CSE168Computer Graphics II, RenderingSpring 2006Matthias ZwickerLast time• Sampling and aliasingAliasing• Moire patternsAliasingSufficiently sampledInsufficiently sampled[R. Cook ]Fourier analysis• Periodic signals can be expressed as a summation of sinusoidal waves• The Fourier transform computes the complex amplitude at each frequencySpatial domainFrequency domain,power spectrumConvolutionSpatial domainFrequency domainConvolutionMultiplicationSampling• Spatial domain: multiplysignal with impulse train• Frequency domain: convolve signal with Fourier transform of impulse trainSpatial 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• SupersamplingBand-limit SampleSample ReconstructSupersampling• Sampling patterns• Reconstruction filtersPoisson Disk Sampling[Hanrahan][Hanrahan]Spatial domain Frequency domain• Random sampling with minimum distance constraint• Dart throwing algorithmPoisson Disk Sampling2x2 Poisson sampling 2x2 uniform sampling[Dippe 85][Dippe 85]Today• Reconstruction filtering• Realistic camera models• High dynamic range imagingReconstruction• Reconstruction filters are weighting functions to compute a weighted average of the samplesSample ReconstructContinuous pixel Sampled pixelBox Filter• Pretending pixelsare little squares• Take the average of samples in eachpixelspatialfrequencyBox filter• Pixels are not little squares…Down-sampled with a 5x5 box filter (uniform weights)Original high-resolution imageHorizontal banding artifactsThe Ideal Reconstruction Filter• Unfortunately it has infinite spatial extent– Every sample contributes to every interpolated point • Expensive/impossible to compute• Ringing (Gibbs phenomenon)spatialfrequencyfrequencySampledsignalIdeal reconstruction filterProblems with Reconstruction Filters• Excessive pass-band attenuation results in blurry images• Excessive high-frequency leakage can accentuate the sampling grid• Filters with a small support in the spatial domain have a large support in the frequency domainfrequencyMitchell-Netravali Filters [1988]Reconstruction from non-uniform samples• Requires normalizationSamplesReconstruction kernelNon-uniform positionsPixel boundaryQuestions?Realistic camera models• So far: ray tracing using the pinhole modelPinhole camerasPinhole camerasProblemsPinhole camerasProblems• Small pinhole gathers little light, requires long exposure• Larger pinhole reduces sharpnessLenses•Gather more light• Need to be focusedLensLensesPinhole Lens6 sec. exposure 0.01 sec exposureThin lens model• Approximative model for well-behaved lenses• All parallel rays converge at focal length•Rays through the center are not deflectedThin lens model• How are arbitrary rays deflected when passing through a thin lens?Thin lens modelThin lens model• Similar trianglesThin lens model• More similar trianglesThin lens model• Thin lens formula• All rays passing through a single point on a plane at distance in front of the lens will pass through a single point at distance behind the lensThin lens model• Focus at infinity:• Closest focusing distance: Film planeObjectThin lens model• Out of focus film plane results in spherical blur Out of focus film planesSpherical blurDepth of field• Blurriness of out of focus objects depends on aperture sizeApertureDepth of fieldRay tracing using a thin lens modelImage plane Image planePrimary raysPrimary raysObjectin focus• Place image plane at distance D from lens plane• Generate primary rays with random origin on lens aperturePinhole Thin lensCamera parametersTypical SLR lens• Focal length 35mm• f-number f-3.5• Aperture is (focal length)/(f-number), i.e. 10mm• Depth of field effects only for very short distances distances, < 5mCamera parametersTypical SLR lens• Field of view depends on focusing distance• Film size is 36mm x 24mm• Focus at infinity, vertical field of view isMore realistic camera models• “A realistic camera model for computer graphics”, Kolb, Mitchell, Hanrahan• Lens distortion• Exposure, motion blurFull simulation Thick lensapproximationThin lensapproximation[Kolb et al.]Questions?HDR and tone mapping• HDR: high dynamic range• Dynamic range: ratio of largest over smallest intensity value in image• The dynamic range of light in real environments is often larger than the range of the sensor• The dynamic range of rendered images is often larger than the range of the displayHDR photography• Acquire several images with different exposures• Recover a HDR intensity for each pixel[Wikipedia]HDR photography•Steve Mannhttp://genesis.eecg.toronto.edu/wyckoff/index.html•Paul Debevechttp://www.debevec.org/Research/HDR/• Mitsunaga, Nayar, Grossberghttp://www1.cs.columbia.edu/CAVE/projects/rad_cal/rad_cal.phpTone mapping• Compress dynamic range of image without losing detail[Wikipedia]Tone mapping• Naïve approach: map fix luminance value to white using linear scalingTone mappingContrast preserving, non-linear scaling• “A contrast based scale factor for luminance display”, Ward, 1994Tone mappingSpatially varying non-linear scaling• “A tone mapping algorithm for high contrast images”, Ahiskmin, 2002Multi-scale methods• Reduce contrast of low-frequencies• Keep high frequenciesMulti-scale methods•HalosEdge-preserving filtering• Do not blur across edges• Non-linear filtering (not a convolution)Bilateral filter• Tomasi and Manduchi 1998http://www.cse.ucsc.edu/~manduchi/Papers/ICCV98.pdfTone mapping with the bilateral filterTone mapping with the bilateral filter•“Fast Bilateral Filtering for the Display of High-Dynamic-Range Images”, Durand et al., 2002Next time• Participating media and subsurface