Graphics Reference
In-Depth Information
which after simple transformations becomes
err
=
n
r
r
i
+
n
g
g
i
+
n
b
b
i
i
i
i
+2
n
r
n
b
+2
n
g
n
b
.
r
i
g
i
r
i
b
i
g
i
b
i
+2
n
r
n
g
i
i
i
For minimalistic implementation, we can use the above equation to compute
all six partial sums depending only on the texel colors. Then we can use a brute
force approach to test a predefined set of potential normal vectors to find the
one minimizing the total approximation error. Because each test is carried out
in linear time, costing only several multiplications and additions, this approach
is still tolerably fast.
The final step after finding the optimal color plane is to revert the color
space distortion caused by the color component weighting by scaling using the
reciprocal weights. Because the plane normal is a surface normal vector, the usual
rule of non-uniform space scaling for normals applies and we have to multiply the
normal by the inverse transpose of the matrix we would use otherwise. While the
transposition does not affect the scaling matrix, the matrix inversion does and
the final scaling operation is using non-reciprocal weights again:
⎛
−
1
⎞
⎠
⎡
⎤
⎡
⎤
T
1
/w
r
0
0
w
r
00
0
w
g
0
00
w
b
⎝
⎣
⎦
⎣
⎦
.
N
=
N
0
/w
g
0
=
N
0
0
1
/w
b
As all subsequent computation is typically done in the linear RGB space, we
do not have to convert into sRGB (which would be nonlinear transform anyway).
2.2.2 Computing Base Colors
The important parameters for the encoding and the decoding process are the two
base colors. The color plane cutting through the RGB unit cube forms a triangle
or a quadrilateral, with one of the corners placed at the point (0,0,0). The two
corners neighboring the point (0,0,0) in this shape are defined as the base colors
for the planar color space, as shown on Figure 2.2. Every other color available
on the plane lies within the angle formed by the point (0,0,0) and the two base
color points. Because the color plane starts at (0,0,0) and enters the unit cube,
the base color points will always lie on the silhouette of the unit cube, as seen
from the point (0,0,0). To find the base colors, we can simply compute the plane
intersection with the silhouette edges, resulting in the desired pair of points. We
have to bear in mind that the plane can slice through the silhouette vertices, or
even embed a pair of silhouette edges. Therefore, to compute the points we can
