1Advanced Computer Graphics Advanced Computer Graphics (Fall 2009)(Fall 2009)CS 294, Rendering Lecture 9: Frequency Analysis and Signal Processing for RenderingRavi Ramamoorthihttp://inst.eecs.berkeley.edu/~cs294-13/fa09MotivationMotivation Signal-processing provides new understanding Methods based on (Spherical) Fourier analysis Allows understanding of sampling rates (in IBR) Frequency-domain algorithms like convolution This lecture high-level, mostly conceptual ideas. Follow original papers for details, applicationsPlenopticPlenopticSamplingSampling Plenoptic Sampling. Chai, Tong, Chan, Shum 00 Signal-processing on light field Minimal sampling rate for antialiased rendering Relates to depth range, Fourier analysis Fourer spectra derived for 2D light fields for simplicity. Same ideas extend to 4DSiggraph’2000, July 27, 2000A Geometrical IntuitionZminZoptCamera i Camera i+1Siggraph’2000, July 27, 2000A Geometrical IntuitionZminZoptCamera i Camera i+1Disparity Error <1 PixelRendering CameraSiggraph’2000, July 27, 2000A Constant PlaneZvttvZ1tvZ12Siggraph’2000, July 27, 2000Two Constant PlanesZvtZ1Z2tvtvZ1Z2Siggraph’2000, July 27, 2000Between Two PlanesZvttvZ1tvZ1Z2Z2Siggraph’2000, July 27, 2000Between Two PlanesZvttvZ1tvZ1Z2Z2Siggraph’2000, July 27, 2000Light Field ReconstructionSiggraph’2000, July 27, 2000Minimum Sampling CurveJoint Image andGeometry SpaceJoint Image andGeometry SpaceMinimum Sampling CurveMinimum Sampling CurveNumber of Depth Layers123612Accurate DepthNumber of Images2x28x84x416x1632x32Frequency Analysis and Sheared Reconstruction for Rendering Motion BlurFrequency Analysis and Sheared Reconstruction for Rendering Motion BlurKevin EganYu-Ting TsengNicolas HolzschuchFrédo DurandRavi RamamoorthiColumbia UniversityColumbia UniversityINRIA -- LJKMIT CSAILUniversity of California, Berkeley3ObservationObservation• Motion blur is expensive• Motion blur removes spatial complexityBasic ExampleBasic Example• Object not movingxySPACEf(x, y) f(x, t)Space-time graphTIMEBasic ExampleBasic Examplexytxf(x, t)• Low velocity, t [ 0.0, 1.0 ]f(x, y)Basic ExampleBasic Examplexytxf(x, t)• High velocity, t [ 0.0, 1.0 ]f(x, y)Shear in Space-TimeShear in Space-Timexytxf(x, t)• Object moving with low velocityf(x, y)shearShear in Space-TimeShear in Space-Timexytx• Object moving with high velocityf(x, y) f(x, t)4Shear in Space-TimeShear in Space-Time• Object moving away from cameraxytxf(x, y) f(x, t)Basic ExampleBasic Example• Applying shutter blurs across timexytxf(x, y) f(x, t)Basic Example – Fourier DomainBasic Example – Fourier Domain• Fourier spectrum, zero velocity txf(x, t)F(Ωx, Ωt)texture bandwidthΩtΩxBasic Example – Fourier DomainBasic Example – Fourier Domain• Low velocity, small shear in both domainsf(x, t) F(Ωx, Ωt)txslope = -speedΩtΩxBasic Example – Fourier DomainBasic Example – Fourier Domain• Large shearf(x, t) F(Ωx, Ωt)txΩtΩxBasic Example – Fourier DomainBasic Example – Fourier Domain• Non-linear motion, wedge shaped spectraf(x, t)ΩtΩxF(Ωx, Ωt)txshutter bandlimits in time-min speed-max speedshutter applies blur across timeindirectly bandlimits in space5Sampling in Fourier DomainSampling in Fourier DomainΩtΩxtx+• Sampling produces replicas in Fourier domain• Sparse sampling produces dense replicasFourier DomainPrimal DomainStandard Reconstruction FilteringStandard Reconstruction Filtering• Standard filter, dense sampling (slow)Ωtno aliasingΩxFourier DomainreplicasStandard Reconstruction FilterStandard Reconstruction Filter• Standard filter, sparse sampling (fast)ΩtFourier DomainaliasingΩxSheared Reconstruction FilterSheared Reconstruction Filter• Our sheared filter, sparse sampling (fast)ΩtΩxNo aliasing!Fourier DomainSheared Reconstruction FilterSheared Reconstruction Filter• Compact shape in Fourier = wide in primaltxPrimal DomainΩtΩxFourier DomainCar SceneCar SceneStratified Sampling4 samples per pixelOur Method,4 samples per pixel6Teapot SceneTeapot SceneOur Method8 samples / pixmotion blurred reflectionReflection as ConvolutionReflection as Convolution My PhD thesis (A signal-processing framework for forward and inverse rendering Stanford 2002) Rewrite reflection equation on curved surfaces as a convolution with frequency-space product form Theoretical underpinning for much work on relighting (next lecture), limits of inverse problems Low-dimensional lighting models for LambertianAssumptionsAssumptions Known geometry Convex curved surfaces: no shadows, interreflection Distant illumination Homogeneous isotropic materialsLater precomputed methods: relax many assumptionsReflectionReflection22id()iLLightingLi()oBReflected Light Field(, )ioBRDFBoReflection as Convolution (2D)Reflection as Convolution (2D)22id()iLLighting()oBReflected Light Field(, )ioBRDFLioBiBLLoReflection as Convolution (2D)Reflection as Convolution (2D)22id()iLLighting()oBReflected Light Field(, )ioBRDF()iL(, )ioid22(, )oBLioBLiBo7Reflection as Convolution (2D)Reflection as Convolution (2D)()iL(, )ioid22(, )oBLioBLiBoConvolutionConvolutionxSignal f(x)Filter g(x)Output h(u)uConvolutionConvolutionxSignal f(x)Filter g(x)Output h(u)u11() ( )()hu gx u f x dxu1ConvolutionConvolutionxSignal f(x)Filter g(x)Output h(u)uu222() ( )()hu gx u f x dxConvolutionConvolutionxSignal f(x)Filter g(x)Output h(u)uu333() ( )()hu gx u f x dxConvolutionConvolutionxSignal f(x)Filter g(x)Output h(u)u() ( ) ()hu gx u f x dxhfggfFourier analysishfg8Reflection as Convolution (2D)Reflection as Convolution (2D)BLFrequency: productSpatial: integral()iL(, )ioid22(, )oB,,2lp l lpBLFourier analysisR. Ramamoorthi and P. Hanrahan “Analysis of Planar Light Fields from Homogeneous Convex Curved Surfaces under Distant Illumination” SPIE Photonics West 2001: Human Vision and Electronic Imaging VI pp 195-208LiBoLioBSpherical HarmonicsSpherical Harmonics-1-2 0 1 2012...(, )lmYxyzxyyz231z
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