Page 143 - Computational Colour Science Using MATLAB

130 CHARACTERIZATION OF CAMERAS
where A is the 363 system matrix. If three suitable samples are available, then
the linear system is exactly determined. Further samples could be used, to over-
determine the system, but are only strictly necessary if a linear transform does
not exist between the two colour spaces. In this situation it is usually preferable
to use a non-linear transform. Johnson (2002) notes that, even if a non-linear
transform is used, it is usually better to perform a linearization process and then
use approximately linear values as input to the non-linear transform.
Various non-linear transforms can be used such as
2
2
2
X ¼ a 11 R þ a 12 G þ a 13 B þ a 14 R þ a 15 G þ a 16 B þ a 17 RGB þ a 18 ,
2
2
2
Y ¼ a 21 R þ a 22 G þ a 23 B þ a 24 R þ a 25 G þ a 26 B þ a 27 RGB þ a 28 , ð8:3Þ
2
2
2
Z ¼ a 31 R þ a 32 G þ a 33 B þ a 34 R þ a 35 G þ a 36 B þ a 37 RGB þ a 38 ,
where, in this case, a total of 24 coefficients need to be determined. In matrix
notation we can write
T ¼ AD, ð8:4Þ
where the system matrix A is now a 368 matrix of the coefficients a –a . The
11
38
matrix D is the 86n column matrix of augmented device responses [in the case of
Equation (8.3) this is given by the terms R, G, B, R , G , B , RGB and 1].
2
2
2
The system is determined by computing the pseudoinverse (see Chapter 2,
Section 2.4) of the augmented matrix; thus
A ¼ D T. ð8:5Þ
þ
As an alternative to linear or non-linear transforms of this type it is also possible
to use a neural network to perform a mapping from C to T. However, it has been
shown that neural networks offer no advantage over polynomial transforms for
camera characterization (Cheung and Westland, 2002) and yet can be difficult
and time-consuming to train.

8.4 Implementations and examples

The first stage in characterizing an input device such as a scanner or a camera is
to linearize the measured RGB values. Table 8.1 lists the camera RGB values and
the mean reflectances for the grey samples of the Macbeth ColorChecker chart
which were measured using a typical high-end digital camera (Cheung and
Westland, 2002). Note that the first row of data in Table 8.1 does not show
measured values but implies that the camera gives a zero response for a zero
signal.
Figure 8.1 shows the relationship between the camera responses and the mean
reflectance P for the neutral patches of the ColorChecker. It is noticeable that
there is an approximately linear relationship between the RGB values and the P