Skip to main content
Share Your Experience: Take the 2024 Developer Survey

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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
28 views

Kalman Filter: handling large covariance matrices with principal-component-like structures

I understand that the estimation of the covariance matrices are the important part of the Kalman filter. However in my use case my covariance matrices are really big but with a pretty neat factor ...
Matt Frank's user avatar
1 vote
0 answers
121 views

Proof that the order of a linear prediction model cannot be reduced by analysing non-adjacent points

I have a signal, for which a linear prediction model of order $M$ can be constructed. This means, that a system of the shape $$ \begin{bmatrix} s_0 & s_1 & \cdots & s_M \\ s_1 & s_2 &...
Tigozawr's user avatar
0 votes
0 answers
13 views

Derive white balance matrix operator in YUV space

Assume that the white balance matrix operator in RGB space is $$W=\begin{bmatrix} \alpha &0 &0 \\ 0 & \beta & 0 \\ 0&0&\gamma \end{bmatrix}$$ where the constants ...
In the blind's user avatar
1 vote
1 answer
80 views

Properties of DFT of Circularly Symmetric 2-D Matrices

I'm very new in image processing and trying to get a grasp in the basic concepts in 2-D DFT. As far as I understood, DFT of a circulant matrix should also be a circulant matrix. But when I define a ...
Deniz Açıkgöz's user avatar
4 votes
2 answers
302 views

Smallest Eigenvalue in the Derivation of the MUSIC Algorithm

I am seeking clarification on a particular step in the derivation of the MUSIC algorithm as presented in a specific paper. Here, there is an intermediate step I cannot follow and I would appreciate ...
Naetmul's user avatar
  • 145
1 vote
1 answer
91 views

To find the unitary matrix which is the null of the results of multiplication with another matrix

I have a matrix $F ∈ \mathbb{C}^{(m × N)}$, where $m < N$, and $F \times F^H$ is a unitary $m × m$ matrix. I need to find a unitary matrix $G$ with a dimension of $N × N$ such as results of $F\...
Fatima_Ali's user avatar
0 votes
0 answers
13 views

How to handle imaginary numbers with total square error questions

I am pretty new to algrebra so if u can use simple terms and explain a bit more it would be helpfull. Ive been trying to use imaginary numbers to get the total square error of a graph of a non linear ...
Joziah's user avatar
  • 1
0 votes
0 answers
133 views

Python - Discrete deconvolution using Toeplitz matrix

Lets say I have 2 vectors (1D signals that are sigmoids): $s$ and $m$, both related through the relation: $m = s * r$, my goal here is to recover the vector $r$ (should be a gaussian $\rightarrow$ ...
Michael's user avatar
  • 75
8 votes
5 answers
329 views

Solving Inverse Problem of Multiple Pulses Over Multiple Channels with Convolution Kernel and Cross Channel Mix

Before I start, let me note that I have 0 experience with signal processing, so please bear with me: My System My system can be represented as an $m \times n$ matrix $X$ (input) where each column ...
Daniel Duque's user avatar
2 votes
1 answer
121 views

How to Solve a Composition of Convolutions from Regularized Least Squares Model in Frequency Domain

Assume we need to solve the model: $$ \arg \min_{\boldsymbol{x}} \frac{1}{2} {\left\| \boldsymbol{h} \ast \boldsymbol{x} - \boldsymbol{y} \right\|}_{2}^{2} + \frac{\lambda}{2} {\left\| \boldsymbol{g} \...
Royi's user avatar
  • 19.7k
2 votes
1 answer
169 views

Solving linear equation of discrete convolution kernels using black box model for the convolution

In Solving inverse problem using black box implementation of the kernel the solution depends on solving the equations of the form: $${\left( {H}^{T} H + \lambda {G}^{T} G \right)} x = y$$ Where $H$ ...
Eric Johnson's user avatar
3 votes
1 answer
124 views

Solving regularized least squares problem using black-box computation of $\mathbf{A}\mathbf{x}$ and $\mathbf{A}^T\mathbf{x}$

Let $\mathbf{A} \in \mathbb{R}^{n \times n}$. I'm working in a problem where I have a black-box algorithmic solution to compute the products $\mathbf{A}\mathbf{x}$ and $\mathbf{A}^T \mathbf{x}$ given ...
mlbj's user avatar
  • 95
2 votes
2 answers
180 views

What is the adjoint of the Goertzel algorithm?

The adjoint of the DFT is the IDFT. But suppose we don't need to calculate a full DFT (just a subset of frequencies), and so are using the Goertzel algorithm instead. What is the adjoint of the ...
Gillespie's user avatar
  • 1,797
6 votes
5 answers
2k views

Under what conditions does DFT(f(x)) = f(DFT(x)) hold?

I have encountered a text (a specification) in which $$\text{FFT}\left(f\left(x\right)\right) = f\left(\text{FFT}\left(x\right)\right)$$ is clearly presumed (for the functions they are applying). Thus,...
Bernd Wechner's user avatar
1 vote
1 answer
80 views

Why is the total noise variance less than the sum of individual noise variances?

I have three random variables: $Y$: my data $Y_n$: my data corrupted by additive white Gaussian noise (AWGN) $Y_{nc}$: my noisy data corrupted by a non-linear transformation $\mathcal{C}$. I have ...
graille's user avatar
  • 57
1 vote
1 answer
78 views

Matrix form of Overlap-add

We know overlap-add of a en-framed signal can be done easily by following code ...
Markus Christensen's user avatar
3 votes
0 answers
85 views

Nuclear norm minimization of convolution matrix (circular matrix) with fast Fourier transform

I am reading a paper Recovery of Future Data via Convolution Nuclear Norm Minimization. Here, I know there is a definition for convolution matrix. Given any vector $\boldsymbol{x}=(x_1,x_2,\ldots,x_n)^...
Xinyu Chen's user avatar
0 votes
1 answer
30 views

Expressing mathematically the number of real addition operation for a vector after dividing it

I assume I have the length of such vector $y$ is $N$. In the first time I divide that vector into two columns and then sum them point-wise summation. The second time, I divide the same vector $y$ ...
Gze's user avatar
  • 640
0 votes
1 answer
48 views

What is the complexity of multiplication a real matrix with real vector

I have a real matrix $Z$ which is following the form as following: $Z = \begin{bmatrix} x_1& 0& 0& 0& 0& 0& 0& 0\\ 0& x_2& 0& 0& 0 & 0&...
Gze's user avatar
  • 640
3 votes
0 answers
223 views

Improving the intuition for the 2d fourier transform

As far as I understand, the 2d fourier transform is calculated as following: ...
dmmpie's user avatar
  • 131
0 votes
1 answer
35 views

How can I express the flipped output of multiplication in function of original inputs?

I have the vector $y = Dx$ where $D$ is a complex matrix with dimension $N \times N$, and $x$ is a complex vector of dimension $N \times 1$. If the vector $y_2 = [y'_N, y'_{N-1}, y'_{N-2},.... , y'_{...
Gze's user avatar
  • 640
1 vote
0 answers
90 views

Convolution between a vector and another symmetric vector

Let's have the vector $y = h * x$ where $*$ is the convolution operation, $h$ is the channel with length $N$ and $x$ is a symmetry vector which means $x = [x_M, x_{M-1}, ....,x_0, 0 , x_0, x_1, .... ...
Gze's user avatar
  • 640
4 votes
1 answer
1k views

The Matrix Form of a 2D Circular Convolution

I have 3 closely related questions regarding 2d convolutions and how they are represented in matrix form. 1. Miming what happens in 1d, I assume the product of a doubly block circulant matrix $A$ by a ...
mlbj's user avatar
  • 95
0 votes
0 answers
73 views

Coherence in compresive sensing

I am starting to write my master thesis, and it's in field of compressive sensing. I have some problems with math behind it. I don't understand the concept of matrix coherence. I know how it is ...
narutouzumaki 99's user avatar
0 votes
1 answer
60 views

Can we recover a vector from one element of resulted vector after multiplication?

I have a matrix $X = \begin{bmatrix} 0.5000 + 0.5000i & 0.5000 - 0.5000i\\ 0.5000 - 0.5000i & 0.5000 + 0.5000i \end{bmatrix}$ multiplied with a column containing a complex number and its ...
Sajjad's user avatar
  • 167
0 votes
0 answers
49 views

Signal Processing on non-Euclidean domains

I have a very simple yet fundamental question. Suppose I have a vector of data $x \in \mathbb{R}^N$. Without additional information, I guess the majority of people think this vector as defined over ...
VanBaffo's user avatar
  • 103
1 vote
1 answer
304 views

Spectral Interpolation vs Linear Interpolation

What is the main edge of using a spectral method (Spectral Intp/Trigonometric Intp) for upsampling or downsampling a signal in comparison to using a linear (Trilinear Intp) method to do the same? I ...
Pranoy Ray's user avatar
1 vote
0 answers
51 views

Reuse channel decoder for a smaller code dimension of linear block codes

I am working with the 5G NR polar code and have implemented the CRC-Aided Successive Cancellation List decoder based on this paper (Tal et al. 2012). As my decoder is only used to decode a limited ...
Rokai's user avatar
  • 63
2 votes
2 answers
374 views

Decomposing Sobel Filter

I am trying to decompose a Sobel filter into two vectors (column and a row) using Matlab. If our Sobel filter is A = [1 0 -1; 2 0 -2; 1 0 -1] we can get the U, S, V ...
100Large's user avatar
4 votes
1 answer
166 views

Converting Hadamard Product into Matrix Multiplication in Image Deconvolution with Total Variation (TV) Using ADMM

I would like to solve the following Image Deconvolution equation by ADMM. $$\mathbf { \min\frac{1}{2}\Vert{Cx-b}\Vert_2^2+\Vert w\circ (D x)\Vert_1 \tag 1}$$ Where, $x$ is a vector of unknown pixel ...
Sushi man in Japan's user avatar
1 vote
2 answers
3k views

Generate the Matrix Form of 1D Convolution Kernel

As a follow up to Generate the Matrix Form of 2D Convolution Kernel, could someone explain how to generate the matrix form of a 1D convolution kernel? How different convolutions shapes are handled? ...
Mark's user avatar
  • 357
0 votes
0 answers
65 views

Matrix multiplication computational complexity based on radix 2

I am wondering, can we use Radix 2 based computational-complexity calculation for any matrix multiplication whose size is $N$ x $N$ ?? where $N$ = $2^K$ and $K > 1$ is an integer ?? Or it can only ...
Fatima_Ali's user avatar
4 votes
1 answer
385 views

Tikhonov Regularization for Complex Matrices

Tikhonov regularization is used to regularize ill-posed inverse problems if the matrix $A \in \mathbb{R}^{n,m}$ to be inversed has a high condition number. For example $$ A=\begin{bmatrix}1&1\\ 1&...
Bulbasaur's user avatar
  • 209
4 votes
1 answer
185 views

How to Solve the Image Dehazing Problem Using ADMM?

I want to solve the image dehazing problem using ADMM. I want to use the proximal algorithm to optimize each element. I refer to this treatise: Efficient image dehazing with boundary constraint and ...
Sushi man in Japan's user avatar
1 vote
2 answers
148 views

Optimization of square matrix multiplied with another matrix to have the final result a unitary matrix

I have a square matrix $D$ whose size is $m \times m$ multiplied with another $m \times m$ square matrix $C$, I need to optimize the matrix $C$ to have a unitary matrix $DC$. I mean optimize the ...
Fatima_Ali's user avatar
3 votes
1 answer
414 views

Why is incoherence important for compressive sensing?

The literature on compressive sensing (CS) frequently notes that CS relies on two principles: sparsity and incoherence. While I understand why the signal of interest should be sparse in some domain ...
blerner's user avatar
  • 31
0 votes
0 answers
45 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 ...
Mohit Lamba's user avatar
1 vote
1 answer
758 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? ...
Sagnik's user avatar
  • 11
5 votes
1 answer
375 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 ...
Luca Romano's user avatar
1 vote
1 answer
451 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,...
Marcel Herbst's user avatar
7 votes
1 answer
352 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 $γ$ ...
Nave Tseva's user avatar
0 votes
1 answer
150 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 ...
New_student's user avatar
2 votes
0 answers
151 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,...
calveeen's user avatar
  • 283
0 votes
1 answer
168 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 ...
calveeen's user avatar
  • 283
0 votes
1 answer
51 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 ...
unfinished_sentenc's user avatar
5 votes
2 answers
86 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.
SOMA REDDY's user avatar
3 votes
1 answer
106 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 ...
Gze's user avatar
  • 640
1 vote
0 answers
127 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. ...
user's user avatar
  • 11
-2 votes
1 answer
35 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 ...
Sergio Cuadros Ibargüen's user avatar
0 votes
4 answers
83 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? ...
Gze's user avatar
  • 640