1MIT EECS 6.837, Durand and CutlerSampling, Aliasing, & MipmapsMIT EECS 6.837, Durand and CutlerLast Time?• 2D Texture Mapping• Perspective Correct Interpolation• Common Texture Coordinate Projections• Bump Mapping• Displacement Mapping• Environment MappingMIT EECS 6.837, Durand and CutlerTexture Maps for IlluminationQuake• Also called "Light Maps"MIT EECS 6.837, Durand and CutlerToday• What is a Pixel?• Examples of Aliasing• Signal Reconstruction • Reconstruction Filters• Anti-Aliasing for Texture MapsMIT EECS 6.837, Durand and CutlerWhat is a Pixel?• A pixel is not:– a box– a disk– a teeny tiny little light• A pixel is a point– it has no dimension– it occupies no area– it cannot be seen– it can have a coordinate• A pixel is more than just a point, it is a sample!MIT EECS 6.837, Durand and CutlerMore on Samples• Most things in the real world are continuous,yet everything in a computer is discrete• The process of mapping a continuous function to a discrete one is called sampling• The process of mapping a continuous variable to a discrete one is called quantization• To represent or render an image using a computer, we must both sample and quantize2MIT EECS 6.837, Durand and CutlerAn Image is a 2D Function•An ideal image is a function I(x,y) of intensities.• It can be plotted as a height field.• In general an image cannot be represented as a continuous, analytic function. • Instead we represent images as tabulated functions. • How do we fill this table?MIT EECS 6.837, Durand and CutlerSampling Grid• We can generate the table values by multiplying the continuous image function by a sampling grid of Kronecker delta functions.MIT EECS 6.837, Durand and CutlerSampling an Image• The result is a set of point samples, or pixels.MIT EECS 6.837, Durand and CutlerQuestions?MIT EECS 6.837, Durand and CutlerToday• What is a Pixel?• Examples of Aliasing• Signal Reconstruction • Reconstruction Filters• Anti-Aliasing for Texture MapsMIT EECS 6.837, Durand and CutlerExamples of Aliasing3MIT EECS 6.837, Durand and CutlerExamples of AliasingMIT EECS 6.837, Durand and CutlerExamples of AliasingMIT EECS 6.837, Durand and CutlerExamples of AliasingTexture Errorspoint samplingMIT EECS 6.837, Durand and CutlerQuestions?MIT EECS 6.837, Durand and CutlerToday• What is a Pixel?• Examples of Aliasing• Signal Reconstruction – Sampling Density– Fourier Analysis & Convolution• Reconstruction Filters• Anti-Aliasing for Texture MapsMIT EECS 6.837, Durand and CutlerSampling Density• How densely must we sample an image in order to capture its essence? • If we under-sample the signal, we won't be able to accurately reconstruct it...4MIT EECS 6.837, Durand and CutlerNyquist Limit / Shannon's Sampling Theorem• If we insufficiently sample the signal, it may be mistaken for something simpler during reconstruction (that's aliasing!)Image from Robert L. Cook, "Stochastic Sampling and Distributed Ray Tracing", An Introduction to Ray Tracing, Andrew Glassner, ed., Academic Press Limited, 1989.MIT EECS 6.837, Durand and CutlerExamples of AliasingTexture Errorspoint samplingmipmaps & linear interpolationMIT EECS 6.837, Durand and CutlerImages from http://axion.physics.ubc.ca/341-02/fourier/fourier.htmlRemember Fourier Analysis?• All periodic signals can be represented as a summation of sinusoidal waves. MIT EECS 6.837, Durand and CutlerRemember Fourier Analysis?• Every periodic signal in the spatial domain has a dual in the frequency domain.• This particular signal is band-limited, meaning it has no frequencies above some thresholdfrequency domainspatial domainMIT EECS 6.837, Durand and CutlerRemember Fourier Analysis?• We can transform from one domain to the other using the Fourier Transform.spatial domainfrequency domainFourier TransformInverse Fourier TransformMIT EECS 6.837, Durand and CutlerRemember Convolution?Images from Mark Meyerhttp://www.gg.caltech.edu/~cs174ta/5MIT EECS 6.837, Durand and CutlerRemember Convolution?• Some operations that are difficult to compute in the spatial domain can be simplified by transforming to its dual representation in the frequency domain.• For example, convolution in the spatial domain is the same as multiplication in the frequency domain.• And, convolution in the frequency domain is the same as multiplication in the spatial domain MIT EECS 6.837, Durand and CutlerSampling in the Frequency Domain(convolution)(multiplication)originalsignalsamplinggridsampledsignalFourier TransformFourier TransformFourier TransformMIT EECS 6.837, Durand and CutlerReconstruction• If we can extract a copy of the original signal from the frequency domain of the sampled signal, we can reconstruct the original signal!• But there may be overlap between the copies.MIT EECS 6.837, Durand and CutlerGuaranteeing Proper Reconstruction• Separate by removing high frequencies from the original signal (low pass pre-filtering)• Separate by increasing the sampling density• If we can't separate the copies, we will have overlapping frequency spectrum during reconstruction → aliasing.MIT EECS 6.837, Durand and CutlerQuestions?MIT EECS 6.837, Durand and CutlerToday• What is a Pixel?• Examples of Aliasing• Signal Reconstruction • Reconstruction Filters– Pre-Filtering, Post-Filtering– Ideal, Gaussian, Box, Bilinear, Bicubic• Anti-Aliasing for Texture Maps6MIT EECS 6.837, Durand and CutlerPre-Filtering• Filter continuous primitives• Treat a pixel as an area• Compute weighted amount of object overlap• What weighting function should we use?MIT EECS 6.837, Durand and CutlerPost-Filtering• Filter samples• Compute the weighted average of many samples• Regular or jittered sampling (better)MIT EECS 6.837, Durand and CutlerReconstruction Filters• Weighting function • Area of influence often bigger than "pixel"• Sum of weights = 1– Each pixel contributes the same total to image– Constant brightness as object moves across the screen. • No negative weights/colors (optional)MIT EECS 6.837, Durand and CutlerThe Ideal Reconstruction Filter• Unfortunately it has infinite spatial extent– Every sample contributes to every interpolated point • Expensive/impossible to computespatialfrequencyMIT EECS 6.837, Durand and CutlerGaussian Reconstruction Filter • This is what a CRTdoes for free!spatialfrequencyMIT EECS 6.837, Durand and CutlerProblems with Reconstruction Filters•
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