$$ \begin{cases} &f_0(x)=1 \\ &f_{n+1}(x)= (\varphi*(f\mathbb{I}_{[a_n,b_n]}))(x)=\int_{-\infty}^{+\infty}\varphi(x-t)f_n(t)\mathbb{I}_{[a_n,b_n]}(t)dt = \int_{a_n}^{b_n}\varphi(x-t)f_n(t)dt \quad n = 0,...,N-1 \\ \end{cases} $$ with $\{a_n\}_{n=1,...,N},\{b_n\}_{n=1,...,N} \in \Bbb R$ and $\varphi(x)$ the standard gaussian function $$\varphi(x) = \frac{1}{\sqrt{2\pi}}\exp{\left(-\frac{x^2}{2} \right)}$$

The functions $f_{n+1}$ are defined as a convolution of $\varphi$ and $f_n\mathbb{I}_{[a_n,b_n]}$ for $n = 0,...,N-1$.

The goal is to compute the value $V$ as quick as possible. $$V=\int_{a_N}^{b_N}f_N(x)dx$$

Numerical absolute error is on the order of $\mathcal{O}(10^{-3})$ is sufficient.

This is some kind of multiple 1D Gaussian blur or multiple Gaussian filters.

I wonder whether this problem is known in signal processing field?

Thank you in advance!

  • $\begingroup$ Are you after a closed form solution or numerical solution? $\endgroup$
    – Thomas
    Jul 24 at 10:23
  • $\begingroup$ I'm seeking a numerical solution (a closed form solution, or even an approximate closed form solution, are better but I doubt that they exist). An acccuracy on the order of $\mathcal{O}(10^{-3})$ is sufficient. However, the algorithm should be as fast as possible because later I need to compute not less than one thousand million values $V$. $\endgroup$
    – NN2
    Jul 24 at 11:48
  • $\begingroup$ Have you tried one of the fastest Gaussian Blur implementations? $\endgroup$
    – Thomas
    Jul 31 at 9:04
  • $\begingroup$ @Thomas Thank you very much for the proposition of Gaussian Blur algorithms. I haven't used them yet but I'll try them. $\endgroup$
    – NN2
    Aug 18 at 18:37

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