= Area-coverage antialiasing
<Area-coverage antialiasing> assigns an edge <pixel> a value based on the fraction of its footprint covered by a geometric shape. In a simple <rasterization> model with constant foreground value $C_f$, constant background value $C_b$, and a uniformly weighted <pixel> region $P$, let $S$ be the shape. Then
$$
\alpha=\frac{\operatorname{area}(P\cap S)}{\operatorname{area}(P)},\qquad C=\alpha C_f+(1-\alpha)C_b.
$$
Equivalently, $\alpha$ is the average of the shape's <indicator function> over $P$. This is a spatial average: a partially covered opaque shape can produce an intermediate <pixel> value without its material being transparent. Filtering the image over a finite region reduces the abrupt changes produced by deciding coverage at one sample point.<Physically Based Rendering: From Theory to Implementation (4th edition) by Matt Pharr and others>[section 8.8, image-function integration and reconstruction filters]{ref}
For a worked example, take the unit-square <pixel> $0\leq x,y\leq1$ and a white shape covering $y\leq x$ over a black background. The diagonal divides the square into equal <areas>, so $\alpha=1/2$ and the <pixel> gets half the white value. A single center sample lies exactly on the edge and must choose one side by a convention; a tiny edge movement can flip that binary decision. The covered <area> instead changes continuously. The arithmetic applies to values proportional to light, before a nonlinear display encoding.
Exact geometric coverage and multiple sample tests are two ways of estimating this average; a finite set of samples can miss a sufficiently thin shape. Uniform footprint averaging also has limits: its abrupt filter boundary does not remove every high-frequency component, so it reduces <aliasing> without guaranteeing a perfectly band-limited image.<Physically Based Rendering: From Theory to Implementation (4th edition) by Matt Pharr and others>[section 8.8.1, box filter and residual aliasing]{ref}
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