CS 445 / 645 Introduction to Computer Graphics

1. CS 445 / 645Introduction to Computer Graphics Lecture 20 Antialiasing

2. Environment Mapping • Used to model a object that reflects surrounding textures to the eye • Polished sphere reflects walls and ceiling textures • Cyborg in Terminator 2 reflects flaming destruction • Texture is distorted fish-eye view of environment • Spherical texture mapping creates texture coordinates that correctly index into this texture map

3. Materials from NVidia • http://developer.nvidia.com/object/Cube_Mapping_Paper.html

4. Sphere Mapping

5. Blinn/Newell Lattitude Mapping

6. Cube Mapping

7. Multitexturing • Pipelining of multiple texture applications to one polygon • The results of each texture unit application is passed to the next texture unit, which adds its effects • More bookkeeping is required to pull this off

9. Antialiasing

10. What 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 a point, it is a sample

11. Samples • Most things in the real world are continuous • 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 • Rendering an image requires sampling and quantization

12. Samples

13. Samples

14. Line Segments • We tried to sample a line segment so it would map to a 2D raster display • We quantized the pixel values to 0 or 1 • We saw stair steps, or jaggies

15. Line Segments • Instead, quantize to many shades • But what sampling algorithm is used?

16. Area Sampling • Shade pixels according to the area covered by thickened line • This is unweighted area sampling • A rough approximation formulated by dividing each pixel into a finer grid of pixels

17. Unweighted Area Sampling • Primitive cannot affect intensity of pixel if it does not intersect the pixel • Equal areas cause equal intensity, regardless of distance from pixel center to area

18. Weighted Area Sampling • Unweighted sampling colors two pixels identically when the primitive cuts the same area through the two pixels • Intuitively, pixel cut through the center should be more heavily weighted than one cut along corner

19. Intensity W(x,y) x Weighted Area Sampling • Weighting function, W(x,y) • specifies the contribution of primitive passing through the point (x, y) from pixel center

20. Images • An image is a 2D function I(x, y) that specifies intensity for each point (x, y)

21. Sampling and Image • Our goal is to convert the continuous image to a discrete set of samples • The graphics system’s display hardware will attempt to reconvert the samples into a continuous image: reconstruction

22. Point Sampling an Image • Simplest sampling is on a grid • Sample dependssolely on valueat grid points

23. Point Sampling • Multiply sample grid by image intensity to obtain a discrete set of points, or samples. Sampling Geometry

24. Sampling Errors • Some objects missed entirely, others poorly sampled

25. Fixing Sampling Errors • Supersampling • Take more than one sample for each pixel and combine them • How many samples is enough? • How do we know no features are lost? 150x15 to 100x10 200x20 to 100x10 300x30 to 100x10 400x40 to 100x10

26. Unweighted Area Sampling • Average supersampled points • All points are weighted equally

27. Weighted Area Sampling • Points in pixel are weighted differently • Flickering occurs as object movesacross display • Overlapping regions eliminates flicker

28. Signal Theory • Convert spatial signal to frequency domain Intensity Pixel position across scanline Example from Foley, van Dam, Feiner, and Hughes

29. Signal Theory • Represent spatial signal as sum of sine waves (varying frequency and phase shift) • Very commonlyused to representsound “spectrum”

30. Fourier Analysis • Convert spatial domain to frequency domain • Let f(x) indicate the intensity at a location in space, x (pixel value) • u is a complex number representing frequency and phase shift • i = sqrt (-1) … frequently not plotted • F(u) is the amplitude of a particular frequency in a signal • In this case the signal is f(x)

31. Fourier Transform • Examples of spatial and frequency domains

32. Nyquist Sampling Theorem • The ideal samples of a continuous function contain all the information in the original function if and only if the continuous function is sampled at a frequency greater than twice the highest frequency in the function

33. Nyquist Rate • The lower bound on the sampling rate equals twice the highest frequency component in the image’s spectrum • This lower bound is the Nyquist Rate

34. Band-limited Signals • If you know a function contains no components of frequencies higher than x • Band-limited implies original function will not require any ideal functions with frequencies greater than x • This facilitates reconstruction

35. Flaws with Nyquist Rate • Samples may not align with peaks

36. Flaws with Nyquist Rate • When sampling below Nyquist Rate, resulting signal looks like a lower-frequency one • With no knowledge of band-limits, samples could have been derived from signal of higher frequency

37. Low-pass Filtering • We know we are limited in the resolution of our screen • We want the screen (sampling grid) to have twice the resolution of the signal (image) we want to display • How can we reduce the high-frequencies of the image? • Low-pass filter • Band-limits the image

38. Low-pass Filtering • In frequency domain • If signal is F(u) • Just chop off parts of F(u) in high frequencies using a second function, G(u) • G(u) == 1 when –k <= u <= k 0 elsewhere • This is called the pulse function

39. Low-pass Filtering • In spatial domain • Multiplying two Fourier transforms in the spatial domain corresponds exactly to performing an operation called convolution in the spatial domain • f(x) * g(x) = h(x)  the convolution of f with g… • The value of h(x) at x is the integral of the product of f(x) with the filter g(x) such that g(x) is centered at x • The pulse (frequency) == sinc (spatial)

40. Low-pass Filtering • In spatial domain • Sinc: sinc(x) = sin (px)/px • Note this isn’t perfect way to eliminate high frequencies • “ringing” occurs

41. Sinc Filter • Slide filter along spatial domain and compute new pixel value that results from convolution

42. Bilinear Filter • Sometimes called a tent filter • Easy to compute • just linearly interpolate between samples • Finite extent and no negative values • Still has artifacts

43. Sampling Pipeline

44. How is this done today?Full Screen Antialiasing • Nvidia GeForce2 • OpenGL: render image 400% larger and supersample • Direct3D: render image 400% - 1600% larger • Nvidia GeForce3 • Multisampling but with fancy overlaps • Don’t render at higher resolution • Use one image, but combine values of neighboring pixels • Beware of recognizable combination artifacts • Human perception of patterns is too good

45. GeForce3 • Multisampling • After each pixel is rendered, write pixel value to two different places in frame buffer

46. GeForce3 - Multisampling • After rendering two copies of entire frame • Shift pixels of Sample #2 left and up by ½ pixel • Imagine laying Sample #2 (red) over Sample #1 (black)

47. GeForce3 - Multisampling • Resolve the two samples into one image by computing average between each pixel from Sample 1 (black) and the four pixels from Sample 2 (red) that are 1/ sqrt(2) pixels away

48. GeForce3 - Multisampling • No AA Multisampling

49. GeForce3 - Multisampling • 4x Supersample Multisampling

50. ATI Smoothvision • ATI SmoothVision • Programmer selects samping pattern • Some supersamplingthrown in