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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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 ...
- 131
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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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vote
1
answer
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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
$$
\begin{bmatrix}
s_0 & s_1 & \cdots & s_M \\
s_1 & s_2 &...
- 11
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Derive white balance matrix operator in YUV space
Assume that the white balance matrix operator in RGB space is $$W=\begin{bmatrix}
\alpha &0 &0 \\
0 & \beta & 0 \\
0&0&\gamma
\end{bmatrix}$$
where the constants ...
- 149
1
vote
1
answer
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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 ...
4
votes
2
answers
312
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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 ...
- 145
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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\...
- 618
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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 ...
- 1
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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 ...
- 81
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votes
1
answer
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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} \...
- 20.1k
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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 $H$ ...
- 324
3
votes
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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 ...
- 95
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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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