## Abstract

In multispectral imaging it is advantageous to compress spectral information with a minimum loss of information in a way that permits accurate recovery of the spectrum. By use of the simple vector-subspace model that we propose, spectral information can be stored and recovered by the use of a few inner products, which are easy to measure optically. Two large data sets, the first consisting of 1257 Munsell colors and the other of 218 naturally occurring spectral reflectances, were analyzed to form two bases for the model. The Munsell basis can be used to represent the natural colors, and the basis derived from the natural data can be used to represent the Munsell data. The proposed vector-subspace model includes a simple relation between the inner products and conventional color coordinates. It also provides a way to estimate the spectrum of an object that has known chromaticity coordinates.

© 1990 Optical Society of America

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### Equations (12)

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(1)
$$\tau ={[\tau ({\mathrm{\lambda}}_{1}),\tau ({\mathrm{\lambda}}_{2}),\dots ,\tau ({\mathrm{\lambda}}_{n})]}^{T},$$
(2)
$$L=L({\upsilon}_{1},{\upsilon}_{2},\dots ,{\upsilon}_{p})=\left\{x|x=\text{\u2211}_{1}^{p}{a}_{i}{\upsilon}_{i}\right\},$$
(3)
$${\tau}^{\u2019}=\text{\u2211}_{1}^{p}({\tau}^{T}{\upsilon}_{i}){\upsilon}_{i}$$
(4)
$${\tau}^{\prime}=\text{\u2211}_{1}^{p}({\upsilon}_{i}{{\upsilon}_{i}}^{T})\tau =P\tau ,$$
(5)
$$\begin{array}{ll}\delta (\tau ,L)\hfill & ={\left[\stackrel{\u02c6}{\parallel}\tau {\stackrel{\u02c6}{\parallel}}^{2}-\text{\u2211}_{1}^{p}\hspace{0.17em}{({\tau}^{T}{\upsilon}_{i})}^{2}\right]}^{1/2}\hfill \\ \hfill & ={\left[\text{\u2211}_{1}^{n}{[\tau ({\mathrm{\lambda}}_{k})-{\tau}^{\prime}({\mathrm{\lambda}}_{k})]}^{2}\right]}^{1/2},\hfill \end{array}$$
(6)
$$\left(\begin{array}{l}X\hfill \\ Y\hfill \\ Z\hfill \end{array}\right)={k}_{1}\mathit{\int}\tau (\mathrm{\lambda})S(\mathrm{\lambda})\left[\begin{array}{l}\overline{x}(\mathrm{\lambda})\hfill \\ \overline{y}(\mathrm{\lambda})\hfill \\ \overline{z}(\mathrm{\lambda})\hfill \end{array}\right]\text{d}\mathrm{\lambda},$$
(7)
$${X}_{m}=k\text{\u2211}_{j=1}^{n}\tau ({\mathrm{\lambda}}_{j}){\mathrm{\Phi}}_{m}({\mathrm{\lambda}}_{j}),$$
(8)
$$\begin{array}{cc}{X}_{m}=k{\tau}^{T}{\mathrm{\Phi}}_{m},& m=1,2,3\end{array}.$$
(9)
$${{\mathrm{\Phi}}_{m}}^{\prime}=\text{\u2211}_{i=1}^{p}({{\mathrm{\Phi}}_{m}}^{T}{\upsilon}_{i}){\upsilon}_{i}.$$
(10)
$${{X}_{m}}^{\prime}=k\text{\u2211}_{i=1}^{p}({{\mathrm{\Phi}}_{m}}^{T}{\upsilon}_{i})\hspace{0.17em}({\tau}^{T}{\upsilon}_{i})={{\tau}^{\prime}}^{T}{\mathrm{\Phi}}_{m}.$$
(11)
$$A=\text{\u2211}_{k=1}^{N}{\tau}_{k}{{\tau}_{k}}^{T}/N$$
(12)
$${\delta}_{1}=\text{\u2211}_{i}\hspace{0.17em}|\tau ({\mathrm{\lambda}}_{i})-{\tau}^{\prime}({\mathrm{\lambda}}_{i})|/n.$$