Computer Graphics

Spring 2007, #4Anti-aliasing

Sampling problem

• For chosen technical reasons a computer screen today consists of finite area pixels with constant properties over the pixel area

• Any mathematically exact object is approximated on the screen by a set of pixels

• Any object is thus undersampled and pixels are not where they should be: aliasing

Anti-aliasing

• Typical appearance of aliasing: a straight line or edge is jagged

• Is there any way to improve the result by changing the sampling?

• In principle no: we are often truely losing information when undersampling

• Often expressed as Nyqvist’s theorem: A continuous time signal containing frequencies f ≤ B can be exactly reproduced from samples taken at sampling frequency fS > 2B.

Anti-aliasing

• Two basic principles of improving the appearance by slightly blurring the drawing: change the intensities of Bresenham pixels and neighbouring pixels– supersampling (postfiltering)

• we calculate subpixels and then integrate to obtain original pixels

– area sampling (prefiltering)• we calculate pixel intensities without pixellated

quantities

Supersampling

• Choose a finer subgrid, 2x2, 3x3, ...• Run your normal line drawing algorithm on

this finer grid.• From the result, calculate the intensity for

the original pixel

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10 11 12

21

22

X

Y

Supersampling

• For each original pixel divided into NxN subpixels, 0 or 1 or 2 or ,.. , or N subpixels will be drawn by the line drawing algorithm. This gives us a natural choice of intensity level [0,N] for each pixel.

• We can also give a relative weight to the subpixel, eg. 2 corresponds to 2/16 = 1/8

1

2 4 2

1 2 1

12

Area sampling

• We try to estimate the area of the pixel covered by a rectangle of width one pixel

• Normalizing the area of a pixel to 1.0: the estimated area = the pixel intensity

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10 11 12

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X

Y

Area sampling

• Area estimation: Calcualte the number of subpixels within the rectangle. A subpixel is inside if its lower lefthand corner is inside the rectangle.

Anti-aliasing Area Boundaries

• Area boundaries are notoroius for their jaggedness

• Estimate pixel intensities at area boundaries from the fraction of the pixel inside the boundary

• Do this concurrently with scan-line algorithms

Anti-aliasing Area Boundaries

• The pixel area covered by the interior of a polygon can sometimes be calculated accurately very quickly: mxk+b-yk+m/2

y = mx + b

xk xk+1

yk

yk+1