Lecture_003_Drawing_a_Triangle_and_an_Intro_to_Sampling

Rasterization: given geometry, render pixel Ray Tracing: given pixel, render geometry

Primitives: whatever small things used to build big things that your graphic API supports.

Historically: it has actually taken the graphics community a long time to decide what the right building blocks for graphics should be. Two of my favorite diversions are swept spheres [which are both efficient to raytrace and can be rasterized reasonably efficiently as well] and implicit surfaces [which remain surprisingly relevant to this day]. (Jim McCann)

Implicit surface is used in Shader Toy but not popular because they are - Non Local: can’t figure out where they are without solving equations - hard to un-wrap/texture: surface not parameterize-able

Rasterization Pipeline

Why use triangle:

• can approximate any shape

• always planar, with well-defined normal vector

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

Two Questions:

1. Coverage: what pixel on screen does a triangle cover
2. Occlusion: if we have multiple triangles cover the same pixel, which one is closest to the camera

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)

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.

Usually for RGB pictures, we sample brightness and color in different frequency because brightness has higher frequency than color.

There are two problems with $\text{sinc}$ filter

1. Encode a hard edge (piecewise discontinuity in the triangle edge) needs infinite series of frequencies.
2. If for each pixel we use a $\text{sinc}$, then every pixel can affect every other pixel, therefore computational power is to expensive.

Implementation Details

Incremental Traversal: Check pixel near each other

Parallel Traversal:

1. bound a triangle by a box
2. check every pixel in the box in parallel

Coarse to Fine:

1. divide big box region into smaller boxes
2. check if the box overlaps with triangle
3. if overlap some, we subdivide more
4. if no overlap, all pixels in boxes are turned off
5. 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.

Diamond Exit Rule (for rasterizing line): Line is drawn if and only if it exit the diamond shape of a pixel. Any point in the line can only be exiting or entering the shape, not at the same time. This prevent redraw the same line when it is made up with 2 continuos segment. Specifically, a diamond will contain two opposite line and 2 points with inner area.

Top Left Rule (for rasterizing triangle): The top-left rule is that a pixel center is defined to lie inside of a triangle if it lies on the top edge or the left edge of a triangle.

For more specification: read Microsoft Rasterization, OpenGL, Vulkan Line Rasterization

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