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# google / spherical-harmonics

Spherical harmonics library

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# 3D Graphics-oriented Spherical Harmonics Library

Spherical harmonics can be a tricky thing to wrap your head around. Even once the basic theories are understood, there's some surprisingly finicky implementation work to get the functions coded properly. This is especially true when it comes to rotations of spherical harmonics (much of the literature is math-dense and contains errata). Additionally, different literature sources use slightly different conventions when defining the basis functions.

This library is a collection of useful functions for working with spherical harmonics. It is not restricted to a maximum order of basis function, using recursive definitions for both the SH basis functions and SH rotation matrices. This library uses the convention of including the Condon-Shortely phase function ((-1)^m) in the definition of the basis function.

This is not an official Google project.

Dependencies

This library depends on Eigen3 to for its underlying linear algebra primitives. Colors are represented as `Eigen::Array3f`, where the components are ordered red, green, and blue. Google Test is used for unit testing.

The Bazel build tool is used to build the library. This is responsible for downloading and configuring Eigen3 and the testing framework. You may build the library by executing in the root directory:

``````bazel build //sh:spherical_harmonics
``````

General Functions

See documentation in `sh/spherical_harmonics.h` for details on specific functions. `sh/image.h` provides a very generic and simple image interface that can be used to adapt this library with any actual imaging toolkit already in use.

Core SH Functions

`EvalSH` - Evaluate the SH basis function of the given degree and order at the provided position on a unit sphere. The position is described as either a unit vector or as spherical coordinates.

`EvalSHSum` - Evaluate the approximation of a spherical function that has already been converted to a vector of basis function coefficients.

Projection Functions

Used to estimate coefficients applied to basis functions to approximate complex spherical functions as a weighted sum of the spherical harmonic basis functions. Once projected, the returned coefficients can be passed into `EvalSHSum`.

`ProjectFunction` - Project an analytic spherical function into every basis function up to the specified order. This uses Monte Carlo integration to estimate the coefficient for each basis function.

`ProjectEnvironment` - Project an environment map image arranged in a latitude-longitude projection into the basis functions up to the specified order. This is a specialization of `ProjectFunction` that is more efficient when the spherical function is described as an image containing an environment.

`ProjectSparseSamples` - Project a spherical function that has only been sparsely evaluated (i.e. 10-50 times). Unlike the analytic function, this uses a least-squares fitting to best estimate the coefficients for each basis function. This works well when fitting to photographic data where there can only be so many photos captured.

Diffuse Irradiance Functions

Diffuse irradiance can be efficiently represented in low-order spherical harmonics. It can be computed quickly by estimating the standard diffuse cosine-lobe as a vector of coefficients, and the environment as spherical harmonics. Diffuse irradiance is simply the dot product of the two coefficient vectors.

`RenderDiffuseIrradiance` - Compute diffuse irradiance for a given unit normal vector and SH coefficients that describe the environment illumination (i.e. from `ProjectEnvironment`).

`RenderDiffuseIrradianceMap` - Compute diffuse irradiance for every normal vector described by the texels of the provided latitude-longitude image. This can be useful for computing a texture map of diffuse irradiance and then transferring it to the GPU for shader-based rendering.

Spherical Harmonic Rotations

If a complex spherical function is rotated, and a set of spherical harmonic coefficients is needed for this new function, it's possible to rotate the spherical harmonic coefficients of the original approximation rather than re-projecting the rotated function. This is often much more efficient and is used in `RenderDiffuseIrradiance` to transform the cosine lobe function for the unit z-axis to any other normal vector.

`Rotation` - Object type that computes the transformation matrices that suitably transform spherical harmonic coefficients given a quaternion rotation.

Utility Functions

`GetCoefficientCount` - Return the total number of coefficients needed to represent all basis functions up to a given order.

`GetIndex` - Return a 1-dimensional index (suitable for accessing the returned vectors from all the project functions) given a degree and order.

`ToVector` - Transform spherical coordinates into a unit vector.

`ToSphericalCoords` - Transform a unit vector into spherical coordinates.

`ImageXToPhi` - Transform a pixel's x coordinate in an image of a specific width to the phi spherical coordinate.

`ImageYToTheta` - Transform a pixel's y coordinate in an image of a specific height to the theta spherical coordinate.

`ToImageCoords` - Transform spherical coordinates into floating-point image coordinates given particular image dimensions. The coordinates can be used to bilinearly interpolate an environment map, or cast to integers to access direct pixels.

Literature The general spherical harmonic functions and fitting methods are from [1], the environment map related functions are based on methods in [2] and [3], and spherical harmonic rotations are from [4] and [5]:

1. R. Green, "Spherical Harmonic Lighting: The Gritty Details", GDC 2003, http://www.research.scea.com/gdc2003/spherical-harmonic-lighting.pdf
2. R. Ramamoorthi and P. Hanrahan, "An Efficient Representation for Irradiance Environment Maps",. , P., SIGGRAPH 2001, 497-500
3. R. Ramamoorthi and P. Hanrahan, “On the Relationship between Radiance and Irradiance: Determining the Illumination from Images of a Convex Lambertian Object,” J. Optical Soc. Am. A, vol. 18, no. 10, pp. 2448-2459, 2001.
4. J. Ivanic and K. Ruedenberg, "Rotation Matrices for Real Spherical Harmonics. Direct Determination by Recursion", J. Phys. Chem., vol. 100, no. 15, pp. 6342-6347, 1996. http://pubs.acs.org/doi/pdf/10.1021/jp953350u
5. Corrections to [4]: http://pubs.acs.org/doi/pdf/10.1021/jp9833350

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