aliasing and antialiasing - unc charlotteanti-aliasing techniques preﬁltering (unweighted/weighted...

Aliasing and Antialiasing

ITCS 4120/5120 1 Aliasing and Antialiasing

What is Aliasing?

“Errors and Artifacts arising during rendering, due to the conversionfrom a continuously defined illumination field to a discrete rastergrid of pixels”


What is Aliasing?


Effects of Aliasing


Anti-aliasing


Anti-aliasing Techniques

� Prefiltering (unweighted/weighted area sampling)

� Postfiltering (supersampling, jittering)


Area Sampling Techniques


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.

Properties

� Intensity of pixel decreases as the distance between the pixel centerand primitive increases.

� A primitive cannot influence a pixel’s intensity if it does not intersectit.

� Equal areas (intersected) contribute equal intensity – not a desirableproperty.


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 thepixel with decreasing radial influence.

� Cone filters are a compromise between computational expense andoptimality.


Postfiltering Techniques


Supersampling (Regular Sampling)

� Very expensive.

� Not very satisfactory.


Regular vs. Jittered Sampling


Filtering


Filtering Example


Aliasing from a Sampling Theory Viewpoint

Sampling(Spatial Domain)


Sampling(Spatial Domain)

� Image is a spatial signal


Frequency Domain

� X axis (position): frequency

� Y axis (height): strength of each frequency

� Examples: sine wave: impulse, square wave: infinite train of im-pulses


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

={φ(x)} = φ(ω) =

∫ ∞

−∞φ(x)e−j2πωx dx

is the Fourier Transform of φ(x), with j =√−1 and,

=−1{φ(ω)} = φ(x) =

∫ ∞

−∞φ(ω)ej2πωx dω

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

What does the Fourier Transform Do to ASpatial Signal?

◦ Signal in frequency domain is an integration of individual sinusoids.ITCS 4120/5120 32 Aliasing and Antialiasing

How does this related to Graphics?

◦ Images are just a 2D signal and jagged edges are due to the pixelsampling rate not being high enough to capture the “real signal.


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


Nyquist Rate


Nyquist Rate:Undersampling

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


Comb Function

� Application: Used to digitize continuous functions.

� Series of impulses (delta functions)

� Identity element of convolution: reproduces an indentical copy of thefunction f(x)

� FT of a comb function is another comb functionITCS 4120/5120 37 Aliasing and Antialiasing

Comb Function(contd)

� Multiplying f (x) with a comb in image space ⇐⇒ convolving theirFourier transforms, resulting in multiple identical copies of ={f (x)}

� Can result in aliasing if copies overlap

� Maximum allowable frequency is the Nyquist Frequency, which ishalf the sampling frequency.


Reconstruction Example(Adequate Sampling)


Reconstruction Example(Inadequate Sampling)


Box Filter

� Reconstruction filter for nearest neighbor interpolation.

� Resampling images/volumes to a higher resolution using nearestneighbor values.

� FT of a box filter is the Sinc function (sinπxπx

)

� Large side lobes continuing at regular intervals will cause aliasing.

� Aliasing in images manifests itself as “jaggies”ITCS 4120/5120 41 Aliasing and Antialiasing

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


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


aliasing and antialiasing - unc charlotteanti-aliasing techniques preﬁltering (unweighted/weighted...

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