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ON THE USE OF THE GEODESXC METRIC IN IMAGE ANALYSIS 1 - INTRODUCTION Many specimens in image analysis have two phases. The first phase contains the objects under study (biological cells, stringers in steel, etc.). The second phase is the background. Two successive steps are usually required in order to obtain measurements from an image : firstly, an image transformation in order to emphasize the salient features of the initial image; secondly, the measure¬ ments on the transformed image. Most of the time9 the transformation does not affect the whole image, but only a single phase, which dépends on the type of measurements to be carried out. For instance, in order to détermine the size distribution of stringers in a sheet of steel, it is sufficient to analyse only the stringers. On the other hand, the détermination of their spatial distribution requires only a study of the background. It is thus possible to limit the field of study to only one of the phases. This can be very useful in certain cases to increase the speed of measurements. Let us return to the particular case of the computation of the stringers size distribution. It is usually obtained by analysing the stringers individually. In this procédure, ail the stringers are studied one after the other and independently of each other. This procédure is of course very time consuming, and we may wonder whether it would be possible to study ail the stringers at the same time. In such a case, ail of them should nevertheless be analysed independently of each other. One way to maintain this independ- ence is to introduce a distance function on the stringers, such that two distinct stringers are considered to be at an infinité distance from each other.