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Let $T$ be a compact, connected, proper subset of $\mathbb{R}^3:\quad T \subset \mathbb{R}^3$.

Further let $\left\{ \boldsymbol{\mu}_i \right\}_{i=1}^n$ be a given finite set of $n$ point in $T$: $$ \left\{ \boldsymbol{\mu}_i \right\}_{i=1}^n \subset T $$

I would like to approximate continuous uniform distribution $u(\boldsymbol{x})$ over $T$ with mixture of Gaussians $g(\boldsymbol{x})$, for which means of components are given by the set $\left\{ \boldsymbol{\mu}_i \right\}_{i=1}^n$, that is: $$ g(\boldsymbol{x}) = \sum_{i=1}^{n} \pi_{i} \mathcal{N}(\boldsymbol{x} \,|\, \boldsymbol{\mu}_{i},\Sigma_{i}) $$

We can assume that, informally speaking, means $\left\{ \boldsymbol{\mu}_i \right\}_{i=1}^n$ are "equispaced" over $T$ or "nice" enough such that good approximation exists.

Any ideas on this?

The way I want to attack this is by minimizing some sort of computationally tractable divergence (Kullback–Leibler divergence, for example) between $u$ and $g$.

I'm curious if something similar was attempted before (searching the web didn't yield anything)?

Or could you point me to the relevant literature on this? Basically, any suggestions would be of help.

Motivation: this problem arises as an inverse problem in radiotherapy treatment planning. $T$ is considered to be a tumor and needs to receive radiation dose that is uniformly distributed over $T$. Dose spillage outside $T$ is undesirable. In this case the problem of designing such treatment can be reduced to estimating values of weights $\; \pi_1,\, \dots,\, \pi_n\;$ and covariance matrices $\; \Sigma_1,\, \dots,\, \Sigma_n$.

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  • $\begingroup$ Is the set $T$ in the first line the same of the set $S$? $\endgroup$ – Liviu Nicolaescu Mar 17 '17 at 18:42
  • $\begingroup$ @LiviuNicolaescu Yes. It was a typo I made. Please, see edit. $\endgroup$ – aberdysh Mar 17 '17 at 19:37
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For arbitrary points $x_1,\dotsc, x_n$ this may not be possible. For example, if the convex hull of the finite set $\{x_1,\dotsc, x_n\}$ dos not contain the barycenter of $X$, then this is not possible.

Here is a possible alternate solution. Choose a sequence of points $p_1,p_2,\dotsc, p_n, \dotsc $ in $T$ such that

$$\lim_{n\to\infty} \frac{1}{n}\Big(\, \delta_{p_1}+\cdots +\delta_{p_n}\,\Big) =\mu_T, $$

where $\mu_T$ denotes the uniform probability distribution on $T$, and $\delta_p$ denotes the Dirac delta concentrated at $p$. (The Strong Law of Large numbers guarantees the existence of such a sequence.)

For any $r>0$ and $p\in\mathbb{R}^3$ we denote by $\Gamma_r(p)$ the Gaussian measure on $\mathbb{R}^3$ centered at $p$ and covariance matrix $rI$. It is known that

$$\lim_{r\to 0}\Gamma_r(p)=\delta_p. $$

Let $r_n= n^{-100}$ so $r_n\to 0$ very very fast as $n\to\infty$. Then I believe that

$$ \lim_{n\to\infty} \frac{1}{n}\Bigg(\, \Gamma_{r_n}(p_1)+\cdots +\Gamma_{r_n}(p_n)\,\Bigg) =\mu_T. $$

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My suggestion would be to first determine a 3D delaunay triangulation of the point set; then determine a B-Spline basis on that triangulation and finally calculate for each B-Spline basis the best-approximating Gaussian.

B-Spline basis functions over triangulation are well studied and have nice properties like compact support and, constitute to a partition of unity.

Being a partition of unity resembles the constant density of the tumor in case of your problem; the spline approach allows however also an approximate solution for tumors with non-constant density.

This article on Expo-Rational B-Splines may also be of interest.

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