1 Introduction

Probabilistic relaxation is a powerful technique that allows the incorporation of contextual information in labelling problems. The theoretical foundations of the approach have been developed in a series of papers by various researchers (e.g. [ 9 , 7 , 10 , 4 , 1 ]), and the method has been applied to a variety of problems, ranging from edge detection to high level object recognition. Probabilistic relaxation is an iterative approach such that at each iteration step contextual information from further afield is utilised to refine the labelling result. One could say that at each iteration step information conveyed by coarser resolutions of the scene is incorporated in the labelling process. Seen that way, probabilistic relaxation seems to work in the opposite direction from standard multiresolution/multiscale techniques. Indeed, in such methods, the coarse resolution information is extracted first and it is then used to guide the solution at the finer resolution. This direction of information flow seems to be more intuitive as it fits with the way we perceive scenes: we see something with our peripheral, low resolution vision and subsequently we foveate on it; we see something from a distance, with low resolution, and subsequently we approach and see it better, in more detail. In this paper we use such a causal approach to the problem of multiscale information incorporation. In particular, we develop a general framework for a probabilistic relaxation scheme that works across the various resolution levels and allows the incorporation of measurements pertaining at all levels. The scheme is used for the segmentation of coloured texture images and involves the introduction of an appropriate dictionary.

Before we proceed further, we wish to clarify the use of the terms ``multiresolution'' and ``multiscale'' in the context of this work. The resolution of an image corresponds to the size of a pixel in physical units. As objects have fixed physical sizes, when a pixel represents a larger physical length, fewer pixels are needed to represent the same object. This is the intuitive basis of multiresolution pyramid creation: successive smoothings and subsamplings. However, if one does not perform the subsamplings, but only the successive smoothings, one may say that the approach is multiscale as the filters used to create the coarser versions of the image become larger and larger in pixel size, creating images that are more and more blurred but have the same number of pixels. That is the approach we use here and the reason we do it is to preserve the correspondence between the pixels across the different levels. Thus, we use the terms multiscale and multiresolution interchangeably, although the terms multilevel or even multiblur would be more appropriate.

In Section  2 we shall present a brief review of multiresolution relaxation schemes developed and used by various researchers. In Section  3 we shall develop our methodology. In Section  4 , the process of extraction of multiscale features used in the segmentation of coloured textures shall be described, followed by a presentation of some results of our segmentation approach. We shall conclude in Section  5 .