To make this work on a sphere, we must instead make our Gaussian a function of the angle between two unit direction vectors. This part of the function essentially makes the Gaussian a function of the cartesian distance between a given point and the center of the Gaussian, which can be trivially extended into 2D using the standard distance formula. The part that we need to change in order to define the function on a sphere is the “(x - b)” term. A 1D Gaussian function always has the following form: Since an SG is defined on a sphere rather than a line or plane, it’s parameterized differently than a normal Gaussian. It ends up looking like this:Ī Spherical Gaussian visualized on the surface of a sphere It ends up looking like what you would get if you took the above graph and revolved it around its axisĪ Gaussian filter applied to a 2D image of a white dot, showing that the impulse response is effectively a Gaussian function in 2DĪ Spherical Gaussian still works the same way, except that it now lives on the surface of a sphere instead of on a line or a flat plane. If you’re having trouble visualizing that, imagine if you took the above image and wrapped it around a sphere like wrapping paper. You’re probably also familiar with how it looks in 2D, since it’s very commonly used in image processing as a filter kernel. This produces the characteristic “hump” that you see when you graph it:Ī Gaussian in 1D centered at x=0, with a height of 3 If you’re reading this, then you’re probably already familar with how a Gaussian function works in 1D: you compute the distance from the center of the Gaussian, and use this distance as part of a base-e exponential. What’s a Spherical Gaussian?Ī Spherical Gaussian, or “SG” for short, is essentially a Gaussian function that’s defined on the surface of a sphere. However it is my hope that the material here will be sufficient to gain a basic understanding of SG’s, and also use them in practical scenarios. I should point out that this article is still going to be somewhat high-level, in that it won’t provide full derivations and background details for all formulas and operations. The concepts introduced here will serve as the core set of tools for working with Spherical Gaussians, and in later articles I’ll demonstrate how you can use those tools to form an alternative for approximating incoming radiance in pre-computed lightmaps or probes. In this article, I’m going cover the basics of Spherical Gaussians, which are a type of spherical radial basis function (SRBF for short). In the previous article, I gave a quick rundown of some of the available techniques for representing a pre-computed distribution of radiance or irradiance for each lightmap texel or probe location. Part 5 - Approximating Radiance and Irradiance With SG’s In that case, the 'volume element' of a '1D sphere' would be just 'dx' and there is no 'surface area element' since the surface is not continuous. If so by '1D sphere' he must an interval which has two points as 'surface'. Part 4 - Specular Lighting From an SG Light Source I think lostidentity is using 'sphere' to mean 'ball' and '2D sphere' to mean 'disk'. Part 3 - Diffuse Lighting From an SG Light Source Part 1 - A Brief (and Incomplete) History of Baked Lighting Representations This is part 2 of a series on Spherical Gaussians and their applications for pre-computed lighting.
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