D^+ = \begin{bmatrix} D_h \\ Dv \end{bmatrix} \in \mathbb{R}...
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77 views

Sparse recovery, Restricted Isometry Property for ILL-POSED problems

if $\mathbf x$ is $N\times 1$ sparse vector, and $\mathbf A$ is an $M\times N$ matrix with $M<<N$, and we measure $\mathbf y=\mathbf{Ax}$, then compressed sensing theory tells us that we can ...
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41 views

Expectation of a constant diagonal matrix

Is the expected value of a diagonal matrix with constant entries equal to the mean value of the entries? My question stems from the following observation in a paper. Given a real diagonal matrix $\...
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1answer
73 views

On the use of Permutation matrix to perform iFFT

I have a question about using the permutation matrix for performing $iFFT$ for such matrix and then reshape it row-wise and column-wise way. Let's say that we have a random matrix $x$ whose size is (...
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2answers
87 views

Proving that a product of matrices invertible

Given $R_x$ a Positive Definite (PD) covariance matrix of size $M\times M$ and $C$ a full rank $M \times N$ matrix, I want to prove that $C^* R_x^{-1} C$ is invertible to derive the Linearly ...
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1answer
150 views

Orthogonal Basis for a 2D Signals (Compressive Sensing)

I have a 2-D signal that is (1536x128) and that is sparse in the Fourier domain (after applying fft2). I want to apply compressive sensing to recover the signal using fewer random elements, but I am ...
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52 views

Does this system of equations have a solution?

My heuristic approaches to find a solution for the following system of equations have failed so far. Does a solution exist? $$ \left[ {\begin{array}{cc} a_k & 0 \\ a_{k-1} & a_k \\ ...
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63 views
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59 views

Linear interpolation formula

In the following lecture: http://www.ece.mcmaster.ca/~xwu/interp_1.pdf the model (formula) for solving the linear interpolation problem (1D) given at p.5 is: $f(x)= a_1x_1 + a_0x_0$ solve for $a1,...
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2answers
212 views

Implementation of Block Orthogonal Matching Pursuit (BOMP) Algorithm [closed]

How would one implement the Block Orthogonal Matching Pursuit (BOMP) Algorithm in MATLAB?
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1answer
58 views

Graph Fourier transform: the adjoint notation for the eigenbasis matrix

I already asked this question here but there is no response. I'd like to ask this question in signal processing domain. It is well-known that for a real symmetric matrix $L$ (here, graph ...
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2answers
1k views

Using Linear Algebra for DSP

I am new to DSP in general, but can one use linear algebra by itself to characterize a signal? My first idea was to transform the signal into a matrix and then use the determinate to characteristic ...
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0answers
29 views

A good reference for matrix completion [closed]

Does anyone know a complete reference or book on matrix completion?
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1answer
59 views

Resources on Solving Convex Optimization Problems in the Compress Sensing Field

When I read papers of compressed sensing, sparse representation and whatever requiring optimization of a cost function, I just find the final results as an iterative equation or so which will converge ...
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2answers
194 views

The Gradient of Least Squares of 2D Image Convolution

Given the objective function: $$ \frac{1}{2} {\left\| h \ast x - y \right\|}_{2}^{2} $$ Where $ h $ is the 2D convolution kernel and $ x $ is the 2D convolution image and $ y $ is a given 2D image. ...
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1answer
51 views

STBC Rank Criteria Explanation

I am learning about space-time block coding and I am trying to understand why the rank criteria is a good measure of diversity gain. I know that we want to maximize the distance between codewords $X_i,...
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4answers
1k views

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

I'm currently working on a project in which I need to find the tilt of a surface. Let's assume I'm only concerned with a single dimension tilt (i.e. slope) to begin. I currently have the ability to ...
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1answer
205 views

How to implement discrete extended kalman filter in matlab

I have a nonlinear system, and I need to use the extended kalman filter to estimate it. I know I need the jacobian, but once I get that, is everything else the same as the normal kalman filter? I ...
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2answers
128 views

Check whether a given equation is linear

$ a = (x, y) \in \mathbb{Z}^2 $ is given as a pixel. My equation in which $g$ is image(matrix) is defined as, $f(x, y) = 56g(x,y)+93g(x−1,y)+92g(x+1, y)−57g(x, y−1)+555g(x, y+1) $ How can we know ...
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1answer
82 views

DFT as an Orthogonal Basis Change

In one of the homeworks that I am dealing with for Linear Systems course, I have encountered with such a statement: Consider $\mathbb{C}^N$ the vector space of N dimensional complex vectors. We can ...
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2answers
192 views

Conversion from stationarity to non-stationarity

Is there any way to convert a non-stationary signal to a stationary one, perform operations on it meant for a stationary signal and then convert it back to the non-stationary one?
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1answer
1k views

How to Check Separability of 2D Signal / Matrix

suppose x(n1,n2) = {1 ,n1=0,n2=0 ; 2 ,n1=1,n2=0 ; 3 ,n1=0,n2=1 ; 6 ,n1=1,n2=1 } then, how do i prove it is separable.
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1answer
1k views

How do I apply a binary mask and STFT to produce an audio file?

So here's the idea: you can generate a spectrogram from an audio file using shorttime Fourier transform (stft). Then some people have generated something called a "binary mask" to generate different ...
2
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1answer
534 views

How Does Mean Centering Affect the Result of Using SVD to Compress Images?

I have been learning about using the Singular Value Decomposition to find low rank approximations to matrices. I had an image which I converted to a matrix. I regarded each row of the matrix as a '...
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2answers
550 views

What is the relation between kernel functions, kernels used in convolution and null spaces of a matrix?

I have recently started learning about machine learning and have come across kernels and null spaces. I understand that null space is the set of all vectors that satisfy the equation A.v = 0 (Where A ...
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1answer
444 views

How Can PCA Be Used in Image Analysis [closed]

I am still a not how PCA can be used in image analysis and where is it is mostly used. For example how can PCA be used in order to differentiate between different faces? Can you please mention other ...
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0answers
153 views

Multilateration of Sound in 3D Space

TL:DR - How can you find the 3D coordinates of a emitter than transmits an impulse signal? STORY: I'm working on something to improve my bird-watching. I've got a camera that can take pictures of ...
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3answers
158 views

Least Squares with Blocks / Updates

I have a continuous-time system that I want to fit via least squares. I just send $N$ digital samples $x[n]$ through the system and receive (via analog signal chain, ADC etc) $N$ digital samples $y[n]$...
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43 views

Required number of measrments for signal recovery in a compressed sensing MMV problem?

For multiple measurement vector (MMV) problem $Y=AX$ where $A$ is $m \times n$ sensing matrix and $X$ is $n \times L$ matrix haveing K non zero rows. What are the necessary conditions on the ...
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1answer
456 views

Signals and systems book with linear algebra approach

I am currently taking a course on discrete-time signals and systems. I am using B. P. Lathi's Signal Processing and Linear Systems, which I don't like at all, as it doesn't draw any paralells to ...
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1answer
157 views

Tikhonov Regularization Alternate Formulation

2.2 Tikhonov Reqularization Tikhonov regularization, named for Russian mathematician Andrey Tikhonov, attempts to fix the issue that arises when the least squares method is used with an ill-posed ...
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1answer
1k views

3D world homography matrix calculation

I have a matrix in (x,y,z) coordinates and also i have a transformed version like (x_new,y_new,z_new). I need to calculate a transformation matrix that can transform (x,y,z) to (x_new,y_new,z_new). ...
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1answer
32 views

Are there analogues to orthogonal transformations in non-orientable surfaces?

I am working on extremely large, symmetric matrices of counts, and attempting to identify patterns/shapes within them. Wavelets are a popular tool in image processing, and have some nice statistical ...
4
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1answer
207 views

Time domain basis

I have some troubles with understanding time domain, not on the intuitive level, but in math terms. For example I have a vector signal $$ x = [x_0,x_1,x_2,...,x_{N-1}]$$ I understand that generally ...