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Tagged with convolution reference-request
9
questions
1
vote
0
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41
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Fourier transform relation for spherical convolution
Let $f$ and $g$ be two functions defined over the 2d sphere $\mathbb{S}^2$.
The convolution between $f$ and $g$ is defined as a function $f * g$ over the space $SO(3)$ of 3d rotations as
$$(f*g)(R) = \...
- 2,226
1
vote
1
answer
341
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Extracting eigenvalues of a circulant matrix using discrete Fourier matrix
The eigenvalues of a circulant matrix $C$ can be extracted as $$
\Lambda=F^{-1} C F
$$
where the $F$ matrix is a discrete Fourier transform matrix and $\Lambda$ is a diagonal matrix of eigenvalues.
...
- 791
1
vote
0
answers
198
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Special function: Pulse peak modified with a power term
PeakFit (Systat, v. 4.12) is a software for fitting experimental peaks obtained in physics or chemical experiments. Under the miscellenous peak functions, it shows the following equations with a name, ...
- 791
2
votes
1
answer
104
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Equivalent of a local limit theorem in the large deviation region and asymptotics of a convolution operator
Let $\{X_i \}_{i \in \mathbb{N}}$ be a sequence of i.i.d. random variables satisfying $\mathbb{E} X_1 = 0$ and $\mathbb{E} X_1 ^2 < \infty$. Assume that $\{S_n \}_{n \in \mathbb{N}}$ is a non-...
- 714
2
votes
1
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319
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Is $g(v)=\mathbb{E}[f(v+W)]$ a differentiable function of $v$ when $f$ is continuous and $W$ is multivariate normal?
Suppose $f$ is a continuous function on $\mathbb{R}^n$, and $W$ has a multivariate normal distribution on $\mathbb{R}^n$. If the expectation
$$g(v)=\mathbb{E}[f(v+W)]$$
is defined for all $v \in \...
user44143
5
votes
1
answer
317
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Is there a name for this type of matrix?
For my thesis in neural networks, I was trying to find a way to generalize a Sobel operator. I quickly thought of this:
$$
\begin{bmatrix}
a&b&c\\
d&0&-d\\
-c&-b&-a
\end{...
- 153
4
votes
2
answers
711
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Earliest use of deconvolution by Fourier transforms
From a previous discussion here Origin of the convolution theorem, it was shown that the property of convolution $y(t)$=$a$*$b$ becoming a multiplication after Fourier transform: $F$$(y(t))$= $F(a)F(b)...
- 791
3
votes
1
answer
672
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Exponential deconvolution using the first derivative
There is an interesting observation using the first derivative to deconvolve an exponentially modified Gaussian:
The animation is here at terpconnect.umd.edu.
The main idea is that if we have an ...
- 791
7
votes
2
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427
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What is $\int_{0}^{z} e^{-a^{2} x^{2}} {\rm erf}(bx)\, dx$?
The integral $$\int_{0}^{z} e^{-a^{2} x^{2}} {\rm erf}(bx)\, dx$$ is related to the convolution of two half-normal distributions. This can be inferred from this question on MSE. The following ...
- 4,655