# Quasi-random sequence on the unit $2$-sphere for Monte-Carlo based method

Please forgive my possible misuse of the appropriate definitions. I'm looking for a quasi-random sequence of directions in the unit $2$-sphere, to be used in a Monte-Carlo method to calculate an integral over the entire $4 \pi$ solid angle.

I'm currently using a Monte-Carlo approach for the calculation of global illumination for a 3d application, using pseudo-random numbers from a standard software package. The thing is that the convergence is too slow. I've read about it in Wikipedia, and found out that by using quasi-random sequences (like the Sobel sequence) I could improve the convergence speed from $\frac{1}{\sqrt{N}}$ to $\frac{1}{N}$. Is this correct?

NOTE: Please if there is some typing error in the math, I'll appreciate an editor's help.

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## 1 Answer

Wouldn't a standard quasi-random sequence on a rectangle (eg Sobol) followed by a area-preserving mapping from the rectangle to the sphere (eg Gall-Peters) do the trick? Surely, we don't have -a priori- guarantees that a good quasi-random sequence will preserve its good properties after the mapping, but it sounds reasonable to expect so.

Another recipe (again, which only sounds reasonable) would be to generate generate the quasi-random sequence in 3D, inside the enclosing cube; if the sample falls outside the sphere, discard it and go for the next; elsewhere, project it over the sphere surface.

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I'll go with the second choice. I already did it once for pseudo-random vectors in the unit sphere. Hope it works. Thanks! –  Alejandro Piad Feb 8 '13 at 20:02