1 Introduction

The sensor responses measured by a camera, or rgb s, depend on the surfaces in a scene and the prevailing illumination conditions. Many computer vision tasks require a descriptor of the surface reflectance which is constant under changing illumination. This requirement has led a number of researchers to develop colour constancy algorithms which aim to process illuminant dependent rgbs to recover illuminant independent descriptors (e.g. [ 10 , 11 , 6 , 2 ]).

The gamut mapping approach to colour constancy begun by Forsyth [ 7 ] and extended by Finlayson [ 4 ] has resulted in the most successful solution to the colour constancy problem developed so far. Forsyth founded his work on the assumption that scenes consist only of flat, matte surfaces, and that though of undetermined colour and intensity, the illumination is spatially constant. The problem is then to recover the rgb descriptor of each surface in the scene which would be recorded by the camera under a standard canonical illuminant.

To see how this might be done let us suppose, as Forsyth supposed, that rgbs measured under a pair of different illuminants are related by a diagonal matrix . It follows then that the colour constancy problem involves solving for the set of diagonal matrices such that:

where denotes an image colour which belongs to the gamut of image colours and denotes the gamut of canonical colours. A diagonal matrix D which takes all image colours into the canonical gamut is a solution to the colour constancy problem. Forsyth developed an algorithm, called CRULE, which finds all solutions to relation (1).

Forsyth presented experimental results which showed that CRULE performs well provided that the assumed world restrictions, flat matte world under uniform lighting, are satisfied. However, when these assumptions do not hold, and this is the case for almost all real images, then CRULE does not work well. The basic problem is that when scenes contain specular highlights, spatially varying illumination, and shape information then the linear relationship defined in (1) does not apply. Importantly, Finlayson [ 4 ] recognised that whilst these factors conspire against accurate recovery of illumination intensity, they do not affect the estimate of the orientation of the illumination vector in colour space. He used this fact to develop an alternative algorithm which recovers only orientation information but which is unaffected by factors such as specular highlights present in real images. Intensity is factored out of the colour constancy problem by mapping colour vectors into a perspective space At a second stage Forsyth's CRULE algorithm is applied to this perspective colour data. This new algorithm, recovers the gamut of perspective solutions to colour constancy (in fact a set of 2-dimensional diagonal matrices).

While, Finlayson's perspective algorithm allowed the gamut mapping idea to be applied to real-world images, it does not generate a single answer to the colour constancy problem. Rather algorithm output, is as before, the set of all maps which take image colours back to canonical viewing conditions. To obtain a single answer Finlayson [ 4 ] adopted the heuristic (first used by Forsyth [ 7 ]) of making the image gamut as colourful as possible. Later Barnard [ 1 ] proposed the mean of the feasible set. Neither of these answers it turns out is acceptable. The basic problem is that in order to make gamut mapping work the 3-d rgbs have been transformed to 2-d perspective colours and this in turn implies that the set of feasible maps is also perspectively distorted.

In this paper we propose a 3-d, non-perspective selection procedure. At the final stage we take the 2-d answer generated by gamut mapping and invert the perspective transform in order to get a 3-dimensional mapping set. The average of this 3-dimensional cone of solutions is shown to be a much more accurate, and stable, selection procedure. Indeed, it supports excellent colour constancy performance for both synthetic and real images; even when images contain few colours.

In section 2 , we review Forsyth's gamut mapping approach to colour constancy, its limitations how these can be overcome. We then address the question of map selection in section 3 and present a modified version of Finlayson's colour in perspective algorithm. A comparison of results obtained by testing the new algorithms with two existing implementations is presented in section 4; the new method delivers far superior performance, especially for images containing few colours. Finally we draw some conclusions from this work in section 5 .