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MIT 6 837 - Sampling, Aliasing, & Mipmaps

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Sampling, Aliasing, & MipmapsLast Time?Slide 3TodayWhat is a Pixel?Slide 6More on SamplesAn Image is a 2D FunctionSampling GridSampling an ImageQuestions?Slide 12Examples of AliasingSlide 14Slide 15Slide 16Slide 17Slide 18Sampling DensitySampling DensitySlide 21Remember Fourier Analysis?Slide 23Slide 24Remember Convolution?Slide 26Sampling in the Frequency DomainReconstructionGuaranteeing Proper ReconstructionSampling TheoremSlide 31Slide 32FiltersSlide 34Pre-FilteringPost-FilteringThe Ideal FilterProblems with Practical FiltersGaussian FilterBox Filter / Nearest NeighborTent Filter / Bi-Linear InterpolationBi-Cubic InterpolationWhy is the Box filter bad?Slide 44Slide 45Sampling Texture MapsSlide 47Linear InterpolationSpatial FilteringMIP MappingMIP Mapping ExampleSlide 52Slide 53Storing MIP MapsAnisotropic MIP-MappingSlide 56Slide 57Slide 58Next Time:MIT EECS 6.837Sampling, Aliasing, & MipmapsMIT EECS 6.837Last Time?•Global illumination“physically accuratelight transport”•The rendering equation•The discrete radiosity equation–Diffuse reflection only–View independentL(x',ω') = E(x',ω') + ∫ρx'(ω,ω')L(x,ω)G(x,x')V(x,x') dAjijiiiBFEBnAiAjj=1MIT EECS 6.837Last Time?•Form factors– Fij = fraction of light energy leaving patch j that arrives at patch i–Hemicube algorithm•Advanced techniques–Progressive radiosity–Adaptive subdivision–Discontinuity meshing–Hierarchical radiosityMIT EECS 6.837Today•What is a Pixel?•Examples of aliasing•Sampling & Reconstruction •Filters in Computer Graphics•Anti-Aliasing for Texture MapsMIT EECS 6.837What is a Pixel?•My research during for my PhD was on sampling & aliasing with point-sampled surfaces, i.e., 3D objects•This lecture is aboutsampling imagesNo triangles, just samples in 3DMIT EECS 6.837What is a Pixel?•A pixel is not:–a box–a disk–a teeny tiny little light•A pixel “looks different” ondifferent display devices•A pixel is a sample–it has no dimension–it occupies no area–it cannot be seen–it has a coordinate–it has a valueMIT EECS 6.837More 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 quantizediscrete positiondiscretevalueMIT EECS 6.837An Image is a 2D Function•An ideal image is a continuous 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 6.837Sampling Grid•We can generate the table values by multiplying the continuous image function by a sampling grid of Kronecker delta functions.MIT EECS 6.837Sampling an Image•The result is a set of point samples, or pixels.MIT EECS 6.837Questions?MIT EECS 6.837Today•What is a Pixel?•Examples of Aliasing•Sampling & Reconstruction •Filters in Computer Graphics•Anti-Aliasing for Texture MapsMIT EECS 6.837Examples of Aliasing•Aliasing occurs because of sampling and reconstructionMIT EECS 6.837Examples of AliasingMIT EECS 6.837Examples of AliasingMIT EECS 6.837Examples of AliasingTexture Errorspoint samplingMIT EECS 6.837Questions?MIT EECS 6.837Today•What is a Pixel?•Examples of Aliasing•Sampling & Reconstruction –Sampling Density–Fourier Analysis & Convolution•Filters in Computer Graphics•Anti-Aliasing for Texture MapsMIT EECS 6.837Sampling 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...MIT EECS 6.837Sampling Density•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 6.837Sampling Density•Aliasing in 2D because of insufficient sampling densityMIT EECS 6.837Images 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 6.837Remember 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 6.837Remember Fourier Analysis?•We can transform from one domain to the other using the Fourier Transform.spatial domainfrequency domainFourier TransformInverse Fourier TransformMIT EECS 6.837Remember Convolution?Images from Mark Meyerhttp://www.gg.caltech.edu/~cs174ta/MIT EECS 6.837Remember 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 domainMIT EECS 6.837Sampling in the Frequency Domain(convolution)(multiplication)originalsignalsamplinggridsampledsignalFourier TransformFourier TransformFourier TransformMIT EECS 6.837Reconstruction•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 6.837Guaranteeing 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 6.837Sampling Theorem•When sampling a signal at discrete intervals, the sampling frequency must be greater than twice the highest frequency of the input signal in order to be able to reconstruct the original perfectly from the sampled version (Shannon, Nyquist)MIT EECS 6.837Questions?MIT EECS 6.837Today•What is a Pixel?•Examples of Aliasing•Sampling & Reconstruction •Filters in Computer Graphics–Pre-Filtering, Post-Filtering–Ideal, Gaussian,


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MIT 6 837 - Sampling, Aliasing, & Mipmaps

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