I have a lot of points that I want to calculate an exponential distance falloff value for, to be used in a subsurface rendering application. However, if I can perform hierarchical clustering and only evaluate this for each level it can be calculated a lot faster. To simplify, I am trying to approximate an integral of the type:
$p(x,\sigma) = \int_0^1 \exp\left(\frac{-|x-s|}{\sigma}\right) \text{d}s $
using the integral:
$q(x,\sigma) = \exp\left( -\frac{\int_0^1 |x-s| \text{d} s}{\sigma}\right) $
Where $\sigma$ would be a scattering distance, x the current location, s the location of a point in the volume. However, I cannot seem to figure out how to relate these two values correctly, as for small scattering distances there is a huge difference between them, but for long scattering distances they're about equal. I am trying to find a function $f(x,\sigma) = \frac{p(x,\sigma)}{q(x,\sigma)}$.
Any help is greatly appreciated
Edit: It seems the simplification is not helping the problem, so I'll try to be more concrete at the risk of it becoming harder to explain. I have a volume in 3d space, somewhat uniformly filled with points. The average position of all these points can be calculated once and saved for the volume. Then for any position in space I want to calculate:
$$ \mathbf{V} = \{v_1,v_2,v_3,\dots,v_n\} $$ $$ p(x) = \frac1n\sum_{i=1}^n \exp\left( -||x-v_i||_2\cdot \sigma^{-1} \right) $$
Of course this takes n operations to calculate. I thought perhaps I could perform a rough approximation using the average location of the points and the borders of the volume. Perhaps I should just use the minimum distance from x to any point in the volume.