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Sampling, Antialiasing, and Aliasing in Images and Animation

This article explores the concepts of sampling, antialiasing, and aliasing in images and animation. It covers topics such as high frequencies aliasing as low frequencies, vector spaces, dot product, Fourier transforms, filtering, and reconstruction.

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Sampling, Antialiasing, and Aliasing in Images and Animation

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  1. CMSC 635 Sampling and Antialiasing

  2. Aliasing in images

  3. Aliasing in animation

  4. High frequencies alias as low frequencies Aliasing

  5. Abstract Vector Spaces • Addition • C = A + B = B + A • (A + B) + C = A + (B + C) • given A, B, A + X = B for only one X • Scalar multiply • C = a A • a (A + B) = a A + a B • (a+b) A = a A + b A

  6. Abstract Vector Spaces • Inner or Dot Product • b = a (A • B) = a A • B = A • a B • A • A ≥ 0; A • A = 0 iff A = 0 • A • B = (B • A)*

  7. Vectors and Discrete Functions

  8. 2t in terms of t0, t1, t2 = [1,.5,.5] Vectors and Discrete Functions 2t 1 t t2

  9. 2t in terms of t0, t0.5, t1, t1.5, t2 Vectors and Discrete Functions 2t 1 t0.5 t t1.5 t2

  10. Vectors and Functions

  11. 2t projected onto 1, t, t2 Vectors and Functions 2t 1 t t2

  12. Time: Polynomial / Power Series: Discrete Fourier: integers (where ) Continuous Fourier: Function Bases

  13. Fourier Transforms

  14. Convolution • Where • Dot product with shifted kernel

  15. Filtering • Filter in frequency domain • FT signal to frequency domain • Multiply signal & filter • FT signal back to time domain • Filter in time domain • FT filter to time domain • Convolve signal & filter

  16. Low pass filter eliminates all high freq box in frequency domain sinc in spatial domain ( ) Possible negative results Infinite kernel Exact reconstruction to Nyquist limit Sample frequency ≥ 2x highest frequency Exact only if reconstructing with sync Ideal

  17. Sampling • Multiply signal by pulse train

  18. Reconstruction • Convolve samples & reconstruction filter • Sum weighted kernel functions

  19. Steps Ideal continuous image Filter Filtered continuous image Sample Sampled image pixels Reconstruction filter Continuous displayed result Filtering, Sampling, Reconstruction

  20. Combine filter and sample Ideal continuous image Sampling filter Sampled image pixels Reconstruction filter Continuous displayed result Filtering, Sampling, Reconstruction

  21. Blur away frequencies that would alias Blur preferable to aliasing Can combine filtering and sampling Evaluate convolution at sample points Filter kernel size IIR = infinite impulse response FIR = finite impulse response Antialiasing

  22. Fold better filter into rasterization Can make rasterization much harder Usually just done for lines Draw with filter kernel “paintbrush” Only practical for finite filters Analytic higher order filtering

  23. Compute “area” of pixel covered Box in spatial domain Nice finite kernel easy to compute sinc in freq domain Plenty of high freq still aliases Analytic Area Sampling

  24. Numeric integration of filter Grid with equal weight = box filter Unequal weights Priority sampling Push up Nyquist frequency Edges: ∞ frequency, still alias Supersampling

  25. Numerical integration with varying step More samples in high contrast areas Easy with ray tracing, harder for others Possible bias Adaptive sampling

  26. Monte-Carlo integration of filter Sample distribution Poisson disk Jittered grid Aliasing  Noise Stochastic sampling

  27. Finite impulse response Finite filter Windowed filter Positive everywhere? Ringing? Which filter kernel?

  28. Windowed sync Windowed Gaussian (~0 @ 6) Box, tent, cubic Which filter kernel?

  29. Image samples Reconstruction filter Continuous image Sampling filter Image samples Resampling

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