Geometrical and topological approaches for image processing

Segmentation algorithms based on an energy minimisation framework often depend on a scale parameter which balances a fit to data and a regularising term. Irregular pyramids are defined as a stack of graphs successively reduced. Within this framework, the scale is often defined implicitly as the height in the pyramid. However, each level of an irregular pyramid can not usually be readily associated to the global optimum of an energy or a global criterion on the base level graph. This last drawback is addressed by the scale set framework designed by Guigues. The methods designed by this author allow to build a hierarchy and to design cuts within this hierarchy which globally minimise an energy. This paper studies the influence of the construction scheme of the initial hierarchy on the resulting optimal cuts. We propose one sequential and one parallel method with two variations within both. Our sequential methods provide partitions near an energy lower bound defined in this paper. Parallel methods require less execution times than the sequential method of Guigues even on sequential machines.

The Combinatorial Pyramids define a new family of irregular pyramids. Compared to older models these pyramids present many advantages : Coding of all boundaries between regions, encoding of adjacency and inclusion relationships( which may be differentiated), possibility to avoid the explicit encoding of all levels without loosing information.

This work result from a long collaboration with the PRIP laboratory in vienna

We present a characterization of topology preservation within generic axiomatized digital surface structures (GADS), a generic theoretical framework for digital topology introduced after a collaboration with Yung Kong and Gabor Herman. This characterization is based on the digital fundamental group that has been classically used for that purpose.

More briefly, we define here simple points within GADS and give the meaning of the words: "preserving the topology within GADS", therefore within any admissible bi-dimensional digital space.