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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Build the Laplacian Matrix of Edge Preserving Multiscale Image Decomposition based on Local Extrema
In the paper Edge Preserving Multiscale Image Decomposition based on Local Extrema (Available at DSpace@MIT - Edge Preserving Multiscale Image Decomposition based on Local Extrema or the writer page ...
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Kalman Filter and Generalized Least Squares
I have read in many places how Kalman Filter is related to generalized least squares algorithms. But there is still a bit I found a bit counterintuitive.
Kalman gain solution is ${K}_k = {{P}}_{k\mid ...
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Computationally Efficient Implementation of Kalman Filter
I know there are many formulations of the Kalman Filter. A few I can name are:
Classical Covariance Form
Informational Filter Form
Square-Root Form or Factor Form
But somehow it's hard for me to ...
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How does camera intrinsic parameters change after undistortion
Here is my goal: I have used OpenCV to estimate a camera's intrinsic parameters and distortion coefficients. In order to avoid having to deal with the camera's distortion going forward, I would like ...
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Understanding Notations of Matrix Calculus in Controller Tuning Article
I have been working my way through the paper "Iterative feedback tuning: theory and applications" (Hjalmarsson, Gevers et al IEEE Control Systems Magazine , vol. 18, no. 4, pp. 26-41, Aug. ...
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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
$$
\begin{bmatrix}
s_0 & s_1 & \cdots & s_M \\
s_1 & s_2 &...
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Derive white balance matrix operator in YUV space
Assume that the white balance matrix operator in RGB space is $$W=\begin{bmatrix}
\alpha &0 &0 \\
0 & \beta & 0 \\
0&0&\gamma
\end{bmatrix}$$
where the constants ...
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1
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Properties of DFT of Circularly Symmetric 2-D Matrices
I'm very new in image processing and trying to get a grasp in the basic concepts in 2-D DFT.
As far as I understood, DFT of a circulant matrix should also be a circulant matrix. But when I define a ...
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Smallest Eigenvalue in the Derivation of the MUSIC Algorithm
I am seeking clarification on a particular step in the derivation of the MUSIC algorithm as presented in a specific paper. Here, there is an intermediate step I cannot follow and I would appreciate ...
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To find the unitary matrix which is the null of the results of multiplication with another matrix
I have a matrix $F ∈ \mathbb{C}^{(m × N)}$, where $m < N$, and $F \times F^H$ is a unitary $m × m$ matrix.
I need to find a unitary matrix $G$ with a dimension of $N × N$ such as results of $F\...
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Python - Discrete deconvolution using Toeplitz matrix
Lets say I have 2 vectors (1D signals that are sigmoids): $s$ and $m$, both related through the relation: $m = s * r$, my goal here is to recover the vector $r$ (should be a gaussian $\rightarrow$ ...
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Solving Inverse Problem of Multiple Pulses Over Multiple Channels with Convolution Kernel and Cross Channel Mix
Before I start, let me note that I have 0 experience with signal processing, so please bear with me:
My System
My system can be represented as an $m \times n$ matrix $X$ (input) where each
column ...
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How to Solve a Composition of Convolutions from Regularized Least Squares Model in Frequency Domain
Assume we need to solve the model:
$$ \arg \min_{\boldsymbol{x}} \frac{1}{2} {\left\| \boldsymbol{h} \ast \boldsymbol{x} - \boldsymbol{y} \right\|}_{2}^{2} + \frac{\lambda}{2} {\left\| \boldsymbol{g} \...
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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 $H$ ...
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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 ...
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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 ...
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5
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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,...
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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 ...
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Matrix form of Overlap-add
We know overlap-add of a en-framed signal can be done easily by following code
...
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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)^...
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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 $y$ ...
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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&...
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245
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Improving the intuition for the 2d fourier transform
As far as I understand, the 2d fourier transform is calculated as following:
...
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1
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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'_{...
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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, .... ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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388
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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 ...
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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 ...
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2
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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?
...
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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 ...
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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&...
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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 ...
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2
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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 ...
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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 ...
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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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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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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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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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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}$$
I want to show that the convolution of $γ$ ...
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Relation between the matrix trace and the amplitude of each element
Assume a diagonal matrix $\mathbf X$ whose size $N\times N$ and its diagonal elements are $0.5 + 0.5i$, and the vector $\mathbf p$ of size $N\times 1$ whose elements have similar amplitude.
I have ...
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Taylor series approximation in Harris corner detection
While watching through the computer vision lecture on interest point detection, computing $E(u,v)$ requires computing the quantity
$$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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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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1
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52
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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 ...
5
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2
answers
89
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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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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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