球面调和函数ppt课件

Sphericalharmonics(球面调和函数),陈方2013.08.13,Outline,球面调和函数的定义球面函数的分解基函数的系数的求解球面调和函数的应用光照,球面调和函数的定义(1/4),球面调和函数的定义(2/4),Technically：,球面调和函数的定义(3/4),Conceptually：,球面调和函数的定义(4/4),Visually:,m=-3m=-2m=-1m=0m=1m=2m=3,球面函数的分解(1/4),Givenasphericalfunctionf,wecanobtainarotationinvariantrepresentationbyexpressingfintermsofitssphericalharmonicdecomposition:whereeachflVl:,where:,球面函数的分解(2/4),球面函数的分解(3/4),+,+,=,+,+,+,+,Constant,1stOrder,2ndOrder,3rdOrder,球面函数的分解(4/4),+,+,+,Constant,1stOrder,2ndOrder,3rdOrder,基函数的系数的求解(1/9),G(x).functiontorepresentB1(x),B2(x),Bn(x)basisfunctionsG(x)isalinearcombinationofbasisfunctions,StoringafinitenumberofcoefficientscigivesanapproximationofG(x),基函数的系数的求解(2/9),sumofscaledbasisfunctions,基函数的系数的求解(3/9),Howtofindcoefficientsci?-MinimizeanerrormeasureWhaterrormeasure?-L2error,基函数的系数的求解(4/9),MinimizingEL2leadsto,Where,基函数的系数的求解(5/9),GivenabasisBi(x)-ComputematrixB-ComputeitsinverseB-1GivenafunctionG(x)toapproximate-Computedotproducts-Computecoefficientsas,基函数的系数的求解(6/9),OrthonormalbasismeansIfbasisisorthonormalthen,基函数的系数的求解(7/9),Ifthebasisisorthonormal,computationofapproximationcoefficientssimplifiesto,基函数的系数的求解(8/9),基函数的系数的求解(9/9),Intergralofproduct=dotproductofcoefficients,球面调和函数的应用光照(1/2),球谐光照用新的光照方程来替代通常所说的光照方程，把方程中的信息