Aliasing and Antialiasing. What is Aliasing? What is Aliasing? What is Aliasing?

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1 What is Aliasing? Errors and Artifacts arising during rendering, due to the conversion from a continuously defined illumination field to a discrete raster grid of pixels 1 2 What is Aliasing? What is Aliasing? 3 4

2 What is Aliasing? Effects of Aliasing 5 6 Effects of Aliasing Effects of Aliasing 7 8

3 Effects of Aliasing Anti-aliasing 9 10 Area Sampling Techniques Anti-aliasing Techniques Prefiltering (unweighted/weighted area sampling) Postfiltering (supersampling, jittering) 11 12

4 Area Sampling Techniques 13 Area Sampling Techniques Area Sampling Techniques Area Sampling Techniques 16

5 Unweighted Area Sampling Pixel intensity is varied in proportion to the area of the pixel intercepted by the primitive. Unweighted equivalent to a box filter of unit height over pixel. Weighted Area Sampling Properties Intensity of pixel decreases as the distance between the pixel center and primitive increases. A primitive cannot influence a pixel s intensity if it does not intersect it. Equal areas (intersected) contribute equal intensity not a desirable property. 17 Equal areas can contribute unequally in terms of pixel intensity. Areas closer to the pixel center contribute more. Essentially results in filtering with a mask that is centered over the pixel with decreasing radial influence. Cone filters are a compromise between computational expense and optimality. 18 Postfiltering Techniques Supersampling (Regular Sampling) Very expensive. Not very satisfactory

6 Regular vs. Jittered Sampling 21 Filtering 23 Filtering Example Filtering 22 24

7 Filtering Example 25 Filtering Example Filtering Example 26 Aliasing from a Sampling Theory Viewpoint Sampling(Spatial Domain) 27 28

8 Frequency Domain Image is a spatial signal Sampling(Spatial Domain) X axis (position): frequency Y axis (height): strength of each frequency Examples: sine wave: impulse, square wave: infinite train of impulses How do we get to the Frequency Domain? What does the Fourier Transform Do to A Spatial Signal? Use the Fourier Transform Let φ(x) be a continuous function of a real variable x. Then I{φ(x)} = φ(ω) = φ(x)e j2πωx dx is the Fourier Transform of φ(x), with j = 1 and, I 1 {φ(ω)} = φ(x) = is the Inverse Fourier Transform. φ(x) is continuous and integrable φ(ω) is integrable x (spatial domain), ω (frequency domain) φ(ω)e j2πωx dω 31 ITCS 4120/5120 Signal in frequency domain is an 32 integration of individual Aliasingsinusoids. and Antialiasing

9 How does this related to Graphics? Sampling Theorem Continuous-time signal can be completely recovered from its samples iff the sampling rate is greater than twice the maximum frequency present in the signal. Claude Shannon Also known as the Nyquist rate Images are just a 2D signal and jagged edges are due to the pixel sampling rate not being high enough to capture the real signal Nyquist Rate Nyquist Rate:Undersampling The lower signal is undersampled and results in an aliased wave (dotted curve)

10 Comb Function Comb Function(contd) Application: Used to digitize continuous functions. Series of impulses (delta functions) Identity element of convolution: reproduces an indentical copy of the function f(x) FT of a comb function is another comb function 37 Multiplying f(x) with a comb in image space convolving their Fourier transforms, resulting in multiple identical copies of I{f(x)} Can result in aliasing if copies overlap Maximum allowable frequency is the Nyquist Frequency, which is half the sampling frequency. 38 Reconstruction Example(Adequate Sampling) Reconstruction Example(Inadequate Sampling) 39 40

11 Box Filter Pyramid Filter Reconstruction filter for nearest neighbor interpolation. Resampling images/volumes to a higher resolution using nearest neighbor values. FT of a box filter is the Sinc function ( sinπx πx ) Large side lobes continuing at regular intervals will cause aliasing. Aliasing in images manifests itself as jaggies 41 Reconstruction filter used in linear interpolation Computationally more expensive, but more accurate FT is much better behaved (side lobes much smaller) Less tendency to produce aliasing 42 Gaussian Filter The optimal filter in terms of avodiding side lobes FT of a Gaussian is another Gaussian Widely used to blur images and the basis for scale space 43