Over the last decades, the evolution of acquisition techniques yields the generalization of detailed 3D objects, represented as huge point sets composed of millions of vertices. The complexity of the involved data often requires to analyze them for the extraction and characterization of pertinent structures, which are potentially defined at multiple scales. Among the wide variety of methods proposed to analyze digital signals, the scale-space analysis is today a standard for the study of 2D curves and images. However, its adaptation to 3D data leads to instabilities and requires connectivity information, which is not directly available when dealing with point sets. In this thesis, we present a new multi-scale analysis framework that we call the Growing Least Squares (GLS). It consists of a robust local geometric descriptor that can be evaluated on point sets at multiple scales using an efficient second-order fitting procedure. We propose to analytically differentiate this descriptor to extract continuously the pertinent structures in scale-space. We show that this representation and the associated toolbox define an efficient way to analyze 3D objects represented as point sets at multiple scales. To this end, we demonstrate its relevance in various application scenarios. A challenging application is the analysis of acquired 3D objects coming from the Cultural Heritage field. In this thesis, we study a real-world dataset composed of the fragments of the statues that were surrounding the legendary Alexandria Lighthouse. In particular, we focus on the problem of fractured object reassembly, consisting of few fragments (up to about ten), but with missing parts due to erosion or deterioration. We propose a semi-automatic formalism to combine both the archaeologist's knowledge and the accuracy of geometric matching algorithms during the reassembly process. We use it to design two systems, and we show their efficiency in concrete cases.
