Questions tagged [convolution]

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3
votes
3answers
223 views

Convolute a gaussian kernel with a large array of off-grid centroids without looping? (how to make "A Thousand (Gaussian) Points of Light" )

For a finite object size diffraction simulator, I need to generate arrays which are the sum of thousands of instances of a Gaussian (or other) 2D kernel at centroids that will not fall in any ...
1
vote
1answer
143 views

How to generate the convolution of f(x, y) with a parametric function g(t), x(t), y(t) in Python? (Something better than this brute-force sum)

The answer to Convolute a gaussian kernel with a large array of off-grid centroids without looping? (how to make "A Thousand (Gaussian) Points of Light" ) involves summing a 3D array over ...
0
votes
1answer
45 views

Compute efficiently a 1D function relying on a 2D convolution

Let $X = [0,1]$, $h$ the Gaussian function (i.e. $\forall x \in X, h(x) = e^{-\frac{x}{2}}$) and $p \in L^2(X^2)$ I would like to compute numerically the following function : $$ \forall x \in X, \...
0
votes
0answers
290 views

How to take convolution of two arrays in Python by using NumPy?

Generally, we know that if we have this relation between Fourier transforms of three functions in frequency domain as: $$\mathfrak{F}\{\mathsf{P}(t)\} = \mathfrak{F}\{\mathsf{Z}(t)\}\mathfrak{F}\{\...
3
votes
1answer
141 views

Problem implementing convolutions exactly with the FFT

I'm trying to perform convolutions as defined mathematically $f \star g (\tau)= \int_{\mathcal{R}}f(t-\tau)g(t) dt$ in a numerical simulation. Hence, my signal is a sampling of points $f(x_i)$. I ...
1
vote
2answers
214 views

Convolution in Python

I have an integral of a convolution between two functions. How can I calculate this in Python? It is a continuum convolution.
4
votes
1answer
155 views

Computation of triple nested loops as a convolution product?

I'm trying to compute efficiently the following \begin{equation} A_j = \sum_{l'=1}^{\infty}\sum_{k= 0}^{K-1} L_{l'}T_ke^{2\pi i \frac{k}{K}j}\epsilon_{l',k} \end{equation} for $j = 0,1, \ldots, K-2,K-...
5
votes
0answers
92 views

Levinson Recursion for Non Square Toeplitz Matrices

Given a rectangular Toeplitz Matrix $ H $, how could one solve: $$ y = H x $$ For instance, $ H $ can be Linear Convolution Matrix of the filter $ h $: $$ H = \begin{bmatrix} {h}_{1} & 0 & ...
2
votes
0answers
36 views

Computing convolution of two characteristic function over a 1D Cartesian mesh

I am trying to compute the convolution of two characteristic functions over a Cartesian mesh. First, I define my Cartesian mesh of the interval $[0,1]$ as follows $$ x_{i} = i \Delta x, i = 0, 1, 2\...
1
vote
0answers
37 views

FFT convolution works only with certain domain length

in my quest to understand how I can use FFT to compute integrals (see my other question click, still no answer there), I came across the fact that a convolution of two functions can be calculated by ...
1
vote
1answer
45 views

Accelerating Conjugate Gradients fitting for small localized kernel (like cubic B-spline)

Question: Is there some pre-conditioner for Conjugate-Gradient (CG) cheap enough, that it is worth using even if my operator is very local (i.e. already has a low number of non-zero elements), as it ...
3
votes
1answer
144 views

Calculating the Convolution Using DFT (FFT)

I have the following convolution as part of a numerical simulation. $$T(r)=\int \mathrm{d}^3r_2\, p(r_2)f(r_2)\alpha(r-r_2)\, .$$ My problem is that the analytical expressions for $f$ and $p$ do ...