International journals

[SPL08] Kevin Loquin and Olivier Strauss,“Histogram density estimators based upon a fuzzy partition”, Statistics & Probability Letters, 78(13):1863-1868, Septembre 2008.

Abstract:
This paper presents a density estimator based upon a histogram computed on a fuzzy partition. We prove the consistency of this estimator in the Mean Squared Error (MSE). We give the optimal bin width of the estimator which minimizes the Asymptotic Integrated Mean Squared Error (AIMSE).

[FSS08] Kevin Loquin and Olivier Strauss, “On the granularity of summative kernels”, Fuzzy Sets and Systems, From Knowledge Representation to Information Processing and Management - Selected papers from the French Fuzzy Days (LFA 2006), 2008, 159, 1952-1972.

Abstract:
In this paper, we propose granularity as a new index to characterize the non- specificity of a summative kernel. This index is intended to reflect the behavior of a kernel in the usual signal processing applications. We show, in different experiments, that two kernels having the same granularity have very similar behavior. This granularity-based adaptation is compared to other adaptation methods. These experiments highlight the ability of the granularity index to measure the spreading and collecting properties of a summative kernel.

International conferences

[SMPS06] Kevin Loquin and Olivier Strauss “Fuzzy histograms and density estimation”, In: Soft Methods in Probability and Statistics 2006 (SMPS’06). Lawry, Miranda, Bugarin, Li, Gil, Grzegorzewski, Hryniewicz Eds, Springer, pp. 45-52

Abstract:
In this paper, we have presented density estimators based upon a fuzzy histogram. This latter being nothing else but a generalization of the popular crisp histogram, when replacing the crisp partition by a fuzzy partition. Those proposed density estimators consist in interpolations of the nodes' values of the density obtained in the usual way.

[ICIP07] Florence Jacquey, Kevin Loquin, Frédéric Comby and Olivier Strauss “Non-additive approach for gradient-based edge detection”, ICIP’07 International Conference on Image Processing, San Antonio, Texas, USA, Vol.III, pp 49-52, September 16-19, 2007.

Abstract:
In this paper, we propose a new method to perform the first derivative estimation of a discrete intensity distribution. This approach is based on a non-additive aggregation process and provides an estimate of the gradient as intervals instead of single values. These intervals are used to threshold a gradient based edge detection and therefore discard spurious detections due to noise.

[SMPS08] Kevin Loquin and Olivier Strauss “Imprecise functional estimation: the cumulative distribution case”, SMPS’08 Soft Methods in Probability and Statistics, Toulouse, France, pp 175-182, September 8-10, 2008.

Abstract:
In this paper, we propose an adaptation of the Parzen Rosenblatt cumulative distribution function estimator that uses maxitive kernels. The result of this estimator, on every point of the domain of F, the cumulative distribution to be estimated, is interval valued instead of punctual valued.We prove the consistency of our approach with the classical Parzen Rosenblatt estimator, since, according to consistency conditions between the maxitive kernel involved in the imprecise estimator and the summative kernel involved in the precise estimator, our imprecise estimate contains the precise Parzen Rosenblatt estimate.

National conferences (papers written in french)

[LFA06] Kevin Loquin and Olivier Strauss “De la granulosité des noyaux d’échantillonnage”, In: Rencontres francophones sur la Logique Floue et ses Applications 2006 (LFA’06). Cépaduès eds, pp. 387-394

Abstract:
In this paper, we propose the granulosity as a characterization of the behavior of a summative sampling kernel. We show that this characterization allows a behavioral adaptation. We compare this adaptation to the adaptation classicaly used in non parametric statistics.

[LFA07] Olivier Strauss and Kevin Loquin “Vers une approche unifiée du filtrage des images”, In: Rencontres francophones sur la Logique Floue et ses Applications 2007 (LFA’07). Cépaduès eds, pp. 41-48

Abstract:
This paper proposes to show that the particular fuzzy extension, based on level cuts, of the mathematical morphology coincide with a non-additive extension of the classical approach for filtering based on convolution.

[LFA08] Bilal Nehme, Kevin Loquin and Olivier Strauss “Estimation Imprécise de la densité de probabilité”, In: Rencontres francophones sur la Logique Floue et ses Applications 2008 (LFA’08), Cépaduès eds, pp. 286-293.

Abstract:
In this paper, we propose an adaptation of the Parzen Rosenblatt density estimator that uses maxitive kernels. The result of this estimator, on every point of the domain of the density to be estimated, is interval valued. We prove the consistency of our approach with the Parzen Rosenblatt estimator, since, according to conditions exposed in this paper, our estimate contains this estimation.

Thesis (written in french)

De l'utilisation des noyaux maxitifs en traitement de l'information / On the use of maxitive kernels in information processing

Abstract:
In this thesis, we propose and develop new methods in statistics and in signal and image processing based upon possibility theory. These new methods are adapted from usual data processing tools. They aim at handling the defects of the usual methods coming from the user's lack of knowledge in the modeling of the observed phenomenon. The precise, punctual outputs of the usual methods become interval, hence imprecise, outputs. The interval outputs thus obtained consistently reflect the arbitrariness in the choice of the parameters of the usual methods.

Many algorithms in signal processing and in statistics use, more or less explicitly, the expectation operator associated to a probabilistic representation of the neighborhood of a point, which we call summative kernel. Thus, we group many data processing methods together under the name of summative extraction of information. Among these methods, there are measure modeling, linear filtering, sampling, interpolation and derivation processes of digital signals, probability density and cumulative distribution functions estimators,...

As an alternative to the summative extraction method, we present the maxitive extraction of information that uses the Choquet integral operator associated to a possibilistic representation of the neighborhood of a point, which we call maxitive kernel. The lack of knowledge on the summative kernel is handled by the fact that a maxitive kernel encodes a family of summative kernels. Moreover, the interval output of the maxitive extraction method is the set of the punctual outputs of the summative extraction methods obtained with the summative kernels encoded by the chosen maxitive kernel. On top of this theoretical justification, we present a series of applications of the maxitive extraction method in statistics and signal processing, which constitutes a toolbox, left to be enriched and used on real cases.