Sang Il Park Sejong University Scan Conversion Sang Il Park Sejong University

Line Attributes Butt cap Round cap Projecting square cap Miter join Round Join Bevel join

Area Filling How can we generate a solid color/patterned polygon area?

How to decide interior -Parity Fill Approach For each pixel determine if it is inside or outside of a given polygon. Approach from the point being tested cast a ray in an arbitrary direction if the number of crossings is odd then the point is inside if the number of crossings is even then the point is outside

How to decide interior -Parity Fill Approach Very fragile algorithm Ray crosses a vertex Ray is coincident with an edge Commonly used in CAD Suitable for H/W

How to decide interior -Winding Number A winding number is an attribute of a point with respect to a polygon that tells us how many times the polygon encloses (or wraps around) the point. It is an integer, greater than or equal to 0. Regions of winding number 0 (unenclosed) are obviously outside the polygon, and regions of winding number 1 (simply enclosed) are obviously inside the polygon. Initially 0 +1: edge crossing the line from right to left -1: left to right use cross product of line and edge vectors No cross vertices

Area Filling (Scan line Approach) Take advantage of Span coherence: all pixels on a span are set to the same value Scan-line coherence: consecutive scan lines are identical Edge coherence: edges intersected by scan line i are also intersected by scan line i+1

Area Filling (Scan line Approach) For each scan line (1) Find intersections (the extrema of spans) Use Bresenham's line-scan algorithm (2) Sort intersections (increasing x order) (3) Fill in between pair of intersections

Area Filling (Scan line method)

Area Filling (Seed Fill Algorithm) basic idea Start at a pixel interior to a polygon Fill the others using connectivity seed

Seed Fill Algorithm (Cont’) 4-connected 8-connected Need a stack. Why?

Seed Fill Algorithm (Cont’) start position

Seed Fill Algorithm (Cont’) 8 6 4 2 0 2 4 6 8 10 0 2 4 6 8 10 8 6 4 2 interior-defined boundary-defined flood fill algorithm boundary fill algorithm

Seed Fill Algorithm (Cont’) 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 hole boundary pixel interior pixel seed pixel The stack may contain duplicated or unnecessary information !!!

Boundary Filling

Scan Line Seed Fill scan line conversion seed filling + Shani, U., “Filling Regions in Binary Raster Images: A Graph-Theoretic Approach”, Computer Graphics, 14, (1981), 321-327

Boundary Filling Efficiency in space! finish the scan line containing the starting position process all lines below the start line process all lines above the start line

Flood Filling : Start a point inside the figure, replace a specified interior color only.

Problems of Filling Algorithm What happens if a vertex is shared by more than one polygon, e.g. three triangles? What happens if the polygon intersects itself? What happens for a “sliver”? Solutions? Redefine what it means to be inside of a triangle Different routines for nasty little triangles

Aliasing in CG Which is the better?

Aliasing in CG Digital technology can only approximate analog signals through a process known as sampling The distortion of information due to low-frequency sampling (undersampling) Choosing an appropriate sampling rate depends on data size restraints, need for accuracy, the cost per sample… Errors caused by aliasing are called artefacts. Common aliasing artefacts in computer graphics include jagged profiles, disappearing or improperly rendered fine detail, and disintegrating textures.

The Nyquist Theorem the sampling rate must be at least twice the frequency of the signal or aliasing occurs

Aliasing Effects

Artifacts - Jagged profiles Jagged silhouettes are probably the most familiar effect caused by aliasing. Jaggies are especially noticeable where there is a high contrast between the interior and the exterior of the silhouette

Artefacts - Improperly rendered detail

Artefacts - Disintegrating textures The checkers should become smaller as the distance from the viewer increases.

Antialiasing Antialiasing methods were developed to combat the effects of aliasing. The two major categories of antialiasing techniques are prefiltering and postfiltering.

Prefiltering Eliminate high frequencies before sampling (Foley & van Dam p. 630) Convert I(x) to F(u) Apply a low-pass filter A low-pass filter allows low frequencies through, but attenuates (or reduces) high frequencies Then sample. Result: no aliasing!

High Frequency

Prefiltering

Prefiltering

Basis for Prefiltering Algorithms

Catmull’s Algorithm A2 A1 AB A3 Find fragment areas Multiply by fragment colors Sum for final pixel color

Prefiltering Example

Prefiltering So what’s the problem? Problem: most rendering algorithms generate sampled function directly e.g., Z-buffer, ray tracing

Supersampling The simplest way to reduce aliasing artifacts is supersampling Increase the resolution of the samples Average the results down Or sometimes, it is called “Postfiltering”.

Supersampling The process: Create virtual image at higher resolution than the final image Apply a low-pass filter Resample filtered image

Supersampling: Limitations Q: What practical consideration hampers super-sampling? A: Storage goes up quadratically Q: What theoretical problem does supersampling suffer? A: Doesn’t eliminate aliasing! Supersampling simply shifts the Nyquist limit higher

Supersampling: Worst Case Q: Give a simple scene containing infinite frequencies A: A checkered ground plane receding into infinity See next slide…

Supersampling Despite these limitations, people still use super-sampling (why?) So how can we best perform it?

Sampling in the Postfiltering method Supersampling from a 4*3 image. Sampling can be done randomly or regularly. The method of perturbing the sample positions is known as "jittering."

Stochastic Sampling Stochastic: involving or containing a random variable Sampling theory tells us that with a regular sampling grid, frequencies higher than the Nyquist limit will alias Q: What about irregular sampling? A: High frequencies appear as noise, not aliases This turns out to bother our visual system less!

Stochastic Sampling An intuitive argument: In stochastic sampling, every region of the image has a finite probability of being sampled Thus small features that fall between uniform sample points tend to be detected by non-uniform samples

Stochastic Sampling Integrating with different renderers: Ray tracing: It is just as easy to fire a ray one direction as another Z-buffer: hard, but possible Notable example: REYES system (?) Using image jittering is easier (more later) A-buffer: nope Totally built around square pixel filter and primitive-to-sample coherence

Stochastic Sampling Idea: randomizing distribution of samples scatters aliases into noise Problem: what type of random distribution to adopt? Reason: type of randomness used affects spectral characteristics of noise into which high frequencies are converted

Stochastic Sampling Problem: given a pixel, how to distribute points (samples) within it? Grid Random Poisson Disc Jitter

Stochastic Sampling Poisson distribution: Completely random Add points at random until area is full. Uniform distribution: some neighboring samples close together, some distant

Stochastic Sampling Poisson disc distribution: Poisson distribution, with minimum- distance constraint between samples Add points at random, removing again if they are too close to any previous points Very even-looking distribution

Stochastic Sampling Jittered distribution Start with regular grid of samples Perturb each sample slightly in a random direction More “clumpy” or granular in appearance

Nonuniform Supersampling Adaptive Sampling

Adaptive Sampling Problem: Final Samples Many more blue samples than white samples But final pixel actually more white than purple! Simple filtering will not handle this correctly Final Samples

Filters

Antialiasing http://www.siggraph.org/education/materials/HyperGraph/aliasing