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UCSD CSE 168 - Lecture

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CSE168Computer Graphics II, RenderingSpring 2006Matthias ZwickerFinal project• Project description due Wednesday May 24Last time• Irradiance cachingGlobal illumination[Wann Jensen]Indirect irradiance[Wann Jensen]Indirect irradiance[Humphreys, Pharr]Irradiance caching• Assume diffuse surfaces• Cache irradiance samples instead of incident radiance as in photon mapping• Interpolate cached samples• Compute new samples only if interpolation failsIrradiance caching algorithmThree components• Irradiance sampling• Irradiance caching• Irradiance interpolation• Similar to photon mapping, but all steps are performed in main rendering passIrradiance sampling• Assign a range for each sample, within which it can be used for interpolation• Where irradiance changes quickly, range should be small• Where irradiance changes slowly, range should be large• Rate of change of irradiance depends on distance to visible surfacesIrradiance sampling[Wojciech Jarosz]Irradiance caching•Store samples in octree• Add sample to each cell that it overlaps• Adaptively subdivide octree such that each cell has limited number of samplesIrradiance caching examples[Wann Jensen]1000 sample rays, w>10Irradiance caching examples[Wann Jensen]1000 sample rays, w>10Irradiance caching examples[Wann Jensen]5000 sample rays, w>10Irradiance caching examples[Wann Jensen]5000 sample rays, w>10Course recap• Part I: Implementing a basic ray tracer• Part 2: Physics of light transport•Part 3: Advanced topicsCourse recapPart I: Implementing a basic ray tracer• Generating primary rays• Ray-surface intersection•Shading• Acceleration structures•TexturingCourse recapPart 2: Physics of light transport• Radiometry• Reflection equation• Monte Carlo integration• Rendering equation• Path tracing• Photon mapping• Irradiance cachingOutlookPart 3: Advanced topics• Sampling and aliasing• Realistic camera models and HDR imaging• Participating media and subsurface scattering• Radiosity• Guest lecture on visualization and VRTodaySampling and aliasing• Introduction• Fourier analysis• AntialiasingIntroduction• Conceptually, an image is a continuous signal describing the radiance arriving at the image plane• Signal is synonymous for functionIntroduction• Digital images are sampled representations of these continuous signals• Sampled means “defined only at discrete locations”, pixel centersIntroduction• Sampling: How to compute the value of the sampled pixel• Reconstruction: How to obtain a continuous image from a set of samples Continuous pixel Sampled pixelSampleReconstructAliasing• Aliasing occurs because of sampling and reconstructionAliasingAliasingAliasing• Moire patternsAliasingSufficiently sampledInsufficiently sampled[R. Cook ]Sampling and aliasing• Is it possible to perfectly sample and reconstruct an image?• If yes, under what circumstances?Continuous pixel Discrete pixelSampleReconstructFourier analysis• All periodic signalscan be representedas a summation ofsinusoidal waves[http://axion.physics.ubc.ca/341-02/fourier/fourier.html]Fourier analysis• The Fourier transform computes the amplitude and phase of the sinusoidal wave at each frequencyFrequencyComplex amplitudeFourier analysisInverse Fourier transformFourier transform• Each signal can be represented in the spatial or the frequency domainFourier analysis• Band-limited signal, no frequencies above a certain thresholdSpatial domainFrequency domain,power spectrumFourier transform exampleSpatial domainFrequency domainDualitySpatialdomainFrequencydomainsincbox ⎯→←FDualityboxsinc ⎯→←FFourier transform examplecosine (even)cos(-ωt) =cos(ωt)sine (odd)sin(-ωt) =-sin(ωt)Spatial domainFrequencydomainDirac delta function• Definition•Sifting propertyDirac delta functionSpatial domainFrequencydomainDirac deltaδ(x)Fourier transformT/20πω=Impulse train∑−=kTkTxx )()(IIIδ∑−=kk )(21)(III000ωωδωπωωImpulse train(shah, comb function)ΙΙΙΤ(x)PeriodPeriod TSpatial domainFrequencydomainConvolution• In the spatial domainConvolution kernel, filterFiltered signalConvolution• In the spatial domainConvolution• In the spatial domain• Corresponds to multiplication in the spatial domain• Convolution leads to the same result as Fourier transform, multiplication, inverse Fourier transformConvolutionSpatial domainFrequency domainConvolutionMultiplicationLow-pass filters• The larger the support in the spatial domain, the smaller the support in the frequency domainSpatial domainFrequencydomainHigh-pass filterHigh-pass filterSampling• Spatial domain: multiply signal with impulse trainSampling• Spatial domain: multiply signal with impulse train• Frequency domain: convolve signal with Fourier transform of impulse trainSampling and reconstructionSpatial Domain Frequency Domain Spatial Domain Frequency DomainSampling and reconstructionSpatial Domain Frequency Domain Spatial Domain Frequency DomainSampling Theorem (Shannon 1949)• A signal can be reconstructed exactly if it is sampled, at least, at twice its maximum frequency• The minimum sampling frequency is called the Nyquist frequencyAnti-aliasing in graphicsImage signals are not band-limited to half the pixel frequency in general• Prefiltering• SupersamplingPrefiltering• Band-limit the continuous signal to the Nyquist frequency before sampling• Theoretically the way to goBand-limit SampleContinuous pixel Sampled pixelPrefiltering• Not applicable as a general solution in graphics, since continuous image signal is not known• Useful for fexture filtering (mip-mapping)Supersampling• Compute several intermediate samples per pixel• Reconstruct final pixel sample from intermediate samplesSample ReconstructContinuous pixel Sampled pixelSupersampling• Sampling patterns• Reconstruction filtersSampling patterns• Two perspectives to assess quality of sampling patterns– Avoid aliasing– Efficient Monte Carlo integration• Today: focus on aliasingUniform supersampling• Increases Nyquist limit• Spectrum of uniform sampling grid is also a uniform grid of Dirac impulses• Aliases are coherent and very noticeable• Can aliasing be made “less visually disturbing”?Distribution of extrafoveal cones• Studied by Yellot (1983)• Visual system is less sensitive to high frequency noisemonkey eye cone distribution Fourier transformBlue noise characteristics• Least conspicuous form of


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