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CS 551 / CS 645

CS 551 / CS 645. Antialiasing. 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. Samples.

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CS 551 / CS 645

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  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. 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

  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 dependssolely on valueat grid points

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

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

  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 movesacross 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 beingsampled

  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 (px)/px Frequency Domain Spatial 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

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