Ruben D. Spies

Terry Herdman, Ruben D. Spies, Karina G. Temperini

The concept of qualification for spectral regularization methods for inverse ill-posed problems is strongly associated to the optimal order of convergence of the regularization error. In this article, the definition of qualification is extended and three different levels are introduced: weak, strong and optimal. It is shown that the weak qualification extends the definition introduced by Mathe and Pereverzev in 2003, mainly in the sense that the functions associated to orders of convergence and source sets need not be the same. It is shown that certain methods possessing infinite classical qualification, e.g. truncated singular value decomposition (TSVD), Landweber's method and Showalter's method, also have generalized qualification leading to an optimal order of convergence of the regularization error. Sufficient conditions for a SRM to have weak qualification are provided and necessary and sufficient conditions for a given order of convergence to be strong or optimal qualification are found. Examples of all three qualification levels are provided and the relationships between them as well as with the classical concept of qualification and the qualification introduced by Mathe and Perevezev are shown. In particular, spectral regularization methods having extended qualification in each one of the three levels and having zero or infinite classical qualification are presented. Finally several implications of this theory in the context of orders of convergence, converse results and maximal source sets for inverse ill-posed problems, are shown.

Francisco J. Ibarrola, Leandro E. Di Persia, Ruben D. Spies

When a signal is recorded in an enclosed room, it typically gets affected by reverberation. This degradation represents a problem when dealing with audio signals, particularly in the field of speech signal processing, such as automatic speech recognition. Although there are some approaches to deal with this issue that are quite satisfactory under certain conditions, constructing a method that works well in a general context still poses a significant challenge. In this article, we propose a method based on convolutive nonnegative matrix factorization that mixes two penalizers in order to impose certain characteristics over the time-frequency components of the restored signal and the reverberant components. An algorithm for implementing the method is described and tested. Comparisons of the results against those obtained with state of the art methods are presented, showing significant improvement.