1. CS 551 / CS 645 Antialiasing

2. 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

3. 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

4. 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

5. Line Segments Instead, quantize to many shades But what sampling algorithms is used?

6. 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

7. Unweighted Area Sampling Pixel intensity decreases as distance betewen the pixel center and edge increases Primitive cannot affect intensity of pixel if it does no intersect the pixel Equal areas cause equal intensity, regardless of distance from pixel center to area

8. 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

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

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

11. 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

12. Point Sampling an Image Simplest sampling is on a grid Sample depends solely on value at grid points

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

14. Some objects missed entirely, others poorly sampled Sampling Errors

15. 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?

16. Unweighted Area Sampling Average supersampled points All points are weighted equally

17. Weighted Area Sampling Points in pixel are weighted differently –Flickering occurs as object moves across display Overlapping regions eliminates flicker

18. Frequency Domain Fourier Transform –Convert signal, f(x), from spatial domain to frequency domain, F (u) –F (u) indicates how much of u frequency is in image

19. Sampling in the Frequency Domain Remember, sampling was defined as multiplying a grid of delta functions by the continuous image This is called a convolution in frequency domain The sampling grid The function being sampled

20. Convolution This amounts to accumulating copies of the function’s spectrum sampled at the delta functions of the sampling grid

21. Nyquist Rate The lower bound on an image’s sample rate is the Nyquist Rate The Nyquist rate is twice the highest frequency component in the spectrum Actually, sampling greater than Nyquist is usually required –Draw special case

22. Aliasing Sampling below Nyquist rate can result in loss of high-frequencies –Draw picture May also result in adding high frequency –Texture map of brick wall (seen on DVD’s)

23. Filtering To lower Nyquist rate, remove high frequencies from image: low-pass filter –Only low frequencies remain Sinc function is common filter: –sinc(x) = sin (  x)/  x Spatial Domain Frequency Domain

24. Sinc function Value of 1 at sample point and 0 at other integer values Exactly matches values at sample points But has infinite extent But has negative values But assumes sample repeats infinitely Truncated version introduces high frequencies again

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

26. Supersampling Techniques Adaptive supersampling –store more points when necesssary Stochastic supersampling –Place sample points at stochastically determined points Eye has harder time detecting aliasing when combined with the noise generated by stochastics