Rasterization

4.1 Display

An image is a 2D array of pixel colors. Display hardware converts these values into visible light.

Cathode Ray Tube (CRT):

• An electron beam is emitted, accelerated, focused, and deflected onto a phosphor-coated screen.
• Beam intensity controls brightness.
• Applications include oscilloscopes and traditional televisions.

Raster Scan: The beam sweeps across the screen line by line while its intensity is modulated. Interlaced scanning alternates odd and even lines in successive fields.

Frame Buffer: Memory storing the color of each pixel. For analog CRT displays, DACs (Digital-to-Analog Converters) convert digital RGB values into analog signals.

Flat-Panel Displays:

Technology	Principle
LCD	Liquid crystals change light polarization; polarizers control how much backlight passes through. Partial rotation produces intermediate brightness levels.
OLED	Organic light-emitting materials produce their own light.
LED array	Individually controlled LEDs form an image.
E-Ink	Electric fields move charged black and white particles to change the visible color.

4.2 Rasterizing a triangle

Triangles as Fundamental Shape Primitives:

• Guaranteed to be planar
• Well-defined interior
• Well-defined method for interpolating vertex values over the triangle

Rasterization as Sampling: Sample a 2D indicator function by testing whether each pixel center lies inside the triangle.

Incremental Triangle Traversal:

Visit candidate pixel centers and update edge-function values using additions instead of recomputing them at every pixel. For each triangle edge:

\[ \begin{aligned} E(x,y) &= Ax + By + C, \\ E(x+1,y) &= E(x,y) + A, \\ E(x,y+1) &= E(x,y) + B. \end{aligned} \]

Compute the three edge functions at the starting pixel, then update them as traversal proceeds. Their signs determine coverage, using a consistent vertex winding and boundary rule.

Hierarchical Rasterization: Test blocks of pixels before testing individual samples.

• Fully outside: Skip the entire block.
• Fully inside: Mark all samples as covered; color and depth still need to be computed.
• Partially intersecting: Subdivide into smaller blocks until individual pixel tests are needed.

For example, subdivide a \(4\times4\) block into \(2\times2\) blocks, then individual pixels. Incremental edge-function updates can be used within the remaining blocks.

4.3 Antialiasing

Sampling Artifacts in Computer Graphics:

Artifacts caused by sampling are called aliasing:

• Jaggies: Sampling in space.
• Moiré patterns: Undersampling images.
• Wagon-wheel effect: Sampling in time.

Aliasing occurs when a signal changes too quickly (high frequency) relative to its sampling rate.

Antialiased Sampling:

Original signal → Pre-filter → Sample

• Remove frequencies above the Nyquist limit before sampling.
• Along antialiased triangle edges, pixel values take intermediate values.

Fourier Transform: Spatial and Frequency Domains:

• Spatial domain: \(f(x)\).
• Frequency domain: \(F(\omega)\).

The Fourier transform decomposes a signal into frequencies.

Forward transform — spatial domain → frequency domain:

\[ F(\omega) = \int_{-\infty}^{\infty} f(x)e^{-2\pi i\omega x}\,dx. \]

Inverse transform — frequency domain → spatial domain:

\[ f(x) = \int_{-\infty}^{\infty} F(\omega)e^{2\pi i\omega x}\,d\omega. \]

Sampling and Frequency Replicas: Sampling repeats the frequency content.

\[ S(f) = \frac{1}{T} \sum_k X(f - k \cdot f_s). \]

Reducing Aliasing Error:

1. Increase the sampling rate: Increase the distance between replicas in the frequency domain.
2. Apply antialiasing: Narrow the frequency content before repeating it.

Filtering: Remove selected frequency components using either approach:

1. Spatial domain: Convolve the signal with a filter kernel.
2. Frequency domain:
   • Apply the Fourier transform.
   • Multiply by the Fourier transform of the convolution kernel.
   • Apply the inverse Fourier transform.

Antialiasing by Computing Average Pixel Values:

For a triangle indicator function \(f(x,y) = \text{inside}(\text{triangle},x,y)\), the average over a pixel gives the fraction of its area covered by the triangle.

Supersample Antialiasing:

Approximate a one-pixel box filter by:

1. Sampling multiple locations within each pixel.
2. Averaging the sample values.

Antialiasing Methods at a Glance:

Method	Core idea	Characteristics
SSAA	Multiple samples per pixel, all fully shaded	Best quality, highest cost
MSAA	Multiple coverage/depth samples, shade once	Only improves geometric edges, high memory cost
FXAA	Post-process edge blurring	Fastest, blurry result
TAA	Accumulate jittered samples across frames	Cheap and effective, may ghost
DLSS	Neural-network reconstruction of high resolution	Improves both performance and quality

4.4 Z-Buffering

Idea: Store the minimum depth value for each sample (pixel).

• Frame buffer: Stores color values.
• Depth buffer (z-buffer): Stores depth values in an additional buffer.

Z-Buffer Algorithm (Pseudocode):

// Initialization
for each sample (x, y):
    zbuffer[x][y]     = +infinity   // farthest depth
    framebuffer[x][y] = background_color

// Rasterization
for each triangle T:
    for each sample (x, y) covered by T:
        z = interpolate_depth(T, x, y)
        if z < zbuffer[x][y]:       // closer than the current one
            zbuffer[x][y]     = z
            framebuffer[x][y] = shade(T, x, y)
        // else: discard the fragment