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Home , Nyquist rate

and Antialiasing

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”

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What is Aliasing? What is Aliasing?

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Effects of Aliasing Effects of Aliasing

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Area Sampling Techniques

Anti-aliasing Techniques

 Prefiltering (unweighted/weighted area sampling)  Postfiltering (, jittering)

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Area Sampling Techniques Area Sampling Techniques

ITCS 4120/5120 15 Aliasing and Antialiasing ITCS 4120/5120 16 Aliasing and Antialiasing Unweighted Area Sampling  Pixel intensity is varied in proportion to the area of the pixel inter- cepted by the primitive.  Unweighted – equivalent to a box filter of unit height over pixel. 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 Properties pixel with decreasing radial influence.  Intensity of pixel decreases as the distance between the pixel center  Cone filters are a compromise between computational expense and and primitive increases. optimality.  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 4120/5120 17 Aliasing and Antialiasing ITCS 4120/5120 18 Aliasing and Antialiasing

Postfiltering Techniques Supersampling (Regular Sampling)

 Very expensive.  Not very satisfactory.

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Filtering Filtering Example

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Filtering Example Aliasing from a Sampling Theory Viewpoint

Sampling(Spatial Domain)

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 X axis (position): frequency  Y axis (height): strength of each frequency Sampling(Spatial Domain)  Examples: sine wave: impulse, square wave: infinite train of im- pulses  Image is a spatial signal

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What does the Do to A How do we get to the Frequency Domain? Spatial Signal?

 Use the Fourier Transform Let φ(x) be a continuous function of a real variable x. Then

∞ j2πωx φ(x) = φ(ω) = φ(x)e− dx ={ } Z−∞ is the Fourier Transform of φ(x), with j = √ 1 and, −

1 ∞ j2πωx − φ(ω) = φ(x) = φ(ω)e dω = { } Z−∞ is the Inverse Fourier Transform. φ(x) is continuous and integrable ◦ φ(ω) is integrable ◦ x (spatial domain), ω (frequency domain) ◦ ITCS 4120/5120 31 Aliasing and Antialiasing ITCS 4120/5120Signal in frequency domain is an32 integration of individualAliasing sinusoids. and Antialiasing ◦ How does this related to Graphics?

Sampling Theorem

“Continuous-time signal can be completely recovered from its sam- ples iff the sampling rate is greater than twice the maximum fre- quency 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.

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Nyquist Rate Nyquist Rate:

 The lower signal is undersampled and results in an aliased wave (dotted curve).

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 Application: Used to digitize continuous functions.  Multiplying f(x) with a comb in image space convolving their Fourier transforms, resulting in multiple identical⇐⇒ copies of f(x)  Series of impulses (delta functions) ={ }  Identity element of convolution: reproduces an indentical copy of the  Can result in aliasing if copies overlap function f(x)  Maximum allowable frequency is the , which is  FT of a comb function is another comb function half the sampling frequency. ITCS 4120/5120 37 Aliasing and Antialiasing ITCS 4120/5120 38 Aliasing and Antialiasing

Reconstruction Example(Inadequate Sampling) Reconstruction Example(Adequate Sampling)

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 Reconstruction filter for nearest neighbor interpolation.  Resampling images/volumes to a higher resolution using nearest  Reconstruction filter used in linear interpolation neighbor values. Computationally more expensive, but more accurate sinπx   FT of a box filter is the Sinc function ( ) πx  FT is much better behaved (side lobes much smaller)  Large side lobes continuing at regular intervals will cause aliasing.  Less tendency to produce aliasing  Aliasing in images manifests itself as “jaggies” ITCS 4120/5120 41 Aliasing and Antialiasing ITCS 4120/5120 42 Aliasing and Antialiasing

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

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