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"QUE, TWO, THREE,... INFINITY". J. SERRA Centre de Morphologie Mathématique Ecole Nationale Supérieure des Mines de Paris 35 rue st. Honoré, 77305, Pontainebleau - France. ABSTRACT This famous title remarkably summarizes the recent évolution of the ideas in quantitative image analysis, and the impact of mathematical morphology on it. "One", for the case of basic stereological measure- ; iments. In fact, they ail involve an implicit image transformation. j"Two" appears when this transformation is preceded by another one, as I a linear érosion (laws of intercepts) or a covariaace. v/ith "Three", we iterate two transformations before the measurements : bi-dimensionali openings, for instance. And immediately after, one ;jumps to infinity with the pattern récognition algorithms, as the Skeletton. In this évolution, the train of thoughts is given by the conditions that the transforma have to satisfy. The quantization is associated v/ith four basic principles : invariance under the translations, corapa- I tibility with homothetics, local knowledge, semi-continuity. Moreover, each criterion corresponds to new iogicai constraints, like the grarm- lometric axioms for example. SET TRANSFORMATIONS AND THEIR ITERATIONS Obvicusly, Gamow had not foreseen the following paper when he found such an evocative title for his book, although it fits our purpose so perfectly that we have "adopted" it. It f antastically describes - in four words - the whole recent évolution of the methods in Texture Ana¬ lysis, such as the quantitative description of histological slides for exemple. Ohe may start the contemporary history of the se methods and techniques at the beginning of the 1960's. A cristallisation of various events occurred. At approximately the sanie period, an interest for quantiza¬ tion came into the scope of micro-biology, petrography and metallogra- phy. The Society for Stereology and the first Q.T.M. date from this time. For the first as well as for the second, to characterize quanti-