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TECHNIQUES FOR MULTISPECTRAL IMAGING 171

10.4 Techniques for multispectral imaging

In this section we consider some typical techniques to allow reflectance recovery
using multispectral imaging.

10.4.1 The Hardeberg method

The method proposed by Hardeberg (1999) assumes a linear camera model
[Equation (10.1)] and has similarities to the method of reflectance recovery
proposed by Maloney and Wandell (1986). For a single surface, the Hardeberg
method is based upon Equation (10.10), so that
r ¼ Ka, ð10.10Þ

where r is an r61 matrix of sensor responses, K is an r6n system matrix and a is
an n61 column matrix of weights that defines the surface in the space of basis
functions. Since it is known (Maloney, 1986; Owens, 2002b) that a linear model
of at least six basis functions is required for the accurate representation of
reflectance spectra, r must be at least size 661. Most practical multispectral
imaging systems therefore consist of at least six separate channels and this may
be achieved by a filter wheel containing a number of different filters and a
monochrome camera system.
If we consider the case where r ¼ n ¼ 3, then the 363 matrix K would be
obtained from the product of the 3631 matrix of the sensor spectral sensitivities
(weighted by the illuminant power distribution) and the 3163 matrix of the basis
functions of the linear model. The entries of K are thus known.
The reflectance of the surface may be recovered by manipulating Equation
(10.10) to yield
1
a ¼ K r, ð10.11Þ
where the inverse K 71 may easily be computed if r ¼ n (when K is a square
matrix). An alternative procedure to using more sensor classes is to use more
than one light source. For example, if an image is taken using a trichromatic
camera using one light source and then the same image is taken using a second
light source, then it allows the construction of a matrix K with six rows and
consequently allows a linear model of reflectance spectra with six basis functions.
It is important, however, that the spectral power distributions of the two or more
light sources are as unrelated as possible and that they contain power throughout
the whole visible spectrum (Connah et al., 2001).
Hardeberg (1999) has also considered the situation where r4n so that the
number of sensor classes exceeds the number of basis functions in the linear
model of reflectance. This leads to Equation (10.12),
a ¼ K r, ð10.12Þ
þ