CISC/CMPE 454 - Nearest and Bilinear Lookup

A texture unit is given floating point coordinates and returns a value. There are three principal methods of computing the return value.

In what follows,

Let $(s,t)$ be floating-point coordinates in $[0,1] \times [0,1]$ .
Let $M$ be a 2D texture map of dimensions $c \times r$ .
Let $\hat{s} = s (c-1)$ and let $\hat{t} = t (r-1)$ .

Then $(\hat{s},\hat{t})$ are in $[0,c-1] \times [0,r-1]$ and are the texel coordinates in the 2D texture array of size $c \times r$ .

When reading the texture array at $M[\hat{s},\hat{t}]$ , the coordinates are taken modulo $c$ and modulo $r$ so that they are always inside the array. This has the effect of "wrapping the coordinates around" when they go out of bounds.

Nearest Lookup

Given $(s,t)$ , this method return the array texel whose centre is closest to $(\hat{s},\hat{t})$ , that is:

$M[ \; \textrm{round}(\hat{s}), \; \textrm{round}(\hat{t}) \; ]$

Bilinear Interpolation

Given $(s,t)$ , this method takes a weighted average of the four array texels closest to $(\hat{s},\hat{t})$ .

The weights are determined by placing a texel-sized square at $(\hat{s},\hat{t})$ , then weighting each of the four texels in proportion to the amount of overlap it has with the square.

For coordinates $(\hat{s},\hat{t})$ , let

$S = \lfloor \hat{s} \rfloor$
$T = \lfloor \hat{t} \rfloor$
$\alpha = \hat{s} - S$
$\beta = \hat{t} - T$

Then the weighted average is

$\begin{array}{rccl} & (1-\alpha) & (1-\beta) & M[S,T] \\ + & (1-\alpha) & \beta & M[S,T+1] \\ + & \alpha & (1-\beta) & M[S+1,T] \\ + & \alpha & \beta & M[S+1,T+1] \end{array}$

Texture Lookup

Nearest Lookup

Bilinear Interpolation