A skin detection approach based on the Dempster–Shafer theory of evidence

Skin detection is an important step for a wide range of research related to computer vision and image processing and several methods have already been proposed to solve this problem. However, most of these methods suffer from accuracy and reliability problems when they are applied to a variety of images obtained under different conditions. Performance degrades further when fewer training data are available. Besides these issues, some methods require long training times and a significant amount of parameter tuning. Furthermore, most state-of-the-art methods incorporate one or more thresholds, and it is difficult to determine accurate threshold settings to obtain desirable performance. These problems arise mostly because the available training data for skin detection are imprecise and incomplete, which leads to uncertainty in classification. This requires a robust fusion framework to combine available information sources with some degree of certainty. This paper addresses these issues by proposing a fusion-based method termed Dempster–Shafer-based Skin Detection (DSSD). This method uses six prominent skin detection criteria as sources of information (SoI), quantifies their reliabilities (confidences), and then combines their confidences based on the Dempster–Shafer Theory (DST) of evidence. We use the DST as it offers a powerful and flexible framework for representing and handling uncertainties in available information and thus helps to overcome the limitations of the current state-of-the-art methods. We have verified this method on a large dataset containing a variety of images, and achieved a 90.17% correct detection rate (CDR). We also demonstrate how DSSD can be used when very little training data are available, achieving a CDR as high as 87.47% while the best result achieved by a Bayesian classifier is only 68.81% on the same dataset. Finally, a generalized DSSD (GDSSD) is proposed achieving 91.12% CDR.     

Dempster–Shafer fusion of evidential pairwise Markov fields

Hidden Markov fields (HMFs) have been successfully used in many areas to take spatial information into account. In such models, the hidden process of interest X is a Markov field, that is to be estimated from an observable process Y. The possibility of such estimation is due to the fact that the conditional distribution of the hidden process with respect to the observed one remains Markovian. The latter property remains valid when the pairwise process $(X,Y)$ is Markov and such models, called pairwise Markov fields (PMFs), have been shown to offer larger modeling capabilities while exhibiting similar processing cost. Further extensions lead to a family of more general models called triplet Markov fields (TMFs) in which the triplet $(U,X,Y)$ is Markov where U is an underlying process that may have different meanings according to the application. A link has also been established between these models and the theory of evidence, opening new possibilities of achieving Dempster–Shafer fusion in Markov fields context. The aim of this paper is to propose a unifying general formalism allowing all conventional modeling and processing possibilities regarding information imprecision, sensor unreliability and data fusion in Markov fields context. The generality of the proposed formalism is shown theoretically through some illustrative examples dealing with image segmentation, and experimentally on hand-drawn and SAR images.     

Joint cumulative distribution functions for Dempster–Shafer belief structures using copulas

We first introduce the Dempster–Shafer belief structure and highlight its role in the representation of information about a random variable for which our knowledge of the probabilities is interval-valued. We investigate the formation of the cumulative distribution function (CDF) for these types of variables. It is noted that this is also interval-valued and is expressible in terms of plausibility and belief measures. The class of aggregation operators known as copulas are introduced and a number of their properties are provided. We discuss Sklar’s theorem, which provides for the use of copulas in the formulation of joint CDFs from the marginal CDFs of classic random variables. We then look to extend these ideas to the case of joining the marginal CDFs associated with Dempster–Shafer belief structures. Finally we look at the formulation CDFs obtained from functions of multiple D–S belief structures.     

Fusion de Dempster–Shafer dans les chaînes triplet partiellement de Markov

Hidden Markov Chains (HMC), Pairwise Markov Chains (PMC), and Triplet Markov Chains (TMC), allow one to estimate a hidden process X from an observed process Y. More recently, TMC have been generalized to Triplet Partially Markov chain (TPMC), where the estimation of X from Y remains workable. Otherwise, when introducing a Dempster–Shafer mass function instead of prior Markov distribution in classical HMC, the result of its Dempster–Shafer fusion with a distribution provided $Y=y$, which generalizes the posterior distribution of X, is a TMC. The aim of this Note is to generalize the latter result replacing HMC with multisensor TPMC. To cite this article: W. Pieczynski, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

A Wiener–Hopf based approach to numerical computations in fluctuation theory for Lévy processes

This paper focuses on numerical evaluation techniques related to fluctuation theory for Lévy processes; they can be applied in various domains, e.g., in finance in the pricing of so-called barrier options. More specifically, with $\bar{X}_t:= \sup _{0\le s\le t} X_s$ denoting the running maximum of the Lévy process $X_t$ , the aim is to evaluate $\mathbb{P }(\bar{X}_t \in \mathrm{d}x)$ for $t,x>0$ . The starting point is the Wiener–Hopf factorization, which yields an expression for the transform $\mathbb E e^{-\alpha \bar{X}_{e(\vartheta )}}$ of the running maximum at an exponential epoch (with $\vartheta ^{-1}$ the mean of this exponential random variable). This expression is first rewritten in a more convenient form, and then it is pointed out how to use Laplace inversion techniques to numerically evaluate $\mathbb{P }(\bar{X}_t\in \mathrm{d}x).$ In our experiments we rely on the efficient and accurate algorithm developed in den Iseger (Probab Eng Inf Sci 20:1–44, 2006). We illustrate the performance of the algorithm with various examples: Brownian motion (with drift), a compound Poisson process, and a jump diffusion process. In models with jumps, we are also able to compute the density of the first time a specific threshold is exceeded, jointly with the corresponding overshoot. The paper is concluded by pointing out how our algorithm can be used in order to analyze the Lévy process’ concave majorant.     

Cartan–Laptev method in the theory of multidimensional three-webs

We show how the Cartan–Laptev method that generalizes Elie Cartan’s method of external forms and moving frames is applied to the study of closed G-structures defined by multidimensional three-webs formed on a C ^s -smooth manifold of dimension 2r, r ≥ 1, s ≥ 3, by a triple of foliations of codimension r. We say that a tensor T belonging to a differential-geometric object of order s of a three-web W is closed if it can be expressed in terms of components of objects of lower order s. We find all closed tensors of a three-web and the geometric sense of one of relations connecting three-web tensors. We also point out some sufficient conditions for the web to have a closed G-structure. It follows from our results that the G-structure associated with a hexagonal three-web W is a closed G-structure of class 4. It is proved that basic tensors of a three-web W belonging to a differential-geometric object of order s of the web can be expressed in terms of an s-jet of the canonical expansion of its coordinate loop, and conversely. This implies that the canonical expansion of every coordinate loop of a three-web W with closed G-structure of class s is completely defined by an s-jet of this expansion. We also consider webs with one-digit identities of kth order in their coordinate loops and find the conditions for these webs to have the closed G-structure.     

Equimultiplicity in Hilbert–Kunz theory

Mathematische Zeitschrift - We study further the properties of Hilbert–Kunz multiplicity as a measure of singularity. This paper develops a theory of equimultiplicity for Hilbert–Kunz...     

Extensions of Perron–Frobenius theory

The classical Perron–Frobenius theory asserts that, for two matrices $A$ and $B$ , if $0\le B \le A$ and $r(A)=r(B)$ with $A$ being irreducible, then $A=B$ . It has been extended to infinite-dimensional Banach lattices under certain additional conditions, including that $r(A)$ is a pole of the resolvent of $A$ . In this paper, we prove that the same result holds if $B$ is irreducible and $r(B)$ is a pole of the resolvent for $B$ . We also prove some other interesting extensions of the theorem for infinite-dimensional Banach lattices.     

Representation theory of finite groups in Wiener–Hopf factorization problem

We consider the Wiener–Hopf factorization problem for a matrix function that is completely defined by its first column: the succeeding columns are obtained from the first one by means of a finite group of permutations. The symmetry of this matrix function allows us to reduce the dimension of the problem. In particular, we find some relations between its partial indices and can compute some of the indices. In special cases, we can explicitly obtain the Wiener–Hopf factorization of the matrix function.     

Bohr–Sommerfeld–Heisenberg theory in geometric quantization

In the framework of geometric quantization we extend the Bohr–Sommerfeld rules to a full quantum theory which resembles the Heisenberg matrix theory. This extension is possible because Bohr–Sommerfeld rules not only provide an orthogonal basis in the space of quantum states, but also give a lattice structure to this basis. This permits the definition of appropriate shifting operators. As examples, we discuss the 1–dimensional harmonic oscillator and the coadjoint orbits of the rotation group.     

Covering techniques in Auslander–Reiten theory

Given a finite dimensional algebra over a perfect field the text introduces covering functors over the mesh category of any modulated Auslander–Reiten component of the algebra. This is applied to study the composition of irreducible morphisms between indecomposable modules in relation with the powers of the radical of the module category.     

Green–Samoilenko operator in the theory of invariant sets of nonlinear differential equations

We establish conditions for the existence of an invariant set of the system of differential equations