資源描述:
《球面調(diào)和函數(shù).pptx》由會(huì)員上傳分享,免費(fèi)在線閱讀,更多相關(guān)內(nèi)容在教育資源-天天文庫(kù)。
1、Sphericalharmonics(球面調(diào)和函數(shù))陳方2013.08.13Outline球面調(diào)和函數(shù)的定義球面函數(shù)的分解基函數(shù)的系數(shù)的求解球面調(diào)和函數(shù)的應(yīng)用—光照球面調(diào)和函數(shù)的定義(1/4)球面調(diào)和函數(shù)的定義(2/4)Technically:球面調(diào)和函數(shù)的定義(3/4)Conceptually:球面調(diào)和函數(shù)的定義(4/4)Visually:m=-3m=-2m=-1m=0m=1m=2m=3球面函數(shù)的分解(1/4)Givenasphericalfunctionf,wecanobtainarotationinvariantrepresentationbyexpressingfinterms
2、ofitssphericalharmonicdecomposition:whereeachfl?Vl:where:球面函數(shù)的分解(2/4)SphericalHarmonicDecomposition++=+球面函數(shù)的分解(3/4)++=++++Constant1stOrder2ndOrder3rdOrder球面函數(shù)的分解(4/4)+++Constant1stOrder2ndOrder3rdOrderΨ基函數(shù)的系數(shù)的求解(1/9)G(x)...functiontorepresentB1(x),B2(x),…Bn(x)…basisfunctionsG(x)isalinearcombinati
3、onofbasisfunctionsStoringafinitenumberofcoefficientscigivesanapproximationofG(x)基函數(shù)的系數(shù)的求解(2/9)sumofscaledbasisfunctions基函數(shù)的系數(shù)的求解(3/9)Howtofindcoefficientsci?-MinimizeanerrormeasureWhaterrormeasure?-L2error基函數(shù)的系數(shù)的求解(4/9)MinimizingEL2leadstoWhere基函數(shù)的系數(shù)的求解(5/9)Givenabasis{Bi(x)}-ComputematrixB-Compu
4、teitsinverseB-1GivenafunctionG(x)toapproximate-Computedotproducts-Computecoefficientsas基函數(shù)的系數(shù)的求解(6/9)OrthonormalbasismeansIfbasisisorthonormalthen基函數(shù)的系數(shù)的求解(7/9)Ifthebasisisorthonormal,computationofapproximationcoefficientssimplifiesto基函數(shù)的系數(shù)的求解(8/9)OriginalfunctionBasisfunctionsCoefficients基函數(shù)的系數(shù)的
5、求解(9/9)Intergralofproduct=dotproductofcoefficientsf(x)=fiBi(x)g(x)=giBi(x)?f(x)g(x)dx=figi球面調(diào)和函數(shù)的應(yīng)用—光照(1/2)球諧光照用新的光照方程來(lái)替代通常所說(shuō)的光照方程,把方程中的信息使用球諧基函數(shù)投射到頻域空間中,在渲染的時(shí)候可以使用SH系數(shù)來(lái)對(duì)原始光照方程還原渲染場(chǎng)景球面調(diào)和函數(shù)的應(yīng)用—光照(2/2)謝謝!