# 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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### Factorizing and rearranging a 2-dimensional convolution integral

Let $t \in (0,T]$ (time) and $x \in \mathbb{R}$ (space), and let $f(t,x)$ and $g(t,x)$ be $L^2([0,T]\times \mathbb{R})$ (or perhaps stronger integrability is needed?). Let $H$ be the convolution of $f$...
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### Convolution and conditional density

I am learning convolution. Here is a problem about convolution of two conditional densities that I've been thinking. Suppose we have two random variables $T$ and $Z$, where $Z=X+Y$ and $X$,$Y$ are ...
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### Computing a 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 ...
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### How to efficiently solve a least squares problem involving Kronecker product and Tikhonov regularization

I have the following regularized least squares problem: $$\min_x \|y - Ax\|_2^2 + \lambda \|Dx\|_2^2,$$ where $y \in \mathbb{R}^m$, $x \in \mathbb{R}^n$, $A \in \mathbb{R}^{m \times n}$, $D$ is a ...
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### Identification of mix signatures/columns in a matrix

There's a Matrix X wholes several columns were chosen as basis to construct new columns by linear combination with positive coefficients (weighted mean). These new columns had been joined to X to form ...
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Suppose $$a_i = x_i+ b_i , \qquad i=1,\ldots,n,$$ where $a_1,\ldots,a_n \in \mathbb{R}$ are known, $x_i\geq 0$ is unknown and $b_1,\ldots,b_n \in \mathbb{R}$ are known to take one of the values $... 0answers 31 views ### Under what conditions can a distribution function be deconvolved with a particular kernel? Let$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$with a kernel that has also full support and ... 0answers 100 views ### Deconvolution of a mean-preserving spread Context I have been working on proving the existence of a mathematical object. After trying several things, I think that if I can show the following, an important step towards proving existence will ... 0answers 44 views ### Can we extract the signal back after convolution with orthogonal code? Assume we have a random signal$h$convoluted with another signal$swhich is assumed to be Walsh code represented by one column of Hadamard-matrix, i.e., $$s = \begin{bmatrix} 1\\ -1\\ -1\\ 1\end{... 0answers 403 views ### Approximation to the n-th derivative using reproducing kernels. For integrable functions defined on the real line, the normalized gaussian function approximates the convolution identity, Dirac Delta, in the sense that if$$g(t):=N_0e^{-x²}$$(denoting the ... 0answers 25 views ### Ensure properties in the result of discrete deconvolution I need to perform deconvolution to obtain information about a vector b. Let c=a*b the convolution operation (the inverse of our operation), I need to calculate b=c/a. The vectors are afflicted by ... 1answer 36 views ### Find effective inverse of Toeplitz matrix I would like to do a deconvolution of a noisy process.$$y_i = \sum_j k_{j-i} x_{j} + \nu$$where k is some well-behaved localized kernel (e.g. gaussian), and \nu is gaussian noise with zero mean ... 1answer 48 views ### Deconvolution with respect to a particular function Let \mathcal L, \mathcal L^*: \Theta \times \mathcal A \to \mathbb R be functions. When can \mathcal L be expressed as the convolution of \mathcal L^* with some third function U? That is, when ... 1answer 28 views ### Estimate Signal by Its Convolution by 2 Different Kernels I have a discrete Signal s that has been convoluted with two functions h_1 and h_2. I measure the result of this convolution:$$y_1=s*h_1, \quad y_2=s * h_2.$$I have a short time segment (for ... 0answers 162 views ### Linear deconvolution using FFT I want to deconvolve a filtered signal with a known input to recover the filter used using FFTs. Let x be a vector of length N and h a filter of length K where N > K. Let x \ast h = y,... 0answers 44 views ### How to calculate real pixel color from a blurred image using n equations in n unknowns? I've been dealing with a big image de-blurrying issue for past months and now I'm stuck with this issue that I want to get original sharp image from a blurred image by using some extra data and math. ... 0answers 46 views ### Convolution of two step functions Consider the probability distribution function$$ \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,...,\... 1answer 156 views ### 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 ... 0answers 127 views ### 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: \... 0answers 76 views ### 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 $$where Y \in \mathbb{R}^{n\... 0answers 24 views ### Verifying the results of Deconvolution using Residual Number System I have been reading the paper Exact Deconvolution Using Number Theoretic Transforms and I think I understand it. However, I am not able to verify the example given at the end of the paper. To ... 1answer 102 views ### Solving minimization problem L_2 IRLS (Iteration derivation) In the article ''' Chartrand, Rick, and Wotao Yin. "Iteratively reweighted algorithms for compressive sensing." Acoustics, speech and signal processing, 2008. ICASSP 2008. IEEE international ... 0answers 51 views ### Deconvolving the convolution of two identical functions I want to deconvolve p(x) from the following expression:$$f(x) = (p\cdot p)(x)$$where f(x) and p(x) are both real functions. Additionally, f(x): has odd symmetry \lim\limits_{x\... 0answers 26 views ### (deterministic) time-varying Gaussian filter I'm a math undergrad working on a psychology question: Assume a person subconsciously estimates a function x(t) where t is time since hearing a beep. However, with time, their estimate of time t ... 0answers 224 views ### Convolution theorem for generalized functions The standard convolution theorem says \mathcal{F}(f*g)=\mathcal{F}(f)\mathcal{F}(g), where f and g are both functions.However, it still works for some generalized function, for example, when f ... 0answers 98 views ### Is this convolution product reversible? I am doing an exercise on Fourier transforms and i have the following questions which i really tried to solve myself Let k > 0 and consider F_k(x) a function with real values where$$... 0answers 23 views ### On the positiveness of a convolution product I've the following question: Assume thatf(t),g(t)\in L^2[0;+\infty)$and$f(t)$is causal and positive$\forall t > 0$. Consider the convolution operation $$\int_0^{+\infty} f(\tau)g(t-\tau) d\... 0answers 77 views ### Discorrespondence between Continous Fourier Transform and Discrete Fourier Transform My goal is to use a deconvolution method to extract a desired signal (delta peak) out of a convoluted measured function. My problem at first concerns the discrete Fourier Transform (DFT) or FFT ... 0answers 819 views ### How to convolve a periodic signal with an aperiodic signal? Basically, when there are two periodic signals, say x(t) and h(t) which are to be convolved, then convolution is carried out over a range of their common time period (which is equal to the least ... 0answers 63 views ### How to get sampled H_i for deconvolution in frequence space (fourier space) I have a function f(t) : [0, 2 \pi] \rightarrow {\Bbb R}. This function is sampled on N points (equidistant in interval [0, 2 \pi], getting the discretized function f_i, i = 1, .., N. The ... 0answers 305 views ### Deconvolution and Curve Fitting I have a function$$g(x) = (f \star f) (x)$$, where \star denotes convolution. g(x) is a piece-wise quadratic polynomial function whose exact closed-form formula I know. I want to deconvolve g(x)... 0answers 172 views ### Fredholm equation with unknown Kernel / De-convolution I'm wondering if the following integral equation has any hope of an algebraic solution: \frac{2}{(x-2)^2}=\int_0^{\frac{1}{2}} f(x-s) f(s) \, ds, where f(\cdot) is unknown. This is a Fredholm-... 0answers 149 views ### Fourier Convolution Inversion Consider a Fourier convolution f(x) = (g * h)(x), where g and h are arbitrary but known functions with reasonable properties. Is there any possibility to determine the inverse function of this ... 0answers 598 views ### The Deconvolution Integral The standard 1D continuous convolution integral is defined as:$$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(... 0answers 67 views ### 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$I_i$: $$S_A = W_A +... 0answers 67 views ### Method of Moment Estimator for Deconvolution The distributions Y, X, Z and W are related as follows:$$Y_1 = X + ZY_2 = X + W,$$that is X (random variable) is a common factor to the random variables Y_1 and Y_2, which ... 1answer 414 views ### Flipped Point Spread Function I was reading on wikipedia about the Lucy-Richardson algorithm and its equivalent iterative function:$$ u^{(t+1)}=u^{(t)}\cdot \Big(\frac{d}{u^{(t)}\otimes p}\otimes \hat{p}\Big)$$where d is the ... 0answers 181 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 ... 0answers 905 views ### 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 ... 1answer 208 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 ... 1answer 247 views ### 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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### 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 &...
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### 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 ...