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Aliasing & Antialiasing

Aliasing & Antialiasing. Recap: Aliasing. Sampling a function = multiplying I(x) by sampling fn S x = convolving F(u) by sampling fn S 1/x = copying F(u) at regular intervals If the copies of F(u) overlap, high frequencies “fold over”, appearing as lower frequencies

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Aliasing & Antialiasing

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  1. Aliasing & Antialiasing David Luebke 13/15/2014

  2. Recap: Aliasing • Sampling a function = multiplying I(x) by sampling fn Sx = convolving F(u) by sampling fn S1/x = copying F(u) at regular intervals • If the copies of F(u) overlap, high frequencies “fold over”, appearing as lower frequencies • This is known as aliasing David Luebke 23/15/2014

  3. Prefiltering • Eliminate high frequencies before sampling (Foley & van Dam p. 630) • Convert I(x) to F(u) • Apply a low-pass filter (e.g., multiply F(u) by a box function) • Then sample. Result: no aliasing! • So what’s the problem? David Luebke 33/15/2014

  4. Prefiltering • Eliminate high frequencies before sampling (Foley & van Dam p. 630) • Convert I(x) to F(u) • Apply a low-pass filter (e.g., multiply F(u) by a box function) • Then sample. Result: no aliasing! • Problem: most rendering algorithms generate sampled function directly • e.g., Z-buffer, ray tracing David Luebke 43/15/2014

  5. Supersampling • The simplest way to reduce aliasing artifacts is supersampling • Increase the resolution of the samples • Average the results down • Sometimes called postfiltering • Create virtual image at higher resolution than the final image • Apply a low-pass filter • Resample filtered image David Luebke 53/15/2014

  6. Supersampling: Limitations • Q: What practical consideration hampers supersampling? • A: Storage goes up quadratically • Q: What theoretical problem does supersampling suffer? • A: Doesn’t eliminate aliasing! Supersampling simply shifts the Nyquist limit higher David Luebke 63/15/2014

  7. Supersampling: Worst Case • Q: Give a simple scene containing infinite frequencies • A: A checkered ground plane receeding into infinity • Show Watt & Watt, Figure 4.9 David Luebke 73/15/2014

  8. Supersampling • Despite these limitations, people still use supersampling (why?) • So how can we best perform it? David Luebke 83/15/2014

  9. Supersampling • The process: • Create virtual image at higher resolution than the final image • Apply a low-pass filter • Resample filtered image David Luebke 93/15/2014

  10. Supersampling • Create virtual image at higher resolution than the final image • This is easy David Luebke 103/15/2014

  11. Supersampling • Apply a low-pass filter • Convert to frequency domain • Multiply by a box function • Expensive! Can we simplify this? • Recall: multiplication in frequency equals convolution in space, so… • Just convolve initial sampled image with the FT of a box function • Isn’t this a lot of work? David Luebke 113/15/2014

  12. Supersampling:Digital Convolution • In practice, we combine steps 2 & 3: • Create virtual image at higher resolution than the final image • Apply a low-pass filter • Resample filtered image • Idea: only create filtered image at new sample points • I.e., only convolve filter with image at new points David Luebke 123/15/2014

  13. = =  1 1 1 2 2 2 1 1 1 2 2 2 4 4 4 2 2 2 1 1 1 2 2 2 1 1 1 … Supersampling:Digital Convolution • Q: What does convolving a filter with an image entail at each sample point? • A: Multiplying and summing values • Example (also F&vD p 642): David Luebke 133/15/2014

  14. Supersampling • Q: What filter should we use in space to effect multiplying by a box function in frequency? • A: The sinc function (sin(x) / x) • Q: Why is this hard? • Sinc function has infinite support • Sinc function has negative lobes David Luebke 143/15/2014

  15. Supersampling • Common filters: • Truncated sinc • Box • Triangle • Guassian • What do the frequency spectra of each of these filters look like? • A: see F&vD, p. 634-635 David Luebke 153/15/2014

  16. Supersampling • Typical supersampling algorithm: • Compute multiple samples per pixel • Combine sample values for pixel’s value using simple average • Q: What filter does this equate to? • A: Box filter -- one of the worst! • Q: What’s wrong with box filters? • Passes infinitely high frequencies • Attenuates desired frequencies David Luebke 163/15/2014

  17. SupersamplingIn Practice • Sinc function: ideal but impractical • One approximation: sinc2 (=what?) • Another: Guassian falloff • Q: How wide (what res) should filter be? • A: As wide as possible (duh) • In practice: 3x3, 5x5, at most 7x7 David Luebke 173/15/2014

  18. Supersampling: Summary • Supersampling improves aliasing artifacts by shifting the Nyquist limit • It works by calculating a high-res image and filtering down to final res • “Filtering down” means simultaneous convolution and resampling • This equates to a weighted average • Wider filter  better results  more work David Luebke 183/15/2014

  19. Summary So Far • Prefiltering • Before sampling the image, use a low-pass filter to eliminate frequencies above the Nyquist limit • This blurs the image… • But ensures that no high frequencies will be misrepresented as low frequencies David Luebke 193/15/2014

  20. Summary So Far • Supersampling • Sample image at higher resolution than final image, then “average down” • “Average down” means multiply by low-pass function in frequency domain • Which means convolving by that function’s FT in space domain • Which equates to a weighted average of nearby samples at each pixel David Luebke 203/15/2014

  21. Summary So Far • Supersampling cons • Doesn’t eliminate aliasing, just shifts the Nyquist limit higher • Can’t fix some scenes (e.g., checkerboard) • Badly inflates storage requirements • Supersampling pros • Relatively easy • Often works all right in practice • Can be added to a standard renderer David Luebke 213/15/2014

  22. Antialiasing in the Continuous Domain • Problem with prefiltering: • Sampling and image generation inextricably linked in most renderers • Z-buffer algorithm • Ray tracing • Why? • Still, some approaches try to approximate effect of convolution in the continuous domain David Luebke 223/15/2014

  23. Pixel Grid Filter kernel Polygons Antialiasing in the Continuous Domain David Luebke 233/15/2014

  24. Antialiasing in the Continuous Domain • The good news • Exact polygon coverage of the filter kernel can be evaluated • What does this entail? • Clipping • Hidden surface determination David Luebke 243/15/2014

  25. Antialiasing in the Continuous Domain • The bad news • Evaluating coverage is very expensive • The intensity variation is too complex to integrate over the area of the filter • Q: Why does intensity make it harder? • A: Because polygons might not be flat- shaded • Q: How bad a problem is this? • A: Intensity varies slowly within a pixel, so shape changes are more important David Luebke 253/15/2014

  26. Catmull’s Algorithm • Find fragment areas • Multiply by fragment colors • Sum for final pixel color A2 A1 AB A3 David Luebke 263/15/2014

  27. Catmull’s Algorithm • First real attempt to filter in continuous domain • Very expensive • Clipping polygons to fragments • Sorting polygon fragments by depth (What’s wrong with this as a hidden surface algorithm?) • Equates to box filter (Is that good?) David Luebke 273/15/2014

  28. The A-Buffer • Idea: approximate continuous filtering by subpixel sampling • Summing areas now becomes simple David Luebke 283/15/2014

  29. The A-Buffer • Advantages: • Incorporating into scanline renderer reduces storage costs dramatically • Processing per pixel depends only on number of visible fragments • Can be implemented efficiently using bitwise logical ops on subpixel masks David Luebke 293/15/2014

  30. The A-Buffer • Disadvantages • Still basically a supersampling algorithm • Not a hardware-friendly algorithm • Lists of potentially visible polygons can grow without limit • Work per-pixel non-deterministic David Luebke 303/15/2014

  31. Recap:Antialiasing Strategies • Supersampling: sample at higher resolution, then filter down • Pros: • Conceptually simple • Easy to retrofit existing renderers • Works well most of the time • Cons: • High storage costs • Doesn’t eliminate aliasing, just shifts Nyquist limit upwards David Luebke 313/15/2014

  32. Recap:Antialiasing Strategies • A-Buffer: approximate prefiltering of continuous signal by sampling • Pros: • Integrating with scan-line renderer keeps storage costs low • Can be efficiently implemented with clever bitwise operations • Cons: • Still basically a supersampling approach • Doesn’t integrate with ray-tracing David Luebke 323/15/2014

  33. Stochastic Sampling • Sampling theory tells us that with a regular sampling grid, frequencies higher than the Nyquist limit will alias • Q: What about irregular sampling? • A: High frequencies appear as noise, not aliases • This turns out to bother our visual system less! David Luebke 333/15/2014

  34. Stochastic Sampling • An intuitive argument: • In stochastic sampling, every region of the image has a finite probability of being sampled • Thus small features that fall between uniform sample points tend to be detected by non-uniform samples David Luebke 343/15/2014

  35. Stochastic Sampling • Integrating with different renderers: • Ray tracing: • It is just as easy to fire a ray one direction as another • Z-buffer: hard, but possible • Notable example: REYES system (?) • Using image jittering is easier (more later) • A-buffer: nope • Totally built around square pixel filter and primitive-to-sample coherence David Luebke 353/15/2014

  36. Stochastic Sampling • Idea: randomizing distribution of samples scatters aliases into noise • Problem: what type of random distribution to adopt? • Reason: type of randomness used affects spectral characteristics of noise into which high frequencies are converted David Luebke 363/15/2014

  37. Stochastic Sampling • Problem: given a pixel, how to distribute points (samples) within it? David Luebke 373/15/2014

  38. Stochastic Sampling • Poisson distribution: • Completely random • Add points at random until area is full. • Uniform distribution: some neighboring samples close together, some distant David Luebke 383/15/2014

  39. Stochastic Sampling • Poisson disc distribution: • Poisson distribution, with minimum-distance constraint between samples • Add points at random, removing again if they are too close to any previous points • Very even-looking distribution David Luebke 393/15/2014

  40. Stochastic Sampling • Jittered distribution • Start with regular grid of samples • Perturb each sample slightly in a random direction • More “clumpy” or granular in appearance David Luebke 403/15/2014

  41. Stochastic Sampling • Spectral characteristics of these distributions: • Poisson: completely uniform (white noise). High and low frequencies equally present • Poisson disc: Pulse at origin (DC component of image), surrounded by empty ring (no low frequencies), surrounded by white noise • Jitter: Approximates Poisson disc spectrum, but with a smaller empty disc. David Luebke 413/15/2014

  42. Stochastic Sampling • Watt & Watt, p. 134 • See Foley & van Dam, p 644-645 David Luebke 423/15/2014

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