This post is going to start off describing what SZ Sequences are, and why I think the 4D shuffled-scrambled SZ Sequence should be the default sample sequence many people turn to when implementing something like a path-tracer. Then I’ll point to the open-source code that has them in Using SZ Sequences in Your Renderer.

After that, I suspect the rest of the post will only be interesting to maybe a dozen people. But it’s also where the real contribution is: the New Matrix Construction Algorithm section defines a procedure that’s much simpler and faster than Ahmed et al’s paper. And at the very end, in Hierarchical Faure-Niederreiter Sequences, I’ll expand the construction to other prime bases. The SZ Sequence is just the base-2 HFN Sequence.

Ever since I first heard of spherical harmonics as an approximation in computer graphics, I have felt that the way they are explained is too complicated. Sometimes the explanations assume a bunch of math knowledge that isn’t really necessary, and they tend to be very abstract. Other times they show something complex and through a lot of effort end up simplifying them to something usable. In addition to making it harder to learn, I think those approaches to SH can lead to misapplication. SH introductions often have discussion about certain SH artifacts (ringing and negative values), but ignore another class of common issues.

The Problem

Say you have an array whose length is a power of some base b. For most of this post we’ll be using powers of two, but we’ll discuss other bases as well. Imagine that there is a b-ary tree over the array. We want to shuffle the array in the following way: randomly shuffle the b sub-trees of the root. Then, recurse into each subtree, and do the same, until we get to the bottom.

That subtitle is a little bit of a mouthful, but I’ve been wanting to talk about Andrew Kensler’s permute() function for a while and I wanted to make it clear right off the bat why it’s useful. permute() is used some in the graphics community, especially in offline rendering, and the techniques are more widely known in cryptography*, but it’s applicable to so much more than graphics and cryptography.