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 ...

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1answer
32 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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0answers
51 views

Wavelet domain Gaussian processes

Maraun et al, "Nonstationary Gaussian processes in wavelet domain: Synthesis, estimation, and significant testing" (2007) defines a class of non-stationary GPs that can be specified by multipliers in ...
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2answers
84 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 $...
4
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1answer
84 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-...
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1answer
34 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
34 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}&...
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0answers
29 views

Independent Component Analysis [closed]

I am doing my project on analysis of the algorithm for cocktail party problem.As a part of the project i have to separate two voice signals.The best possible way to do is using independent component ...
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0answers
28 views

Eigenvalues of correlation matrix which have the form of an harmonic function [duplicate]

As a continuation to this question, I took the matrix $C_{2 \times 2}$ which is: $$C=\left[ \begin{array}{} a& ace^{-\frac{|\phi_1-\phi_2|}{2}}\\ ace^{-\frac{|\phi_1-\phi_2|}{2}} &...
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0answers
26 views

How to reduce polynomial?

I have some 4 polynomials like this. these equation is made by polyfit() from MATLAB tool. ...
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0answers
44 views

In which way do image transformations change camera extrinsics?

Given a pinhole camera model, we have an image and according camera intrinsics K and extrinsic parameters R. When we scale the ...
4
votes
1answer
103 views

New Horizons at Pluto

I'm teaching an undergraduate linear algebra course, and looking for applications. The New Horizons spacecraft is approaching Pluto, and returning images I find fascinating: http://pluto.jhuapl.edu/...
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2answers
155 views

Improvement of Minimum description length (MDL) estimate

I earnestly request apology if this question is inappropriate for the forum. The question has two parts one technical and the other is not technical. I would appreciate any response. Let me consider ...
3
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0answers
164 views

Weighted Nuclear Norm Minimization for Image Denoising

Recently, I saw new published papers like this paper about denoising images using Weighted Nuclear Norm Minimization (WNNM) approach and I am wondering what is the physical intuition behind it. The ...
1
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1answer
286 views

Extending 1D window functions to 3D (or higher)

(cross post of SO: http://stackoverflow.com/questions/27345861/extending-1d-function-across-3-dimensions-for-data-windowing ) For the sake of image (volume) registration, I'd like to apply a ...
0
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1answer
85 views

Looking for integral-free DSP textbooks

I am looking for a treatment of fundamental DSP theory where only discrete/finite signal models are used. To make a concrete example, every time standard textbooks write a convolution as a continuous ...
2
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0answers
97 views

Least Angle without matrix inversion?

Sorry if this is too damned long. I did what I could to abbreviate it. I'm new to numerical work with matrices. I believe I have a way to compute Least Angle without explicit matrix inversion. I'm ...
5
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2answers
124 views

How to estimate the poles of a linear combination of exponentials?

Consider a signal that can be expressed as a linear combinations of exponentials: $x(n) = \sum_{k=1}^K c_k z_k^n$ I would like to estimate the poles $z_k$ and I was told that I could estimate them ...
1
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1answer
286 views

Using a Wiener Filter to Estimate a Transfer Function

As a follow-on to this question about estimating a transfer function of an unknown system using a Wiener filter, How would you put a minimum MSE criteria on how well the estimated filter weights ...
4
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2answers
1k views

Estimate the transfer function of an unknown system?

Suppose you have a system, H, that you want to estimate its transfer function. You have a finite number of complex input samples, x, and noisy complex (magnitude and phase) output samples, y: In ...
3
votes
1answer
717 views

Relationship between Discrete Deconvolution and Toeplitz Matrices

I have 2 vectors, $a$ & $c$, both of length M. I know they are related by $a*b=c$. My goal is to recover $b$. Obviously $b=$deconv$(c,a)$. I am only interested in the first M elements of the ...
0
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1answer
572 views

How to understand relationships between ellipse and second moment matrix of Harris corner detector?

guys. I am really lacking in knowledge of linear algebra. I am reading slides of Harris corner. But I am really confused about one of them. I know that I can find corners by two large eigenvalues but ...
0
votes
1answer
100 views

Calculate the entropy of Turkish alphabet and compare with the entropy of English alphabet with using Matlab? [closed]

Calculate the entropy of Turkish alphabet and compare with the entropy of English alphabet with using Matlab? How can ı do this question anybody help please?..
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1answer
163 views

What are the mathematical consequences of a discrete signal living in Hilbert space?

I am doing the Coursera course on DSP by Prandoni and Vetterli, and was excited to learn that we can use the tools of vector spaces to analyze discrete signals. At this early point in the course the ...
3
votes
1answer
90 views

Error bounds in signal compression represented by truncated Moore-Penrose biorthogonal bases using von Neumann wavelets

I was reading and trying to reproduce the results in the arXiv preprint of Periodic Gabor Functions with Biorthogonal Exchange: A Highly Accurate and Efficient Method for Signal Compression by Asaf ...
2
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0answers
34 views

Recovering eigenvalues from singular values [closed]

I am dealing with a problem similar to principal component analysis. Aka, I have a matrix and i want to recover the 'most efficient basis' to exaplin the matrix variability. With a square matrix these ...
4
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0answers
64 views

Ideas on matrix factorizations and/or transformations for $\ell_1$ minimization

I am starting with a typical $\ell_1$ basis pursuit problem: $$ \min_{\mathbf{x}} \Vert \mathbf{x} \Vert_1 \quad \mathrm{s.t.} \quad \Vert \mathbf{ERx} - \mathbf{y} \Vert_2 \leq \epsilon, $$ where $\...
2
votes
1answer
860 views

why eigenvalues concerned in Harris corner detection?

In Harris corner detection, as I see, the goal is to find out those points $(x,y)$ which makes $S(x,y)$ a large value for any direction of shifting, $$ S(x,y) = [\Delta x , \Delta y]M\left[ \begin{...
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0answers
20 views

How do you combine 2 least squares estimators? [duplicate]

Say you want to compute the least squares estimate of $w$ from a data-set: $$ \begin{bmatrix}d_1 \\d_2 \\\vdots\\d_N \end{bmatrix} =\begin{bmatrix} x_1 \\x_2 \\ \vdots \\x_N\end{bmatrix}w + \begin{...
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0answers
19 views

Panorama global registration

I'm trying to do global registration algorithm for panorama like described here. But I don't understand how they deal with false pairs. Also if expand the system u1*I*P1=u1*I => P1=I u2*I*P2=u2*...
0
votes
3answers
170 views

How can a linear operator on DFT vector produce the same vector using only half of the DFT vector?

Suppose there is a DFT vector $\mathbf{X}$ with length N, which presents complex conjugate symmetry around its middle point, i.e., $X(1) = X(N-1)^*$, $X(2) = X(N - 2)^*$ and so forth. $X(0)$ and $X(N/...
6
votes
2answers
383 views

Conditions for precoding matrix to preserve complex conjugate symmetry on DFT vector

Suppose there is a DFT vector $\mathbf{X}$ with length N, which presents complex conjugate symmetry around its middle point, i.e., $X(1) = X(N-1)^*$, $X(2) = X(N - 2)^*$ and so forth. $X(0)$ and $X(N/...
1
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1answer
2k views

How to compute generator matrix from a parity check matrix?

I have a parity matrix ("H") that is not in canonical form (the identity matrix is not on the right side). I'm trying to programatically calculate the generator matrix ("G") from it. The Wikipedia ...
2
votes
1answer
130 views

Doubt on Weighted Least Square Estimation

This is a page from the book linear algebra,geodesy and gps by Gilbert Strang.... the page explains about the justification of the inverse of the of the co variance matrix of measurement vector $b$ in ...
2
votes
2answers
141 views

Least Square Error Estimation doubt when can we write $(A^TA)^{ -1} = A^{-1}(A^T)^{-1}$?

I am new to linear algebra and have this simple question... in least sqaure estimation...the best estimation of the equation $Ax = b$ is $x_{Estimated} = A (A^t A )^{-1} A^t b$...the projection of $b$ ...
6
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5answers
3k views

Good book or reference to learn Kalman Filter

I am totally new to the Kalman filter. I've had some basic courses on conditional probability and linear algebra. Can someone suggest a good book or any resource on the web which can help me can ...
19
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1answer
2k views

Mathematics of Harris corner point detection

This is the mathematical expression for Harris corner detection: But I have the following doubts: What is the physical significance of $u$ and $v$? Many references say it is the magnitude by ...
14
votes
1answer
251 views

Fitting new images from a SVD/PCA calculation

I'm trying to replicate the ideas from the Eigenface page on wikipedia. From a hundred sample images represented by a data matrix $\bf X$ (where each image flattened to a vector of length $n$, thus $\...