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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.

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How does scaling in frequency domain affect real space?

I have a 3 dimensional array of real data corresponding to measurements in physical 3D space, and its corresponding data in spectral space. I want to scale certain specific frequencies in the ...
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Convolution of two step functions

Consider the probability distribution function $$ \Delta(x; \lambda, \mu)=\sum_{j=1}^J \lambda_j 1\{x\geq \mu_j\} \hspace{1cm} \forall x \in \mathbb{R} $$ where $\lambda\equiv (\lambda_1,...,\...
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Recovering original image from its edges

Suppose we read an image $X$ with $1\times P$ dimensions (a single row and $P$ columns) and apply to it the simplest edge detector, that calculates the horizontal derivative say, $F = [1, 0, −1]$ to ...
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Deconvolution of accumulated values

Introduction I have an unknown function of time $f(t)$ that I would like to learn based on experimental observations. I have an observable $g(t)$, which, to the best of my knowledge, is given by the ...
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Convolution of functions with disjoint supports

Here is the question: Given an arbitrary function $f(t)$, what would be sufficient condition so that $f(t)$ can be deconvoluted into two functions with disjoint support?
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How can I slove those multivalue nonlinear equations with n+m-1 equations and n+m variables?

Does this following system(1) of 4 simultaneous equations in 5 variables $x_1,x_2,y_1,y_2,y_3$ and 4 constants $c_1,c_2,c_3,c_4$ have general solution? (It's simillar to A system of nonlinear ...
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Deconvolution of $(u * u^*)$. What information can I recover about $u$?

I have data corresponding to $(u * u^*)(x)$, where $u(x)$ is a complex function of a real variable. I'm looking to recover as much information about the features of $u$ or even $|u|^2$. For example, ...
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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 theorem: \...
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Regression with embedded convolution

I have a problem where the data I am getting is a convolution of the original data with some function and I am trying to solve the following equation for $A$ $$ Y = AX $$ where $Y \in \mathbb{R}^{n\...
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Verifying the results of Deconvolution using Residual Number System

I have been reading this paper and I think I understand it. However, I am not able to verify the example given at the end of the paper. To summarize, given matrices ...
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Solving minimization problem $L_2$ IRLS (Iteration derivation)

In the article ''' Chartrand, Rick, and Wotao Yin. "Iteratively reweighted algorithms for compressive sensing." Acoustics, speech and signal processing, 2008. ICASSP 2008. IEEE international ...
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Deconvolving the convolution of two identical functions

I want to deconvolve $p(x)$ from the following expression: $$f(x) = (p\cdot p)(x)$$ where $f(x)$ and $p(x)$ are both real functions. Additionally, $f(x)$: has odd symmetry $\lim\limits_{x\...
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(deterministic) time-varying Gaussian filter

I'm a math undergrad working on a psychology question: Assume a person subconsciously estimates a function $x(t)$ where $t$ is time since hearing a beep. However, with time, their estimate of time $t$ ...
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114 views

Convolution theorem for generalized functions

The standard convolution theorem says $\mathcal{F}(f*g)=\mathcal{F}(f)\mathcal{F}(g)$, where $f$ and $g$ are both functions.However, it still works for some generalized function, for example, when $f$ ...
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Is this convolution product reversible?

I am doing an exercise on Fourier transforms and i have the following questions which i really tried to solve myself Let $k > 0$ and consider $F_k(x)$ a function with real values where $$...
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On the positiveness of a convolution product

I've the following question: Assume that $f(t),g(t)\in L^2[0;+\infty)$ and $f(t)$ is causal and positive $\forall t > 0$. Consider the convolution operation $$\int_0^{+\infty} f(\tau)g(t-\tau) d\...
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Discorrespondence between Continous Fourier Transform and Discrete Fourier Transform

My goal is to use a deconvolution method to extract a desired signal (delta peak) out of a convoluted measured function. My problem at first concerns the discrete Fourier Transform (DFT) or FFT ...
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How to convolve a periodic signal with an aperiodic signal?

Basically, when there are two periodic signals, say x(t) and h(t) which are to be convolved, then convolution is carried out over a range of their common time period (which is equal to the least ...
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How to get sampled $H_i$ for deconvolution in frequence space (fourier space)

I have a function $f(t) : [0, 2 \pi] \rightarrow {\Bbb R}$. This function is sampled on $N$ points (equidistant in interval $[0, 2 \pi]$, getting the discretized function $f_i$, $i = 1, .., N$. The ...
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207 views

Deconvolution and Curve Fitting

I have a function $$g(x) = (f \star f) (x)$$, where $\star$ denotes convolution. $g(x)$ is a piece-wise quadratic polynomial function whose exact closed-form formula I know. I want to deconvolve $g(x)...
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Fredholm equation with unknown Kernel / De-convolution

I'm wondering if the following integral equation has any hope of an algebraic solution: $\frac{2}{(x-2)^2}=\int_0^{\frac{1}{2}} f(x-s) f(s) \, ds$, where $f(\cdot)$ is unknown. This is a Fredholm-...
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Fourier Convolution Inversion

Consider a Fourier convolution $f(x) = (g * h)(x)$, where $g$ and $h$ are arbitrary but known functions with reasonable properties. Is there any possibility to determine the inverse function of this ...
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280 views

The Deconvolution Integral

The standard 1D continuous convolution integral is defined as: $$y(t) = h(t)*x(t) = \int^{+\infty}_{-\infty}h(\tau)\cdot x(t-\tau)\ d\tau$$ Using fourier transform, $$Y(j\omega) = X(j\omega)\cdot H(...
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Source estimation for identification of anomalous events

I’m stuck on the following problem. There are two sources $S_A$ and $S_B$ at the ends of a channel. Both are made up of a white noise component $W_i$ plus an impulsive component $I_i$: $S_A = W_A + ...
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Method of moment estimator. 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 ...
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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 ...
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122 views

Deconvolution and Polynomial factoring using the FFT

I've been trying to implement a general N dimensional deconvolver for various engineering applications and some math curiosities. For speed and simplicity I've decided to try and do this with help of ...
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Curve Fitting Including Convolution in MATLAB

I would like to fit two parameters $K_1$ and $k_2$ in the problem $f(t)*C_a(t) = C_E(t)$ where $*$ represents the convolution operator and $f(t) = K_1 e^{-k_2 t}$. $C_a(t)$ and $C_E(t)$ are given ...
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131 views

Deconvolution of two delta functions (solving $y(t) = A x(t-a) + B x(t-b)$)

I would like to calculate $x(t)$, when only $y(t)$ with $y(t) = A x(t-a) + B x(t-b)$ is known. Since this is a linear shift invariant operation (convolution), the inverse relation must be of the ...
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Convolution/Deconvolution $\stackrel{?}{=}$ Coding/Decoding

In a strict mathematical sens, can a convolution/deconvolution be equivalent to a coding/decoding process ? I just got the remark from a reviewer that it's strictly different, it's a little surprising ...
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282 views

Pseudo-inverse of an underdetermined Toeplitz matrix

I have an undetermined Toeplitz matrix (more columns than rows). For example: \begin{equation*} T = \begin{pmatrix} 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 ...
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781 views

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 ...
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240 views

Decomposition of exponential random variable

I know that sum of independent Exponential random variables follows Gamma distribution. But Is it possible to decompose ...
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How to choose a phase for the deconvolution of an autocorrelation?

Say I have a function, $C=C\left(x\right)$, whose fourier transform is denoted by $c=c\left(k\right)$, i.e. $C\left(x\right)=\sum_{k=-\infty}^{\infty}c\left(k\right)\chi\left(x\right)$, where $\chi\...