Page 141 - Computational Colour Science Using MATLAB

128 CHARACTERIZATION OF CAMERAS

8.2 Correction for non-linearity

Although the response of charge-coupled diode (CCD) material is approximately
linearly related to the intensity of the light falling on it, it is unlikely that the RGB
outputs of a scanner or digital camera will be linearly related to the XYZ
tristimulus values of the surfaces in the scene. The raw channel responses are
invariably processed by on-board software in an attempt to generate RGB
responses that are more closely matched to the colour-matching functions than
are allowed by current methods for producing filters. Typically, the raw RGB
values may be transformed by a 363 linear matrix to give the output RGB
values. Furthermore, many manufacturers impose a non-linearity during this
‘matrix-mixing’ stage to approximately match the inverse of the non-linearity of
display systems or as part of the solution to provide high signal-to-noise ratios. It
is therefore common to consider a correction for non-linearity as the first stage of
a camera- or scanner-characterization process. For a digital camera we may, for
example, consider a relation of the form
0 p
C c ¼ðC c Þ , ð8:1Þ
where C’ is the raw response of the camera channel c, p is an exponent for the
c
channel, and C is a transformed camera response for that channel that is linear
c
to the channel input. A set of grey-scale samples is often used to empirically
determine the exponent p. Thus, the raw camera responses are determined for a
range of grey samples under a constant and known light source. The XYZ values
then can easily be computed for the grey samples and linearization is achieved by
finding the value of the exponent p such that there is a linear relationship
between C and Y for the set of grey samples.
c
However, in order to correctly determine p [according to Equation (8.1)] we
need to know the values of C and C’ for each of the grey patches. The value of
c c
C’ is the input to the channel and the value of C is the channel response
c c
following non-linear processing (Thomson and Westland, 2002). The input to the
channel may easily be computed if the spectral reflectance of the sample, the
spectral power of the illumination and the channel sensitivity are all known.
Unfortunately, it is usually the case that the channel sensitivity is not known. If p
is determined so that there is a linear relationship between C and Y, then it is
c
clear that the Y colour-matching function or photopic sensitivity is being used as
a crude estimate of the channel sensitivity. For grey samples imaged under an
equal-energy illuminant, Y and C’ are approximately linearly related but this
c
proportionality decreases for increasingly chromatic samples and light sources. It
is interesting to note that if the channel sensitivity was known, and the true
values of C’ could be computed, then it would not be necessary to use grey
c
samples to linearize the channel responses; any samples could be used. Since the
spectral sensitivities generally are not known, however, linearization is typically
established using achromatic samples. The grey samples of the Macbeth