Page 190 - Computational Colour Science Using MATLAB

IMPLEMENTATIONS AND EXAMPLES 177

Figure 10.7 The basis functions computed without subtracting the mean reflectance (*) are
compared with the basis functions computed after subtracting the mean reflectance (+)

T
a ¼ B P, ð10.17Þ
where B denotes the transpose of the matrix B. A set of vectors is called an
T
orthogonal set if all pairs of distinct vectors in the set are orthogonal. An
orthogonal set in which each vector has norm 1 is called orthonormal (Anton,
1994). Two non-zero vectors are orthogonal if and only if their dot product is
zero. If b is a 16w row matrix representing the first basis function and b is a
2
1
w61 column matrix representing the second basis function, then we can say that
b and b are orthogonal if Equation (10.18) is satisfied,
2
1
b 1 b 2 ¼ 0. ð10.18Þ
The norm of a matrix can be computed by the MATLAB function norm. The
norm of a matrix is also called the length of the matrix. Matrix b will be of
1
length 1 if Equation (10.19) is satisfied,
T
b b 1 ¼ 1.
1 ð10:19Þ
The special property of orthonormality allows Equation (10.17) to be used
instead of Equation (10.16) because the transpose of a matrix of length 1 is equal
to its inverse. Equation (10.17) is clearly easier to implement in a programming