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