Energetic-lattice based optimization. (L'optimization par trellis-énergetique)

Hierarchical segmentation has been a model which both identifies with the construct of extracting a tree structured model of the image, while also interpreting it as an optimization problem of the optimal scale selection. Hierarchical processing is an emerging field of problems in computer vision and hyperspectral image processing community, on account of its ability to structure high-dimensional data. Chapter 1 discusses two important concepts of Braids and Energetic lattices. A braid of partitions is a richer hierarchical partition model that provides multiple locally nonnested partitioning, while being globally a hierarchical partitioning of the space. The problem of optimization on hierarchies and further braids are non-tractable due the combinatorial nature of the problem. We provide conditions, of h-increasingness, scaleincreasingness on the energy defined on partitions, to extract unique and monotonically ordered minimal partitions. Furthermore these conditions are found to be coherent with the Braid structure to perform constrained optimization on hierarchies, and more generally Braids. Chapter 2 demonstrates the Energetic lattice, and how it generalizes the Lagrangian formulation of the constrained optimization problem on hierarchies. Finally in Chapter 3 we apply the method of optimization using energetic lattices to the problem of extraction of segmentations from a hierarchy, that are proximal to a ground truth set. Chapter 4 we show how one moves from the energetic lattice on hierarchies and braids, to a numerical lattice of Jordan Curves which define a continuous model of hierarchical segmentation. This model enables also to compose different functions and hierarchies. Chapter 5 compiles the scale-climbing algorithms by Guigues and Salembier-Garrido, over the hierarchies of partitions, and provides the new dynamic program for the Braids of partitions. Further it discusses a perspective on using intersection graphs to solve the optimal cut problem, and identities “Partition Graphs” to be one of the good graph structures to model partition selection. It finally concludes by formulating the optimal cut problem on hierarchies as a flow-maximization on a tree structure, the case of braids are also discussed.