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David Luebke 1 4/28/2015 Aliasing & Antialiasing

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David Luebke 2 4/28/2015 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 ● This is known as aliasing

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David Luebke 3 4/28/2015 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?

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David Luebke 4 4/28/2015 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

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David Luebke 5 4/28/2015 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

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David Luebke 6 4/28/2015 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

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David Luebke 7 4/28/2015 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

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David Luebke 8 4/28/2015 Supersampling ● Despite these limitations, people still use supersampling (why?) ● So how can we best perform it?

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David Luebke 9 4/28/2015 Supersampling ● The process: ■ Create virtual image at higher resolution than the final image ■ Apply a low-pass filter ■ Resample filtered image

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David Luebke 10 4/28/2015 Supersampling ● Create virtual image at higher resolution than the final image ■ This is easy

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David Luebke 11 4/28/2015 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?

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David Luebke 12 4/28/2015 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

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David Luebke 13 4/28/2015 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): 121 242 121 121 242 121 = 121 242 121 = …

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David Luebke 14 4/28/2015 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

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David Luebke 15 4/28/2015 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

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David Luebke 16 4/28/2015 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

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David Luebke 17 4/28/2015 Supersampling In Practice ● Sinc function: ideal but impractical ● One approximation: sinc 2 (=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

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David Luebke 18 4/28/2015 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

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David Luebke 19 4/28/2015 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

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David Luebke 20 4/28/2015 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

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David Luebke 21 4/28/2015 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

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David Luebke 22 4/28/2015 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

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David Luebke 23 4/28/2015 Antialiasing in the Continuous Domain Pixel Grid Polygons Filter kernel

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David Luebke 24 4/28/2015 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

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David Luebke 25 4/28/2015 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

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David Luebke 26 4/28/2015 Catmull’s Algorithm ABAB A1A1 A2A2 A3A3 ● Find fragment areas ● Multiply by fragment colors ● Sum for final pixel color

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David Luebke 27 4/28/2015 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?)

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David Luebke 28 4/28/2015 The A-Buffer ● Idea: approximate continuous filtering by subpixel sampling ● Summing areas now becomes simple

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David Luebke 29 4/28/2015 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

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David Luebke 30 4/28/2015 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

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David Luebke 31 4/28/2015 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

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David Luebke 32 4/28/2015 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

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David Luebke 33 4/28/2015 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!

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David Luebke 34 4/28/2015 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

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David Luebke 35 4/28/2015 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

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David Luebke 36 4/28/2015 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

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David Luebke 37 4/28/2015 Stochastic Sampling ● Problem: given a pixel, how to distribute points (samples) within it?

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David Luebke 38 4/28/2015 Stochastic Sampling ● Poisson distribution: ■ Completely random ■ Add points at random until area is full. ■ Uniform distribution: some neighboring samples close together, some distant

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David Luebke 39 4/28/2015 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

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David Luebke 40 4/28/2015 Stochastic Sampling ● Jittered distribution ■ Start with regular grid of samples ■ Perturb each sample slightly in a random direction ■ More “clumpy” or granular in appearance

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David Luebke 41 4/28/2015 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.

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David Luebke 42 4/28/2015 Stochastic Sampling ● Watt & Watt, p. 134 ● See Foley & van Dam, p 644-645

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(c) 2002 University of Wisconsin, CS 559

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