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.

Filter by
Sorted by
Tagged with
0
votes
0answers
16 views

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 $f$...
0
votes
0answers
16 views

calculate or decompose a Fourier transform signal amplitudes with unknown weights on sources

I am trying to calculate , or approximate the solution of following Fourier-sine transform problem that can be expressed as a contributions of periodic sources $f_i(x)$ and weights $a_i(x)$ : $$F(k)...
0
votes
0answers
21 views

Multiplying a Fourier space quotient by its denominator to remove its effect

I am materials science graduate student without a strong mathematics background. I have been trying to determine if I am reasoning correctly about a mathematical operation (related to the convolution ...
2
votes
0answers
45 views

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 ...
0
votes
0answers
21 views

How regularization term turn ill-possed to well possed problem?

Lets say we have following non-linear convolution equation of F and X with V noise $$ Y = F \ast X + V $$ to solve inverse of it, we use following equation with L2 norm $$ X = arg min ||F \ast X-Y||^...
0
votes
0answers
15 views

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 ...
1
vote
0answers
33 views

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 ...
0
votes
0answers
60 views

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 ...
0
votes
0answers
15 views

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 ...
0
votes
0answers
32 views

a special class of matrix decomposition

$$ \newcommand{\Mat}{\boldsymbol} \newcommand{\Set}{\mathcal} \newcommand{\real}{\mathbb{R}} \newcommand{\complex}{\mathbb{C}} $$ My question starts from solving the following equation: $$ \Mat{A}^2 \...
1
vote
0answers
57 views

Extracting a Function from Inside an Integral

I have an imaging system that produces a signal in a pixel, in part, by responding to incident radiance $L(\lambda)$ that passes through a filter with transmission $T(\lambda)$. Lumping all the other ...
6
votes
2answers
533 views

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 ...
0
votes
1answer
45 views

Lower bound and deconvolution

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 $ ...
2
votes
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 ...
1
vote
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 ...
0
votes
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 $s$ which is assumed to be Walsh code represented by one column of Hadamard-matrix, i.e., $$s = \begin{bmatrix} 1\\ -1\\ -1\\ 1\end{...
1
vote
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
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 ...
2
votes
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 ...
0
votes
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 $x \ast h = y$,...
1
vote
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. ...
1
vote
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,...,\...
0
votes
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 ...
4
votes
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: \...
0
votes
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\...
0
votes
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 ...
0
votes
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 ...
0
votes
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\...
1
vote
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 $t$ ...
2
votes
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 $f$ ...
1
vote
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 $$...
1
vote
0answers
23 views

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\...
0
votes
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
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)...
1
vote
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-...
1
vote
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 ...
1
vote
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(...
2
votes
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 +...
3
votes
0answers
67 views

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 ...
2
votes
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
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 ...
2
votes
0answers
328 views

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 &...
5
votes
2answers
1k 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 ...