Page 183 - Computational Colour Science Using MATLAB

170 MULTISPECTRAL IMAGING

10.3 Surface and illuminant estimation algorithms

Most algorithms for reflectance estimation rely on two essential components
(Brill, 1979). First, we need a method of representing our knowledge about the
likely surface and illuminant functions (for example, linear models). Secondly,
most modern estimation methods assume that the illumination varies either
slowly or not at all across the image. This assumption is important because it
means that the illumination adds very few extra parameters that need to be
estimated.
Consider an image with p distinct spatial positions. We expect to obtain three
cone excitations at each position in the image so the number of measurements is
3p in total. If we can use a three-dimensional model for the surfaces, then there
are a total of 3p unknown surface coefficients. If the illuminant is known, then
the problem is easy to solve since we have as many measurements (3p)as
unknowns (3p). If the illuminant is not known and can vary from point to point,
then there will be 6p unknown parameters (at each point, three parameters for
the surface and three parameters for the illuminant) and the problem cannot be
solved. If the illuminant is constant across the image we have only three
additional parameters (thus 3p+3 unknowns and 3p measurements) and by
making some modest assumptions we can proceed with the estimation algorithm.
Modern estimation algorithms work by finding a method to overcome the
mismatch between the measurements and the unknowns. The majority of
algorithms infer the illumination parameters by making one additional
assumption about the image contents. For example, if we know the reflectance
function of just one object in the scene, then we can use the sensor responses
from that object to estimate the illuminant. This is often implemented in terms of
the assumption that the average of all the surfaces in the image is grey – the so-
called grey-world assumption ( Land, 1986; Wandell, 1995). Other algorithms are
based on the assumption that the brightest surface in the image is a uniform
perfect reflector (Wandell, 1995). Another interesting idea is that we can identify
specularities in the image from glossy surfaces (D’Zmura and Lennie, 1986;
Tominaga and Wandell, 1990).
A second group of estimation algorithms compensates for the mismatch in
measurements and parameters by suggesting ways to acquire more data.
Maloney and Wandell (1986) showed that by adding a fourth sensor one can
estimate the surface and illuminant. Similarly, D’Zmura and Iverson (1993a,
1993b) explored the possibility of observing the same surface under different
illuminants. However, even if the illuminant is known the number of unknowns
may be greater than 3p if it is assumed that a linear model with greater than three
dimensions is required to represent the reflectance spectra. Multispectral imaging
is a technique that uses more than three channels so that sufficient information
about the scene is captured to allow spectral recovery to an accuracy greater than
that which would be possible using a three-dimensional linear model.