spherical harmonics gdce
DESCRIPTION
Spherical Harmonics for beginnersTom Forsyth presentationTRANSCRIPT
What they are…
• Representation of an image over a sphere– Something like a cube-map
• Not like a cube-map:• Frequency-space representation
– Spherical equivalent of post-Fourier-transform data– A bit like a JPEG, but a 2D sphere, not a 2D plane
• Continuous, rotationally invariant– No sharp edges, no rotational “lumpiness”
• Easy for vector units to manipulate
What they are not…
• Not an algorithm– Just a data representation
• Not Precomputed Radiance Transform– That is an algorithm that happens to use SHs
• Not actually that complex– Just the maths language barrier– Read the Ramamoorthi & Hanrahan paper – it
makes sense eventually!
The Basics• “Looking up a direction in the cube-map”• A set of N basis functions – used by all SHs
– Components of the direction being looked up– 1, X, Y, Z, XY, YZ, ZX, X2-Y2, 1-3Z2,…
• A set of N weights – this stores the data– Each weight is usually an RGB colour
• Multiply each basis function by its weight• And sum – result is the looked-up value• Number of basis functions goes on for ever, but
first 9 is good enough for diffuse lighting
Features of Harmonics
• Smooth over the sphere– No seams or “hot spots”
• Easy to reconstruct in vector unit• Rotation loses no data, has no aliasing
– Rotate then convert to SH is identical to convert to SH then rotate
• Basis functions used are orthonormal– To add two SHs, add their weights– To multiply two SHs, multiply their weights
Irradiance Concept• Ignore SH for now – think of cubemaps• Representing incoming distant lights• Representing the diffuse cosine response
– Cosine response = max(0,N.L), summed for all lights• Only normals matter, not positions• Draw response to single light on cubemap
– The “picture” of a lit sphere• Add together each light’s cubemap• To shade any object, look up normal on the
resulting cubemap
Irradiance Concept
Irradiance Concept
Irradiance Concept
Using SH
• Final cubemap is smooth, low frequency• Encode as SH weights• Feed weights to vertex unit• Vertex unit shades normals with SH• 9 RGB values to feed to vertex unit rather
than a cubemap• 9 weights represents diffuse shading with
maximum 6% error
Using SH Better
• But wait! Why use a cubemap at all?• Go from light description to SH weights• Adding two SHs = add weights together• So:
– For each light, find the SH weights– Sum weights– Feed final weights to vertex unit– No cubemaps anywhere!
CPU Code per Object per Lightvoid AddLightToSH ( Colour *sharm, Colour &colour, Vector &dirn ){
sharm[0] += colour * fConst1;sharm[1] += colour * fConst2 * dirn.x;sharm[2] += colour * fConst2 * dirn.y;sharm[3] += colour * fConst2 * dirn.z;sharm[4] += colour * fConst3 * (dirn.x * dirn.z);sharm[5] += colour * fConst3 * (dirn.z * dirn.y);sharm[6] += colour * fConst3 * (dirn.y * dirn.x);sharm[7] += colour * fConst4 * (3.0f * dirn.z * dirn.z - 1.0f);sharm[8] += colour * fConst5 * (dirn.x * dirn.x - dirn.y * dirn.y);
}
Vertex Processor Codeconst Colour sharm[9];Colour LightNormal ( Vector &dirn ){
Colour colour = sharm[0];colour += sharm[1] * dirn.x;colour += sharm[2] * dirn.y;colour += sharm[3] * dirn.z;colour += sharm[4] * (dirn.x * dirn.z);colour += sharm[5] * (dirn.z * dirn.y);colour += sharm[6] * (dirn.y * dirn.x);colour += sharm[7] * (3.0f * dirn.z * dirn.z - 1.0f);colour += sharm[8] * (dirn.x * dirn.x - dirn.y * dirn.y);return colour;
}
In Practice
• Find brightest n lights (n=1 to 3 usually)• Shade those conventionally
– Specular, bumpmapped, self-shadowing, etc• Rest of lights accumulated into SH• Last real light fades between real and SH
– Provides smooth transition as object moves
Gotchas
• Only vertex diffuse lighting– No bumpmaps (PS2.0 shaders can!)– No specular, only diffuse– So do brightest 2 lights normally
• Only directional lights– Point lights must be approximated– Discontinuities on abutting objects (walls, etc)
Cool Stuff
• Take (HDR) skybox, convert to SH offline– Add to all computed SHs– Perfect sunset lighting– Artists can tweak by drawing on bitmaps!
• Presampled “radiosity”– Pick some sample points in environment– Render cubemaps at those points, convert to SH– At runtime, for any object, linearly interpolate between
nearest sample point SHs.– Gives lighting off walls, through doorways, etc
Demo
• Compare many real lights to just 2– Too dark – lights are discarded.
• Compare many real lights to 2 + ambient– Too washed out
• Compare many real lights to just SH– Pretty good – slight differences
• Compare many lights to 2 real + SH– Excellent, plus bumpmapping works
Questions
Conclusions
• SH is excellent for images on a sphere• SH irradiance makes lights cheap• Standard lighting has per-vertex cost• Using SHs converts it to a per-object cost
– Per-vertex cost is constant• Artists can put down far more lights
– And the game is still faster! (10-20% on PS2)
References
• Ramamoorthi & Hanrahan– “An Efficient Representation for Irradiance
Environment Maps”• Peter-Pike Sloan
– Siggraph 2002 & 2003 papers• Robin Green
– “Spherical Harmonic Lighting: The Gritty Details”• Google for “spherical harmonic irradiance”
– Without “irradiance” you get lots of chemists