Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [deconvolution]

For questions on deconvolution, the resolution of a convolution function into the functions from which it was formed in order to separate their effects.

Deconvolution is the resolution of a convolution function into the functions from which it was formed in order to separate their effects.

63 questions
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4 answers

Determine $f(x)$ knowing $f(x)+f(x+\varepsilon)$

I encountered a problem at work that, in my opinion, has a fundamental mathematical reasoning to determine its solvability. Due to an unwitting software configuration, my associate recorded the audio of our meeting twice, resulting in the audio…
nickh
  • 485
6
votes
2 answers

How to obtain a solution for the following IBVP

I am trying to solve the following advection-diffusion equation for transient flow conditions for radial flow. The governing equation is as follows. $$\frac{\partial T}{\partial t} = \frac{\partial^2 T}{\partial x^2} + \frac{1-2v(t)}{x}…
Engineer
  • 127
5
votes
2 answers

Decomposition of exponential random variable

I know that sum of independent Exponential random variables follows Gamma distribution. But Is it possible to decompose exponential random variate into independent and identically gamma random variates?
ABC
  • 1,477
5
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0 answers

Solving for a function inside a convolution

I have this relationship: \begin{align} \frac{1}{|x|}=f(x)*f(x)\ , \end{align} where $*$ denotes the convolution. I want to solve for $f(x)$. My first instinct was to apply the convolution…
user85503
  • 741
4
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2 answers

Can FFT be adapted for deconvolution of non-periodic functions?

Can a non-periodic function be padded at the boundaries and deconvolved with inverse FFT? Since a Toeplitz matrix can be embedded in a circulant matrix to perform the deconvolution, is there an analogy with the inverse FFT? Clarification Not sure…
hatmatrix
  • 263
3
votes
0 answers

Pseudo-inverse of a fat Toeplitz matrix

I have a fat Toeplitz matrix, say, \begin{equation*} T = \begin{pmatrix} 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\ …
hatmatrix
  • 263
3
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2 answers

How to invert a "tent" convolution?

Consider the following convolution $$ f(x)=(k * g)(x)=\int_{-\infty}^{\infty}k(t)g(x-t) \,{\rm d} t $$ where $g>0$, and $k$ is the "tent" function $$ k(t) = (1 - |t|)\chi_{[-1,1]}(t) $$ where $\chi$ is the characteristic function. Is this transform…
sam wolfe
  • 3,465
3
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0 answers

Finding inverse of self-convolution

I am interested in any insight on inverting a self-convolution. In other words, given a function $f : [0,L] \to \Bbb R$, find a function $g : [0, \frac{L}{2}] \to \Bbb R$ such that $$f = g \ast g$$ The context I am most immediately interested in is…
PatrickComposerPhysicist
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Gaussian deconvolution for rapidly decreasing functions.

Gaussian convolution with variance $v$ is defined as $$ {\cal G}_v[f](x):=\int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi v}}f(y) e^{-\frac{(y-x)^2}{2v}}dx. $$ Given a function $g$, does there exist a a bounded continuous function $f$ to satisfy $${\cal…
Masahito Hayashi
  • 127
3
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1 answer

Computing the Least Squares Least Norm Solution to Image Deconvolution

I want to deconvolve an image $h$ by a kernel $f$. More precisely, let $$G = \operatorname*{argmin}_g \|f \ast g - h\|_2$$ be the set of least-squares solutions. I want to find the least-norm solution in this set: $$ \hat{g} \in…
user76284
  • 6,408
3
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0 answers

Under what conditions can a distribution function be deconvolved with a particular kernel?

Let $X$ be a random variable that has full support and is continuously distributed on $\mathbb{R}$ according to the density $f$. I want to "deconvolve" $f$ with a kernel that has also full support and is continuously distributed on $\mathbb{R}$…
pegasus
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Method of Moment Estimator for Deconvolution

The distributions $Y, X, Z$ and $W$ are related as follows: $$Y_1 = X + Z$$ $$Y_2 = X + W,$$ that is $X$ (random variable) is a common factor to the random variables $Y_1$ and $Y_2$, which otherwise differ for the terms $Z$ and $W$ (random…
Mic
  • 31
3
votes
1 answer

Flipped Point Spread Function

I was reading on wikipedia about the Lucy-Richardson algorithm and its equivalent iterative function: $$ u^{(t+1)}=u^{(t)}\cdot \Big(\frac{d}{u^{(t)}\otimes p}\otimes \hat{p}\Big) $$ where d is the observed pixel, $u^{(t)}$ is the current iteration,…
Woody1193
  • 161
2
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0 answers

When does an exponential deconvolution exist

Say we observe the CDF $F_X(x)$ of the random variable $X$. Is it possible to tell whether $X$ can be deconvoluted as $X = Y + e$, where $e$ is a random variable with a standard exponential distribution $exp(1)$ (and where $e$ and $Y$ are…
Wilbur
  • 35
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Fourier transform of [something]*Gaussian

It has been 20 years since failed to understand Fourier transforms during my formal education. I am trying to solve an optical problem, and it looks as though the Fourier transform should be helpful. I have been reading up, but I don't understand…
W_vH
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