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Code Generation and Factoring for Fast
Evaluation of Low-order Spherical
Harmonic Products and Squares
May 1, 2006
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Download Document
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BibTex
Authors
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John Snyder
Publication Type
TechReport
Pages
9
Number
MSR-TR-2006-53
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Abstract
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Related Info
Abstract
We present a method for fast evaluation of spherical harmonic (SH) products or, more generally,
any binary product of vectors yielding a vector, where the product is governed by a fixed, sparse,
symmetric, order 3 tensor, denoted Γijk. The method is given the nonzero entries of Γ as input
(they can be computed analytically or by numerical integration for the SH basis) and makes use
of an offline code generator to perform the necessary array indexing using constants rather than
variables. Factoring is performed by collecting the tensor’s nonzero components, represented
by index triples (i,j,k), into groups (i,j,k1), (i,j,k2), …,(i,j,kNij) which share a common pair of
indices (i,j) in the triple, and which vary only in the third (completion) index km and its
corresponding coefficient dm = Γijkm where m ∈ \1,2,…,Nij$. The collection is done using a
greedy method that successively chooses the index pair (i,j) maximizing the number Nij of
different km needed to complete the tensor’s nonzero index triples. The greedy method then
continues to the next best initial pair, generates its contribution, and so on, until all nonzero
triples have been accounted for. The combination of “greedy pair― factoring and
generating constant array indices produces code that is significantly faster than naïve
evaluation methods.
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 Mathematics
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 Microsoft Research Lab - Redmond
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