We have previously shown that it is possible to construct a coordinate system in the space of illumination spectra such that the coordinate vectors of the illuminants are located in a cone. Changes in the space of illuminants can then be described by an intensity related scaling and a transformation in the Lorentz group SU(1,1). In practice it is often difficult and expensive to measure these coordinate vectors. Therefore it is of interest to estimate the characteristics of an illuminant from an RGB image captured by a camera. In this paper we will investigate the relation between sequences of illuminants and statistics computed from RGB images of scenes illuminated by these illuminants. As a typical example we will study sequences of black body radiators of varying temperature. We have shown earlier that black body radiators in the mired parametrization can be described by one-parameter groups of the Lorentz group SU(1,1). In this paper we will show that this group theoretical structure of the illuminant space induces a similar structure in spaces of statistical descriptors of the resulting RGB images. We show this relation for mean vectors of RGB images, for RGB histograms and for histograms of images obtained by applying certain spatiospectral linear filters to the RGB images. As a result we obtain estimates of the color temperature of the illuminant from sequences of RGB images of scenes under these illuminants.

It is known that for every selection of illumination spectra there is a coordinate system such that all coordinate vectors of these illumination spectra are located in a cone. A natural set of transformations of this cone are the Lorentz transformations. In this paper we investigate if sequences of illumination spectra can be described by one-parameter subgroups of Lorentz-transformations. We present two methods to estimate the parameters of such a curve from a set of coordinate points. We also use an optimization technique to approximate a given set of points by a one-parameter curve with a minimum approximation error. In the experimental part of the paper we investigate series of blackbody radiators and sequences of measured daylight spectra and show that one-parameter curves provide good approximations for large sequences of illumination spectra. © 2005 Springer Science + Business Media, Inc.

Many applications lead to signals with nonnegative function values. Understanding the structure of the spaces of nonnegative signals is therefore of interest in many different areas. Hence, constructing effective representation spaces with suitable metrics and natural transformations is an important research topic. In this thesis, we present our investigations of the structure of spaces of nonnegative signals and illustrate the results with applications in the fields of multispectral color science and content-based image retrieval.

The infinite-dimensional Hilbert space of nonnegative signals is conical and convex. These two properties are preserved under linear projections onto lower dimensional spaces.

The conical nature of these coordinate vector spaces suggests the use of hyperbolic geometry. The special case of three-dimensional hyperbolic geometry leads to the application of the SU(1,1) or SO 2,1) groups.

We introduce a new framework to investigate nonnegative signals. We use PCA-based coordinates and apply group theoretical tools to investigate sequences of signal coordinate vectors. We describe these sequences with oneparameter subgroups of SU(1,1) and show how to compute the one-parameter subgroup of SU(1,1) from a given set of nonnegative signals.

In our experiments we investigate the following signal sequences: (i) blackbody radiation spectra; (ii) sequences of daylight/twilight spectra measured in Norrk¨oping, Sweden and in Granada, Spain; (iii) spectra generated by the SMARTS2 simulation program; and (iv) sequences of image histograms. The results show that important properties of these sequences can be modeled in this framework. We illustrate the usefulness with examples where we derive illumination invariants and introduce an efficient visualization implementation.

Content-Based Image Retrieval (CBIR) is another topic of the thesis. In such retrieval systems, images are first characterized by descriptor vectors. Retrieval is then based on these content-based descriptors. Selection of contentbased descriptors and defining suitable metrics are the core of any CBIR system. We introduce new descriptors derived by using group theoretical tools. We exploit the symmetry structure of the space of image patches and use the group theoretical methods to derive low-level image filters in a very general framework. The derived filters are simple and can be used for multispectral images and images defined on different sampling grids. These group theoretical filters are then used to derive content-based descriptors, which will be used in a real implementation of a CBIR.

In applications of principal component analysis (PCA) it has often been observed that the eigenvector with the largest eigenvalue has only nonnegative entries when the vectors of the underlying stochastic process have only nonnegative values. This has been used to show that the coordinate vectors in PCA are all located in a cone. We prove that the nonnegativity of the first eigenvector follows from the Perron-Frobenius (and Krein-Rutman theory). Experiments show also that for stochastic processes with nonnegative signals the mean vector is often very similar to the first eigenvector. This is not true in general, but we first give a heuristical explanation why we can expect such a similarity. We then derive a connection between the dominance of the first eigenvalue and the similarity between the mean and the first eigenvector and show how to check the relative size of the first eigenvalue without actually computing it. In the last part of the paper we discuss the implication of theoretical results for multispectral color processing. © 2005 Optical Society of America.

Traditional color science is based on human color vision. For many applications (for example remote sensing, medical imaging, industrial inspection) where spectral properties of the radiation are of interest, human color vision based systems are too restrictive. Therefore multispectral approaches are becoming more and more popular.

In such applications, the investigation of spectral data sets requires different color systems. These systems are often built with a small number of color channels that are not necessarily related to human color perception. Of those, systems based on Principal Component Analysis (PCA) are widely used.

Considering spectra as non-negative signals, we show that PCA based systems lead to conical structures in the coordinate space. Several geometrical properties of this coordinate space such as convexity, boundaries, and different partitions are also discussed. Spaces with conical structures arc usually seen as models of non-Euclidean geometries. The special case of three dimensional conical structures is often investigated with the help of the SU(1,1) or SO(2,1) groups.

Motivated by these observations, we introduce a new framework to investigate illumination spectra. In this framework we start by using a PCA based system to describe spectra by their coordinates in a conical vector space. In this conical space, we apply group theoretical tools to investigate sequences of coordinate vectors. We describe these sequences of spectra coordinate vectors with special continuous subgroups of SU(1,1) acting on this non-Euclidean space. Then we show how to compute the one-parameter subgroup of SU(1,1) from a given set of illumination spectra.

In our experiments we investigate the following spectra sequences: black-body radiation spectra, daylight/twilight spectra sequences measured in Norrköping, Sweden and in Granada, Spain, and spectra generated by the SMARTS2 simulation program. The results show that important properties of sequences of illumination spectra can be modelled in this framework.

Finally, we illustrate the usefulness of the framework by deriving illumination invariants and an efficient visualization implementation.