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172 MULTISPECTRAL IMAGING
where the pseudoinverse of the non-square matrix K is computed. Equation
(10.12) refers to an over-determined system, but this technique can actually lead
to improved estimates when compared with the case r ¼ n. There are two reasons
why an over-determined system may be useful. First, for a system based upon r
sensors the r rows of K may not be independent. This can happen if the spectral
sensitivities of the channels are correlated with each other (or, for a system using
two light sources, if the spectral power distributions of the light sources are
correlated). Secondly, estimates of a when r ¼ n may suffer if the system is noisy
so that the matrix r is known with low precision.

10.4.2 The Imai and Berns method

Imai and Berns (1999) have developed a method for reflectance recovery based
directly upon Equations (10.10) and (10.11). They assume a linear relationship
between the sensor outputs r of the imaging system and the representation of the
surface in an r-dimensional basis space by the weights a. However, unlike the
Hardeberg method, Imai and Berns find the entries of K using an empirical least-
squares analysis. The method is simple and effective because for the Hardeberg
method it is necessary to determine the space of basis functions in which the
reflectance spectra will be represented, to measure the spectral power distribution
of the light source and to determine the spectral sensitivities of the imaging
system. The method proposed by Imai and Berns, however, requires only the first
of these steps, namely the determination of the basis functions, and the entries of
K are then found by optimization.

10.4.3 Methods based on maximum smoothness

One problem with methods for reflectance recovery that use basis functions is
that the recovered reflectance cannot be guaranteed to be within the range [0, 1].
The methods described in Sections 10.4.1 and 10.4.2 do not always yield
physically reasonable solutions. An alternative approach to reflectance recovery
is to replace the constraint imposed by the linear model of basis functions with
some other constraint. One possibility is to employ a constraint of maximum
smoothness (Li and Luo, 2001).

10.5 Implementations and examples

10.5.1 Deriving a set of basis functions

Principal Component Analysis (PCA) may be performed using MATLAB’s
singular value decomposition function svds. Consider the 100 observations of