We introduce the reader to the mathematics behind projection of n-dimensional vectors into a basis on n-dimensional space, where n can be anything upto and including infinity. We show how ideally one wants to project into the duals of a basis if this basis is not orthonormal, and provide the mathematics to formulate this operation in matrix form. The second part of the article discusses spheric...

In this paper we present the first practical method for importance sampling functions represented as spherical harmonics (SH). Given a spherical probability density function (PDF) represented as a vector of SH coefficients, our method warps an input point set to match the target PDF using hierarchical sample warping. Our approach is efficient and produces high quality sample distributions. As a...

In these notes I will describe the separation of variable technique for solving Laplace’s equation, using spherical polar coordinates. The solutions will involve Legendre polynomials for cases with azimuthal symmetry, and more generally they will involve spherical harmonics. I will construct these solutions using traceless symmetric tensors, but in Lecture Notes 8 I describe how the solutions i...

Figure 1 compares our mixture of spherical Gaussians (SG) basis to spherical harmonics (SH) and zonal harmonics (ZH) [Sloan et al. 2005]. A simple 2-lobe fit with our basis provides good accuracy; error decreases further with more lobes. Asymptotic error decrease in our model compares favorably with increasing order of spherical harmonics and with increasing order and number of lobes in the ZH ...

Our concern is with the construction of a frame in L 2 (S) consisting of smooth functions based on kernels of spherical harmonics. The corresponding decomposition and reconstruction algorithms utilize discrete spherical Fourier transforms. Numerical examples connrm the theoretical expectations. x1. Introduction Traditionally, wavelets were tailored to problems on the Euclidean space IR d. Howev...

In this paper, we deal with the problem of spherical interpolation of discretely given data of tensorial type. To this end, spherical tensor elds are investigated and a decomposition formula is described. It is pointed out that the decomposition formula is of importance for the spectral analysis of the gra-vitational tensor in (spaceborne) gradiometry. Tensor spherical harmonics are introduced ...

The spherical harmonics are the angular portion of the solution to the Laplace equation in spherical coordinates and provide a frequency-basis for representing functions on the sphere. The spherical harmonic lighting, as defined by Robin Green at Sony Computer Entertainment in 2003, is a family of real-time rendering techniques that may produce certain realistic shading and shadowing with relat...