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Parametric Investigation of Multispectral Imaging
David Connor, Stephen Westland and Mitchell GA Thomson
Introduction
It is possible to recover estimates of the spectral reflectance of surfaces from sensor responses obtained from a
trichromatic imaging system captured under a known light source [1-3]. This is possible because the reflectance spectra
of most surfaces are highly constrained to be smooth functions of wavelength [4] and the most natural light sources are
similarly constrained [5]. Thus reflectance spectra can be represented by linear models with few parameters. For
example, there exist a set of N basis functions Bj that can be linearly combined to approximate all reflectance spectra Pi
in a large set:
Pi   ai,jBj
(Eqn 1)
where ai,j are a set of coefficients or weighting functions.
If only three basis functions were required to represent exactly the reflectance spectra then it would be possible to
recover exactly the reflectance spectra of surfaces captured by a suitable trichromatic imaging system. However, whilst
three basis functions capture a surprising degree of the variance of many sets of reflectance spectra recent studies
suggest that at least six [4] and possibly more [6] basis function are required to even closely approximate reflectance
spectra of natural and man-made surfaces. The deficiencies of trichromatic imaging systems has led towards the
development of multispectral imaging system with more than three wavelength-selective sensors or channels [7-8].
Recovery of reflectance spectra is possible using a multispectral imaging system by solving Eqn 2 which represents the
response of the system r as a function of a system matrix M and the spectral reflectance p, thus
r = Mp.
(Eqn 2)
Eqn 2 has no unique solution for p since M is a 3  3 matrix and r is a 31  1 matrix. However, Eqn 2 can be rearranged
to form Eqn 3 which represents the response as a function of a set of weights  and a matrix L thus
r = L.
(Eqn 3)
Matrix L is a 3  3 matrix that includes the sensitivity of the imaging system, the spectral distribution of the light
source, and three basis functions Bi that form a linear model of the reflectance spectra. The solution of Eqn 3 is trivial to
yield  and an estimate for p is then deduced using Eqn 1. Eqn 3 can be extended to include more than three sensor
responses N in which case L becomes an N  N matrix and the number of basis functions in the linear model is
correspondingly N.
Research Questions
The mathematics for recovering reflectance spectra from multispectral imaging systems is well established [9].
However, several decisions are critical to the design and development of a multispectral imaging system and the basis
for these decisions is rather less well understood. For example: (i) how many sensor channels are required to allow
recovery of reflectance spectra to a target degree of accuracy? (ii) for a given number N of sensors what are the spectral
sensitivity distributions for the sensors? (iii) what are the optimum light sources for capturing colour signals in a
multispectral imaging system?
Furthermore, the solution of Eqn 3 assumes that the spectral sensitivities of the channels are known. In many practical
applications this is not the case and an analogous linear system can be constructed to allow estimation [10] of the
spectral sensitivities based upon sensor responses of a set of known surfaces (such as the ColorChecker). How many
surfaces are required in order to obtain reliable estimates of the channel sensitivities and how should these surfaces be
selected?
Although some partial answers to these questions have been published elsewhere they are usually specific to a given
imaging system, light source, or collection of known surfaces. The approach that we have adopted is to develop a
virtual model of an imaging system to allow a thorough investigation of the parameters of multispectral imaging.
Experimental
We have simulated the operation of trichromatic and multispectral imaging systems using MATLAB software and
mathematical descriptions of the sensor responses, the light source, and the spectral reflectances of surfaces.
Performance of the imaging system has been quantified by reporting the statistics of colour differences between
reconstructed spectra and real spectra. The following parameters are investigated: (i) number of channels; (ii) spectral
sensitivity of channels; (iii) spectral distributions of light sources. In addition, estimates of spectral sensitivity are made
for varying sets of known surface reflectances. The performance of the model is validated using a real imaging system
comprising of a monochromatic digital camera and a set of filters that can be placed in front of the lens to emulate a
multispectral imaging system.
Results and Discussion
The virtual imaging system has been constructed and work is underway to investigate the parameters of multispectral
imaging and to validate the performance of the virtual system using a real imaging system. An initial analysis of the
data suggest that reconstruction errors of reflectance spectra decrease as the number of channels in the virtual system is
increased and as the spectral power distribution of the light source becomes progressively smoother. The performance
of the imaging system will be discussed in terms of its spectral modulation transfer function [11].
References
1.
Maloney LT & Wandell BA (1986), Color constancy: A method for recovering surface spectral reflectance, JOSA,
A 3 (1), pp29-33.
2. Marrimont DA & Wandell BA (1992), Linear models of surface and illuminant spectra, JOSA, A 9 (11), pp19051913.
3. D’Zmura M & Iverson G (1993a), Color constancy. I. Basic theory of two-stage linear recovery of spectral
descriptions for lights and surfaces, JOSA, A (10), pp2148-2165.
4. Maloney LT (1986), Evaluation of linear models of surface spectral reflectance with small numbers of parameters,
JOSA A, 3, pp1673-1683.
5. Judd DB, MacAdam DL & Wyszecki GW (1964), Spectral distribution of typical daylight as a function of
correlated colour temperature, JOSA, 54, pp1031-1040.
6. Westland S & Thomson MGA (2000), Spectral colour statistics of surfaces: recovery and representation, Colour
Image Science 2000 (Derby University), Proceedings, pp33-40.
7. Sugiura H, Kuno T, Watanabe N, Matoba N, Hayashi J & Miyake Y (1999), Development of high accurate
multispectral cameras, Multispectral Imaging and Color Reproduction for Digital Archives (Chiba University),
Proceedings, pp73-80.
8. Imai FH & Berns RS (1999), Spectral estimation using trichromatic digital cameras, Multispectral Imaging and
Color Reproduction for Digital Archives (Chiba University), Proceedings, pp42-49.
9. Wandell BA (1995), Foundations of Vision, Sinauer Associates.
10. Finlayson G, Hordley S & Hubel PM (1998), Recovering device sensitivities with quadratic programming,
Proceedings of the 6th color imaging conference (Scottsdale, Arizona), pp90-95.
11. Hearn DR (1998), Characterization of instrument spectral resolution by the spectral modulation transfer function,
SPIE, 3439, pp400-407.
Correspondence: Dr Stephen Westland, [email protected]
Colour Imaging Institute, Kingsway House, Derby, DE22 3HL, UK.