easy to interpolate data at corners (blend color using barycentric coordinates)

Two Questions:

Coverage: what pixel on screen does a triangle cover

Occlusion: if we have multiple triangles cover the same pixel, which one is closest to the camera

Sampling

Sampling Coverage

Breaking Ties

Reconstruction

Aliasing: When the frequency is higher than sampling rate, then reconstructed frequency can only be as high as sampling rate. (High frequencies in the original signal masquerade as low frequencies after reconstruction due to undersampling)

Image Reconstruction Example

Sinc Filter

Nyquist-Shannon Theorem: A band-limited signal (ie. no frequencies above threshold w_0) can be perfectly reconstructed if sampled with period T = \frac{1}{2} w_0 and reconstruct (interpolation) using \text{sinc} = \frac{1}{\pi x}\sin(\pi x) filter.

There are two problems with \text{sinc} filter

Encode a hard edge (piecewise discontinuity in the triangle edge) needs infinite series of frequencies.

If for each pixel we use a \text{sinc}, then every pixel can affect every other pixel, therefore computational power is to expensive.

Supersampling

Implementation Details

Incremental Traversal: Check pixel near each other

Parallel Traversal:

bound a triangle by a box

check every pixel in the box in parallel

Coarse to Fine:

divide big box region into smaller boxes

check if the box overlaps with triangle

if overlap some, we subdivide more

if no overlap, all pixels in boxes are turned off

if all overlap, all pixels in boxes are turned on

Note: Coarse to Fine can also be improved with Incremental Traversal since boxes near each other share similar pixel.