Because compressing a spherical function sampled in cubemap form requires us to do 2D Haar Wavelet transform 6 times, one for each cubemap face, to simplify this transformation step, Wang et al.  propose using a single 2D octahedral map to represent a spherical function. So by using octahedral map to store our per-vertex visibility, we only need to do Haar Wavelet transform (# of vertices) times, whereas cubemap representation requires (6 x # of vertices) times.
In my implementation, I simply use the converting method provided in  and  to convert 2D coordinates in an octahedral map to 3D direction vectors.
The following screenshot shows the visibility cubemap of a vertex, and the octahedral map converted from that cubemap.
It’s not easy to imagine what the transformation looks like on a black and white image; I also made a screenshot showing the converted environment cubemap.
1. Rui Wang, Ren Ng, David Luebke, Greg Humphreys, Efficient Wavelet Rotation for Environment Map Rendering, Eurographics Symposium on Rendering (2006)
2. Cigolle, Donow, Evangelakos, Mara, McGuire, Meyer, A Survey of Efficient Representations for Independent Unit Vectors, Journal of Computer Graphics Techniques (JCGT), vol. 3, no. 2, 1-30, 2014
3. Quirin Meyer, Jochen Süßmuth, Gerd Sußner, Marc Stamminger, and Günther Greiner. 2010. On floating-point normal vectors. In Proceedings of the 21st Eurographics conference on Rendering