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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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 $ ...
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21 views

Solving set of sets of mutually exclusive equations derived from convolution and max-pooling

For a research project, I am trying to reconstruct an image by its convoluted and max-pooled result. The result and the kernel weights of various convolution filters are known. I thought to reduce ...
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25 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 ...
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87 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 ...
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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{...
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20 views

Practical implementation of Wiener deconvolution

I am currently trying to reduce noise on my output using Wiener deconvolution, as this is the simplest approach I found. I however lack the Mathematical background in this particular field, so I am ...
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24 views

The convolution of two functions in finding h=g*g and H(k)

I understand that the convolution of two functions f and g is defined to be h = f ∗ g. If I have $g(x)=e^{-|x|}$ How would I find the function $h=g*g$ as well as that the fourier transform is $H(k)=\...
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390 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 ...
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27 views

Deconvolution of probability density functions

I am trying to implement a code that yields the probability density function $p(f)$, given knowledge of two other PDFs, $p(\omega)$ and $p(\theta)$. The whole procedure is based on the paper ...
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19 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 ...
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1answer
25 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 ...
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19 views

How do iterative deconvolution equations account for noise?

I have been working with the Richardson-Lucy deconvolution method and I notice that in matlab, there is an option that allows you to specify readout noise in the input. The richardson-Lucy ...
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37 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 ...
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19 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 ...
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97 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 $x \ast h = y$,...
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35 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. ...
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44 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,...,\...
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146 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 ...
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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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72 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\...
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22 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 ...
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76 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 ...
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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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22 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 $t$ ...
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182 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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74 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 $$...
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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\...
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72 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 ...
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647 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 ...
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60 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 ...
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275 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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156 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-...
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134 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 ...
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479 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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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 +...
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64 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 ...
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1answer
336 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 ...
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154 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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850 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 ...
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1answer
177 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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1answer
221 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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319 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 &...
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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 ...
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259 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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1answer
138 views

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\...
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3answers
5k views

Exact deconvolution of two matrices using numerical techniques

Suppose that I am given two $n \times m$ matrices $\bf{A}$ and $\bf{C}$, and let $\bf{B}$ be a matrix that is convolved with $\bf{A}$, such that: $\bf{A} * B = C$ In the above, $*$ is the ...