## Abstract

It is often possible and desirable to reduce the dimensionality of variables describing data, e.g. when the original number of variables is large and when errors introduced by approximation can be tolerated. This is particularly the case with respect to spectral measurements of illuminants and surfaces. In computer graphics and computer vision, it is often the case that multiplications of whole spectra, component by component, must be carried out. For example, this is the case when light reflects from matter, or is transmitted through materials. This statement particularly holds for spectrally-based ray tracing or radiosity in graphics. There, many such multiplications must be carried out, making a full-spectrum method prohibitively expensive. However, using full spectra is attractive because of the many important phenomena that can only be modelled using all the physics at hand. Here we apply to the task of spectral multiplication a method previously used in modeling RGB-based light propagation. We will in fact show that we can often multiply spectra without carrying out spectral multiplication.

In previous work we developed a method called “spectral sharpening” which took camera RGBs to a special sharp basis that was designed to render illuminant change simple to model. Specifically, it is well known that a 3 × 3 linear transform suffices to map RGBs for a set of surfaces across different viewing illuminants. With respect to the sharp basis, the mapping is to a first approximation a diagonal matrix. That is, illuminant change is modelled by three simple scale factors. Here, we apply this idea of “sharpening” to the set of principal component vectors derived from a representative set of spectra which might reasonably be encountered in a given application. With respect to the sharp spectral basis we show that spectral multiplications can be modelled as the multiplication of the basis coefficients. These new product coefficients applied to the sharp basis serve to accurately reconstruct the spectral product. As such, the number of multiplications required is not dependent on the number of spectral sampling points but rather on the dimension of the spectral basis.

Although the method is quite general and can be applied in several different fields, as an illustration of its utility we show how the method of modeling spectral image formation applies to tasks in graphics. Since we can efficiently model the appearance of phenomena that demand spectra for accurate rendering, we take advantage of metameric surfaces, ones that match under one light but not another, for tasks such as volume rendering. There we can use metamers to allow a user to pick out or merge different volume structures in real time, simply by changing the lighting.

In previous work we developed a method called “spectral sharpening” which took camera RGBs to a special sharp basis that was designed to render illuminant change simple to model. Specifically, it is well known that a 3 × 3 linear transform suffices to map RGBs for a set of surfaces across different viewing illuminants. With respect to the sharp basis, the mapping is to a first approximation a diagonal matrix. That is, illuminant change is modelled by three simple scale factors. Here, we apply this idea of “sharpening” to the set of principal component vectors derived from a representative set of spectra which might reasonably be encountered in a given application. With respect to the sharp spectral basis we show that spectral multiplications can be modelled as the multiplication of the basis coefficients. These new product coefficients applied to the sharp basis serve to accurately reconstruct the spectral product. As such, the number of multiplications required is not dependent on the number of spectral sampling points but rather on the dimension of the spectral basis.

Although the method is quite general and can be applied in several different fields, as an illustration of its utility we show how the method of modeling spectral image formation applies to tasks in graphics. Since we can efficiently model the appearance of phenomena that demand spectra for accurate rendering, we take advantage of metameric surfaces, ones that match under one light but not another, for tasks such as volume rendering. There we can use metamers to allow a user to pick out or merge different volume structures in real time, simply by changing the lighting.

Original language | English |
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Pages (from-to) | 1181-1193 |

Number of pages | 13 |

Journal | Journal of the Optical Society of America A |

Volume | 20 |

Issue number | 7 |

DOIs | |

Publication status | Published - 2003 |