Antialiasing

Han-Wei Shen

Wh t i li ?What is alias? Alias A false signal in telecommunication links from Alias - A false signal in telecommunication links from

beats between signal frequency and sampling frequency(from dictionary.com)(from dictionary.com) Not just telecommunication, alias is everywhere in

computer graphics because rendering is essentially a lisampling process

Examples: Jagged edges Jagged edges False texture patterns

Ali d b d liAlias caused by under-sampling

1D signal sampling example 1D signal sampling example

Actual signalSampled signal

Ali d b d liAlias caused by under-sampling

2D texture display example 2D texture display example

Minor aliasing worse aliasingMinor aliasing worse aliasing

H ft i h?How often is enough?

What is the right sampling frequency? What is the right sampling frequency? Sampling theorem (or Nyquist limit) - the

li f h t b t l t t isampling frequency has to be at least twice the maximum frequency of the signal to be sampledsampled

Need two samples in this cycle

R t tiReconstruction

After the (ideal) sampling is done we need to After the (ideal) sampling is done, we need to reconstruct back the original continuous signalsignal

The reconstruction is done by reconstruction filterfilter

R t ti FiltReconstruction Filters

Common reconstruction filters: Common reconstruction filters: Box filter Tent filter

Sinc filter = sin( x)/ xSinc filter = sin(x)/x

A ti li d t t iAnti-aliased texture mapping

Two problems to address Two problems to address – Magnification Minification

R liRe-sampling

Minification and Magnification resample the Minification and Magnification – resample the signal to a different resolution

Minification

Magnification

(note the minification is done badly here)

M ifi tiMagnification

Simpler to handle just resample the Simpler to handle, just resample the reconstructed signal

Reconstructed signal Resample the reconstructed signalg p g

M ifi tiMagnification

Common methods: nearest neighbor (box Common methods: nearest neighbor (box filter) or linear interpolation (tent filter)

Nearest neighbor bilinear interpolation

Mi ifi tiMinification

Harder to handle Harder to handle The signal’s frequency is too high to avoid

li ialiasing A possible solution: Increase the low pass filter width of the ideal sinc

filter – this effectively blur the imageBlur the image first (using any method) and then Blur the image first (using any method), and then sample it

Mi ifi tiMinification Several texels cover one pixel (under

sampling happens) Solution:

Either increase samplingrate or reduce the textureFrequency

Under sample artifactone pixel We will discuss:

1. Mip-mapping p pp g2. Rip-mapping3. Sum Area Table

Mi iMip-mapping By Lance Williams

average 2x2 texels By Lance Williams

(SIGGRAPH ’83) Mip – many things in a

ll l

texels

d=3

…

small place The original texture is

augmented with a set of low d=2

resolution textures Which resolution to use

depends on the screen d=1depends on the screen projection size – try to maintain pixel to texel ration close to 1

d=1

ration close to 1

Mi iMipmapping

Two common measures are used to compute d Two common measures are used to compute d. Use the longer edge of the quadrilateral

U th l t b l t l f th f Use the largest absolute value of the four differentials (u/x, v/x, u/y, v/y)

I l b l l Interpolate between levels

Mi i blMipmapping problem

Overblurring! When a pixel covers many u Overblurring! – When a pixel covers many u texels but few v texels, we always choose the largest pixel coverage to decide the levellargest pixel coverage to decide the level

Non mipmapping

mipmapping

Ri iRipmapping Correct the over-blurring problem of Correct the over blurring problem of

mipmapping – separating x and y filtering

• Scale by half by x across a row.

• Scale by half in yScale by half in y going down a column.

• The diagonal has the i l t iequivalent mip-map.

• Four times the amount of storage is required.

• Four different texel values to average

S d A T blSummed Area Table

Another way to perform anisotropic filtering Another way to perform anisotropic filtering -can be used to compute the average color for any arbitrary rectangular region in the textureany arbitrary rectangular region in the texture space at a constant speed

A two dimensional array that has the same A two dimensional array that has the same size as the texture

Every pixel stores the sum of all the texel colors in thelower left corner

S d A T bl (SAT)Summed Area Table (SAT)

How to calculate the sum of texels in the area How to calculate the sum of texels in the area between A and B?

BCR = SAT[B] – SAT[C] – SAT[D] + SAT[A]

RA

BC

DThe average value can be computed by

R / num of texels in R

S d A T bl (SAT)Summed Area Table (SAT)

is used to filter the texture is used to filter the texture

polygon Texture spacep

y

pixel

Compute the rectangular bounding box of the pixel area in the texture and thend t bl t t th l (i filt th t l h

x

use summed area table to compute the average color (i.e., filter the texels when magnification takes place)

S d A T blSummed Area Table

Less blurry than Mipmapping Less blurry than Mipmapping Still blurry when the texture is viewed along

it di l thi i b th b diits diagonal – this is because the bounding box of the pixel area is larger than the actual pixel areapixel area

polygon Texture spacenot quit fit

y

pixel x

C iComparison

Non filtering

Mipmapping

Summed area table

Note that only mipmap is implemented by hardware and supported by OpenGL