Iterative regularization methods for ill-posed problems

This Ph.D thesis focuses on iterative regularization methods for regularizing linear and nonlinear ill-posed problems. Regarding linear problems, three new stopping rules for the Conjugate Gradient method applied to the normal equations are proposed and tested in many numerical simulations, including some tomographic images reconstruction problems. Regarding nonlinear problems, convergence and convergence rate results are provided for a Newton-type method with a modified version of Landweber iteration as an inner iteration in a Banach space setting

Symmetric functions for fast image retrieval with persistent homology

Persistence diagrams, combining geometry and topology for an effective shape description used in pattern recognition, have already proven to be an effective tool for shape representation with respect to a certain filtering function. Comparing the persistence diagram of a query with those of a database allows automatic classification or retrieval, but unfortunately, the standard method for comparing persistence diagrams, the bottleneck distance, has a high computational cost. A possible algebraic solution to this problem is to switch to comparisons between the complex polynomials whose roots are the cornerpoints of the persistence diagrams. This strategy allows to reduce the computational cost in a significant way, thereby making persistent homology based applications suitable for large\u2010scale databases. The definition of new distances in the polynomial framework poses some interesting problems, both of theoretical and practical nature. In this paper, these questions have been addressed by considering possible transformations of the half\u2010plane where the persistence diagrams lie onto the complex plane, and by considering a certain re\u2010normalisation the symmetric functions associated with the polynomial roots of the resulting transformed polynomial. The encouraging numerical results, obtained in a dermatology application test, suggest that the proposed method may even improve the achievements obtained by the standard methods using persistence diagrams and the bottleneck distance

Shortened persistent homology for a biomedical retrieval system with relevance feedback

This is the report of a preliminary study, in which a new coding of persistence diagrams and two relevance feedback methods, designed for use with persistent homology, are combined. The coding consists in substituting persistence diagrams with complex polynomials; these are "shortened", in the sense that only the first few coefficients are used. The relevance feedback methods play on the user's feedback for changing the impact of the diferent filtering functions in determining the output

A Feasibility Study for a Persistent Homology-Based k-Nearest Neighbor Search Algorithm in Melanoma Detection

Persistent homology is a fairly new branch of computational topology which combines geometry and topology for an effective shape description of use in Pattern Recognition. In particular, it registers through “Betti Numbers” the presence of holes and their persistence while a parameter (“filtering function”) is varied. In this paper, some recent developments in this field are integrated in a k-nearest neighbor search algorithm suited for an automatic retrieval of melanocytic lesions. Since long, dermatologists use five morphological parameters (A (Formula presented.) asymmetry, B (Formula presented.) boundary, C (Formula presented.) color, D (Formula presented.) diameter, E (Formula presented.) evolution) for assessing the malignancy of a lesion. The algorithm is based on a qualitative assessment of the segmented images by computing both 1 and 2-dimensional persistent Betti Number functions related to the ABCDE parameters and to the internal texture of the lesion. The results of a feasibility test on a set of 107 melanocytic lesions are reported in the section dedicated to the numerical experiments