The aim of my PhD is to adapt existing digital signal processing methods, so as to obtain robust estimations of values computed from the physical signal. Values like its derivative, a transformed signal, or a filtered signal. What we call a robust estimation is an imprecise estimation, i.e. an estimation whose result is an interval instead of a point. An interval is more likely to contain the real physical value to be estimated than a point (which has a null probability of being the real physical value).
Based on previous researches of Olivier Strauss, my PhD supervisor, a first way for performing these estimations, was the study of fuzzy partitions of the universe, because of its robustness to the partitioning modifications (like translation or bin width changes) as qualitatively exposed in paper [3]. We then showed convergence results in non-parametric statistics based on this fuzzy partition in paper [5].
It was already obvious, at this point, that the heart of our problem was the sampling process of a physical signal, whether by considering the partitioning of the domain, as we studied or by considering the modeling of the signal acquisition. Actually the sensor involved in the acquisition is modeled by a summative kernel and the acquisition is modeled by the convolution of this summative kernel with the underlying real signal. We proposed and studied an index that quantifies the non-resolution power of the sensor modeled by the summative kernel. This index is a continuous counterpart of the discrete peakedness index proposed by Birnbaum and studied by Dubois and Hüllermeier. We called this index the granularity of a summative kernel [1] [6], as Pawlak called the granularity of a rough set, its power of indiscernibility between objects, which is a notion close to the resolution power of a sensor.
A summative kernel is formally similar to a probability distribution and can be interpreted, especially when used for sampling, as an uncertainty model. Indeed, randomness appears when the acquisition of a signal is performed. The acquired value is subject to uncertainty, the sensor is not perfect. I made a survey of alternative uncertainty models, taking into account randomness but also incompleteness, since it is obviously naive to pretend that we have all the information about the sensor to assess a unique probability distribution. This survey, going from the coherent lower previsions of Peter Walley to possibility theory, led me to consider the maxitive kernels as an alternative to the summative kernels for the modeling of sensors.
A maxitive kernel is formally similar to a possibility distribution which encodes a family of probability distributions. In other words, a maxitive kernel encodes a family of summative kernels. Most of the operators used to perform punctual estimations with summative kernels can be adapted, by means of the possibility theory operators, to the maxitive kernels. In that case, we obtain consistent interval estimates. The consistency, in this context of estimation, is the fact the interval estimate obtained with a maxitive kernel contains all the punctual estimates obtained with all the summative kernels encoded by the maxitive kernel.
This approach led to very interesting results in image processing. We developed an edge detector in the paper [4] and established a link between imprecise filtering of image and morphology in the paper [2]. We are now exploring some statistical methods of functionnal estimation like the Parzen Rosenblatt cumulative distribution function estimator.
The uncertainty theories survey led me to attend to the 2nd SIPTA School in Madrid 2006 and to keep contact with the SIPTA society. I am now organizer of the 3rd SIPTA School in 2008 in Montpellier, with Jean-Marc Bernard, Olivier Strauss and Céline Berger.
