Monte Carlo ILighting and Soft ShadowsPenumbras and UmbrasMonte Carlo LightingMonte Carlo AlgorithmsRandom VariablesDiscrete Probability DistributionsContinuous Probability DistributionsSampling Continuous DistributionsExample: Power FunctionSampling a CircleSlide 12Rejection MethodsSampling a Circle: RejectionMonte Carlo IntegrationOver Arbitrary DomainsNon-Uniform DistributionsUnbiased EstimatorDirect Lighting – Directional SamplingDirect Lighting – Area SamplingExamplesSlide 22Slide 23Slide 24Slide 25VarianceSlide 27Sampling Projected Solid AngleSlide 29Variance ReductionSampling a TriangleSlide 32CS348B Lecture 6 Pat Hanrahan, Spring 2005Monte Carlo IPrevious lectureAnalytical illumination formulaThis lectureNumerical evaluation of illuminationReview random variables and probabilityMonte Carlo integration Sampling from distributionsSampling from shapesVariance and efficiencyCS348B Lecture 6 Pat Hanrahan, Spring 2005Lighting and Soft ShadowsChallenges Visibility and blockers Varying light distribution Complex source geometry2( ) ( , )cosiHE x L x dw q w=�Source: Agrawala. Ramamoorthi, Heirich, Moll, 2000CS348B Lecture 6 Pat Hanrahan, Spring 2005Penumbras and UmbrasCS348B Lecture 6 Pat Hanrahan, Spring 2005Monte Carlo Lighting1 eye ray per pixel1 shadow ray per eye rayFixed RandomCS348B Lecture 6 Pat Hanrahan, Spring 2005Monte Carlo AlgorithmsAdvantagesEasy to implementEasy to think about (but be careful of statistical bias)Robust when used with complex integrands and domains (shapes, lights, …)Efficient for high dimensional integralsEfficient solution method for a few selected pointsDisadvantagesNoisySlow (many samples needed for convergence)CS348B Lecture 6 Pat Hanrahan, Spring 2005Random Variables is chosen by some random process probability distribution (density) functionX~ ( )X p xCS348B Lecture 6 Pat Hanrahan, Spring 2005Discrete Probability DistributionsDiscrete events Xi with probability piCumulative PDF (distribution)Construction of samples To randomly select an event, Select Xi if0ip �11niip==�1i iP U P-< �ip1jj iiP p==�U10Uniform random variable3XiPCS348B Lecture 6 Pat Hanrahan, Spring 2005Continuous Probability DistributionsPDF CDF10( ) Pr( )P x X x= <Pr( ) ( )( ) ( )X p x dxP Pbaa bb a� � == -�( )p x10Uniform( ) 0p x �0( ) ( )xP x p x dx=�(1) 1P =( )P xCS348B Lecture 6 Pat Hanrahan, Spring 2005Sampling Continuous DistributionsCumulative probability distribution function Construction of samplesSolve for X=P-1(U)Must know:1. The integral of p(x)2. The inverse function P-1(x)UX10( ) Pr( )P x X x= <CS348B Lecture 6 Pat Hanrahan, Spring 2005Example: Power FunctionAssume( ) ( 1)np x n x= +1110011 1nnxx dxn n+= =+ +�1( )nP x x+=11~ ( ) ( )nX p x X P U U-+� = =1 2 1max( , , , , )n nY U U U U+= L111Pr( ) Pr( )nniY x U x x++=< = < =�TrickCS348B Lecture 6 Pat Hanrahan, Spring 2005Sampling a Circle12 1 1 22200 0 0 002rA r dr d r dr dp ppq q q p� �= = = =� �� ��� � �( , ) ( ) ( )p r p r pq q=2( ) 2( )p r rP r r==12 Uq p=rdqdr1( )21( )2pPqpq qp==1( , ) ( , )rp r dr d r dr d p rq q q qp p= � =2r U=CS348B Lecture 6 Pat Hanrahan, Spring 2005Sampling a Circle122 Ur Uq p==122 Ur Uq p==RIGHT Equi-ArealWRONG Equi-ArealCS348B Lecture 6 Pat Hanrahan, Spring 2005Rejection MethodsAlgorithmPick U1 and U2Accept U1 if U2 < f(U1)Wasteful? 10( )( )y f xI f x dxdx dy<==���( )y f x=Efficiency = Area / Area of rectangleCS348B Lecture 6 Pat Hanrahan, Spring 2005Sampling a Circle: RejectionMay be used to pick random 2D directionsCircle techniques may also be applied to the spheredo {X=1-2*U1Y=1-2*U2while( X2+ Y2 >1 )CS348B Lecture 6 Pat Hanrahan, Spring 2005Monte Carlo IntegrationDefinite integralExpectation of fRandom variablesEstimator10( ) ( )I f f x dx��10[ ] ( ) ( )E f f x p x dx��~ ( )iX p x( )i iY f X=11NN iiF YN==�CS348B Lecture 6 Pat Hanrahan, Spring 2005Over Arbitrary DomainsbadxxfI )(abxpUba1)(11)()( babaabxpxPbadxxpxfabI )()()()()( fEabI badxxpxfIE )()()(fabI )( )(1)(1iNiXfNabIabab 1CS348B Lecture 6 Pat Hanrahan, Spring 2005Non-Uniform DistributionsbadxxfI )(badxxpxpxfI )()()()()()(xpxfxg badxxpxgI )()()(gEI NiiXgNI1)(1CS348B Lecture 6 Pat Hanrahan, Spring 2005Unbiased Estimator11 1110110101[ ] [ ]1 1[ ] [ ( )]1( ) ( )1( )( )NN iiN Ni ii iNiNiE F E YNE Y E f XN Nf x p x dxNf x dxNf x dx== ===== ====�� ������[ ] ( )NE F I f=Assume uniform probability distribution for now[ ] [ ]i ii iE Y E Y=� �[ ] [ ]E aY aE Y=PropertiesCS348B Lecture 6 Pat Hanrahan, Spring 2005Direct Lighting – Directional Sampling2( ) ( , ) cosHE x L x dw q w=�*( ( , ), ) cos 2i i i iY L x x w w q p= -*( , )x x wRay intersectionA�w*xxqSample uniformly byWwCS348B Lecture 6 Pat Hanrahan, Spring 2005Direct Lighting – Area Sampling22cos cos( ) ( , ) cos ( , ) ( , )oAHE x L x d L x V x x dAx xq qw q w w��� � � �= =�-� �2cos cos( , ) ( , )i ii o i i iiY L x V x x Ax xq qw�� � �=�-A�w�x�xqx xw� �= -Ray directionSample uniformly byA�x�0( , )1visibleV x xvisible���=��q�CS348B Lecture 6 Pat Hanrahan, Spring 2005Examples4 eye rays per pixel1 shadow ray per eye rayFixed RandomCS348B Lecture 6 Pat Hanrahan, Spring 2005Examples4 eye rays per pixel16 shadow rays per eye rayUniform grid Stratified randomCS348B Lecture 6 Pat Hanrahan, Spring 2005Examples4 eye rays per pixel64 shadow rays per eye rayUniform grid Stratified randomCS348B Lecture 6 Pat Hanrahan, Spring 2005Examples4 eye rays per pixel100 shadow rays per eye rayUniform grid Stratified randomCS348B Lecture 6 Pat Hanrahan, Spring 2005Examples4 eye rays per pixel16 shadow rays per eye ray64 eye rays per pixel1 shadow ray per eye rayCS348B Lecture 6 Pat Hanrahan, Spring 2005VarianceDefinitionPropertiesVariance decreases with sample size22 22 2[ ] [( [ ]) ][ 2 [ ] [ ] ][ ] [ ]V Y E Y E YE Y YE Y E YE Y E Y� -= - += -[ ] [ ]i ii iV Y V Y=� �2[ ] [ ]V aY a V Y=21 11 1 1[ ] [ ] [ ]N Ni ii iV Y V Y V YN N N= == =� �CS348B Lecture 6 Pat Hanrahan, Spring 2005Direct Lighting – Directional Sampling2( ) ( , ) cosHE x L x dw q w=�*( ( , ), ) cos 2i i i iY L x x w w q p= -*( ( , ), )i i iY L x x w w p= -*( , )x x wRay intersectionA�w*xxqSample uniformly byWwSample uniformly byW%wCS348B Lecture 6 Pat Hanrahan, Spring 2005Sampling Projected Solid
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