Aliasing and Antialiasing. ITCS 4120/ Aliasing and Antialiasing
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1 Aliasing and Antialiasing ITCS 4120/ Aliasing and Antialiasing
2 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 ITCS 4120/ Aliasing and Antialiasing
3 What is Aliasing? ITCS 4120/ Aliasing and Antialiasing
4 What is Aliasing? ITCS 4120/ Aliasing and Antialiasing
5 What is Aliasing? ITCS 4120/ Aliasing and Antialiasing
6 Effects of Aliasing ITCS 4120/ Aliasing and Antialiasing
7 Effects of Aliasing ITCS 4120/ Aliasing and Antialiasing
8 Effects of Aliasing ITCS 4120/ Aliasing and Antialiasing
9 Effects of Aliasing ITCS 4120/ Aliasing and Antialiasing
10 Anti-aliasing ITCS 4120/ Aliasing and Antialiasing
11 Anti-aliasing Techniques Prefiltering (unweighted/weighted area sampling) Postfiltering (supersampling, jittering) ITCS 4120/ Aliasing and Antialiasing
12 Area Sampling Techniques ITCS 4120/ Aliasing and Antialiasing
13 Area Sampling Techniques ITCS 4120/ Aliasing and Antialiasing
14 Area Sampling Techniques ITCS 4120/ Aliasing and Antialiasing
15 Area Sampling Techniques ITCS 4120/ Aliasing and Antialiasing
16 Area Sampling Techniques ITCS 4120/ Aliasing and Antialiasing
17 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. 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. ITCS 4120/ Aliasing and Antialiasing
18 Weighted Area Sampling 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. ITCS 4120/ Aliasing and Antialiasing
19 Postfiltering Techniques ITCS 4120/ Aliasing and Antialiasing
20 Supersampling (Regular Sampling) Very expensive. Not very satisfactory. ITCS 4120/ Aliasing and Antialiasing
21 Regular vs. Jittered Sampling ITCS 4120/ Aliasing and Antialiasing
22 Filtering ITCS 4120/ Aliasing and Antialiasing
23 Filtering ITCS 4120/ Aliasing and Antialiasing
24 Filtering Example ITCS 4120/ Aliasing and Antialiasing
25 Filtering Example ITCS 4120/ Aliasing and Antialiasing
26 Filtering Example ITCS 4120/ Aliasing and Antialiasing
27 Filtering Example ITCS 4120/ Aliasing and Antialiasing
28 Aliasing from a Sampling Theory Viewpoint Sampling(Spatial Domain) ITCS 4120/ Aliasing and Antialiasing
29 Sampling(Spatial Domain) Image is a spatial signal ITCS 4120/ Aliasing and Antialiasing
30 Frequency Domain X axis (position): frequency Y axis (height): strength of each frequency Examples: sine wave: impulse, square wave: infinite train of impulses ITCS 4120/ Aliasing and Antialiasing
31 How do we get to the Frequency Domain? 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 ITCS 4120/ Aliasing and Antialiasing
32 What does the Fourier Transform Do to A Spatial Signal? ITCS 4120/5120 Signal in frequency domain is an 32 integration of individual Aliasingsinusoids. and Antialiasing
33 How does this related to Graphics? 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. ITCS 4120/ Aliasing and Antialiasing
34 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 ITCS 4120/ Aliasing and Antialiasing
35 Nyquist Rate ITCS 4120/ Aliasing and Antialiasing
36 Nyquist Rate:Undersampling The lower signal is undersampled and results in an aliased wave (dotted curve). ITCS 4120/ Aliasing and Antialiasing
37 Comb Function 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 ITCS 4120/ Aliasing and Antialiasing
38 Comb Function(contd) 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. ITCS 4120/ Aliasing and Antialiasing
39 Reconstruction Example(Adequate Sampling) ITCS 4120/ Aliasing and Antialiasing
40 Reconstruction Example(Inadequate Sampling) ITCS 4120/ Aliasing and Antialiasing
41 Box 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 ITCS 4120/ Aliasing and Antialiasing
42 Pyramid Filter 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 ITCS 4120/ Aliasing and Antialiasing
43 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 ITCS 4120/ Aliasing and Antialiasing
44 ITCS 4120/ Aliasing and Antialiasing
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