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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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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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81 views

Intuition behind using basis functions to optimize discretization of a 2D image obtained via projections

If I am not mistaken, the technique used to avoid a pixelated appearance of CAT scan images described as basis functions on page 4032 of this paper is not unique to CAT scan, and probably applied to ...
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26 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
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Implementation of Block Orthogonal Matching Pursuit (BOMP) Algorithm [closed]

How would one implement the lock Orthogonal Matching Pursuit (BOMP) Algorithm?
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1answer
28 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
136 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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24 views

A good reference for matrix completion [closed]

Does anyone know a complete reference or book on matrix completion?
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1answer
40 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
41 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
24 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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23 views

Linear equation set construction for Brox et al. optical flow optimization

I'm trying to implement an optical flow calculation program for two successive images based on this article by Brox et al. The Euler-Lagrange combined with a fixed point iteration loop yields two ...
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4answers
337 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
91 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
92 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
36 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
113 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
608 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
639 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
69 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
199 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
191 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 ...
3
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0answers
133 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
106 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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0answers
41 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
294 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
116 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
25 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 ...
3
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1answer
148 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 ...
4
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1answer
121 views

Underdetermined deconvolution of windowed output

Consider a discrete 'blurred' output $h[t]$ given by the convolution of filter $f[t]$ and signal $g[t]$. This question considers recovering $g[t]$ from a window (subset) of $h[t]$. This causes the ...
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1answer
61 views

Can linear transforms of vector space be seen as LTI systems? [duplicate]

Linear transforms of vector spaces has the linearity (i.e., homogeneity and superimposability), which is shared by LTI system. There are also shared concepts such as eigenvectors/eigenvalue... I am ...
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0answers
164 views

Deterministic method to compute “Process noise covariance matrix, Q” for a Kalman filter when parameter variations of the model is known apriori

I am implementing a Kalman filter (for a linear ODE system for now). My model represents a physical device that has 6 "parameters", i.e. those values of the device do not evolve over time (within a ...
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3answers
245 views

Group delay of $H(\omega)= 1- re^{j \theta}e^{ - j \omega} $

I'm studying chapter 5 of Discrete-Time Signal Processing 3rd edition by Alan Oppenheim and I'm having serious difficulties understanding how he obtained equation 5.57. For those who don't have this ...
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6answers
376 views

The Least Norm Solution of Under Determined Linear System

Suppose I have a matrix $$A= \begin{pmatrix} 1 & 0 & 1 & 0\\ 0 & 1 & 1& 0\\ \end{pmatrix} $$ where the variables are channel information like assume $X_1$, $X_2$, $X_3$ ...
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0answers
63 views

Linear Systems, Sparse Solutions, and $4 \times 4$ Sudoku Algorithm [closed]

I am unable to understand the paper Linear Systems, Sparse Solutions, and Sudoku. I have to form a $4 \times 4$ Sudoku using the algorithm in this paper. Can somebody please provide me the algorithm ...
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1answer
35 views

Help with Determinants [closed]

Okay so i understand the first det, which is why i got the correct solution. However for the second det, im not sure how to solve it since because of the different coefficents (0,2,4) and i tried of ...
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2answers
411 views

Deriving Frequency Response for 2-pole Zero-Delay Feedback State Variable Filter

I have an existing zero-delay feedback (ZDF) 2-pole state variable filter implementation (along the lines of the theory presented in VA Filter Design by V. Zavalishin), and I wish to determine the ...
3
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1answer
100 views

Restriction of Fourier Transform

I am currently reading Candes et. al.'s 2006 paper[1] on recovery of sparse signals from incomplete frequency samples. I am having trouble figuring out what is the form of the Fourier transform ...
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1answer
1k views

Sensing matrix for compressed sensing

What are the differences between random binary sensing matrix  and random Gaussian sensing matrix? What the advantages and disadvantages of each matrix? How can I choose the suitable matrix for a ...
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3answers
4k views

Proof of complex conjugate symmetry property of DFT

According to the Proof : \begin{align} X_n &= \sum_{k=0}^{N-1}x_ke^{-j\frac{2\pi k n}{N}}\\ X_{N-n} &= \sum_{k=0}^{N-1}x_ke^{-j\frac{2\pi k (N-n)}{N}}\\ &=\sum_{k=0}^{N-1}x_k e^{-j 2\pi ...
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2answers
233 views

What are the practical constraints on designing Sensing matrix in compressed Sensing?

In a typical compressed sensing scenario, $y=Ax$, where $x$ is a sparse signal and $A$ is the sensing matrix. To reconstruct the sparse signal $x$ from $y$, $A$ should posses the Restricted Isometry ...
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2answers
500 views

Alternative to Orthogonal Matching Pursuit (OMP) Algorithm

In the Compressed Sensing context, assume there is a signal $ x \in {\mathbb{R}}^{n} $ which is $ k $ sparse. Namely its Pseudo $ {\ell}_{0} $ Norm is $ {\left\| x \right\|}_{0} = k $ (The signal has ...
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1answer
2k views

How to find out if a transform matrix is separable?

In image processing, when we have a series of basis images, how could we know if the transformation is separable or not? For example, I know that following bases are separable and transformation can ...
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1answer
119 views

Deriving the Matrix Inversion Lemma for RLS Equations vs the Woodbury Derivation

Can any one help me in deriving the matrix inversion lemma rule for RLS algorithm? I don't know how to start with. Many books have just stated but they haven't derived it.
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1answer
202 views

Finding a good inverse for an ill-conditioned matrix transformation

I have a time-series observation dataset that has been distorted. I want to recover the best approximation of the original signal as possible. Disclaimer:: I know only the basics of linear algebra, so ...
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2answers
164 views

The mathematical interpretation of DFT [closed]

We have the DFT(matrix form) $X = Wx$ ($W$ is the Fourier basis matrix, $x$ is the original signal in time domain, $X$ is in the frequency domain). In mathematics, $x$ represents the coordinates of $...
3
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1answer
105 views

Geometric explanation of a methodology in the article about Image Denoising

In article Ghimpeteanu G., et al. - A Decomposition Framework for Image Denoising Algorithms, I found as below: Let $\displaystyle I : \Omega \subset \mathbb{R}^2\mapsto \mathbb{R}$ be a gray-level ...
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2answers
73 views

Proving conditions for controllability

Let's say I have the following LTI system: $$\dot{x}(t) = \mathbf{A} x(t) + \mathbf{B} u(t)$$ I need to somehow show the following is true or false (proof): This system is controllable if and only ...
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1answer
87 views

Showing a system is always controllable?

I need to show that the following system is always controllable: \begin{align}A &= \begin{bmatrix} -\alpha_1I_{k\times k}& -\alpha_2I_{k\times k}& \cdots &-\alpha_{n-1}I_{k\times k}&...