School of Engineering, Computing and Mathematics

In this paper we propose a geometric approach to the theory of evidence based on convex geometric interpretations of its two key notions of belief function and Dempster's sum. On one side, we analyze the geometry of belief functions as points of a polytope in the Cartesian space called belief space, and discuss the intimate relationship between basic probability assignment and convex combination. On the other side, we study the global geometry of Dempster's rule by describing its action on those convex combinations. By proving that Dempster's sum and convex closure commute we become able to depict the geometric structure of conditional subspaces, i.e. sets of belief functions conditioned by a given function b. Natural applications of these geometric methods to classical problems like probabilistic approximation and canonical decomposition are outlined.

In this paper, we analyze from a geometric perspective the meaningful relations taking place between belief and probability functions in the framework of the geometric approach to the theory of evidence. Starting from the case of binary domains, we identify and study three major geometric entities relating a generic belief function (b.f.) to the set of probabilities P: 1) the dual line connecting belief and plausibility functions; 2) the orthogonal complement of P; and 3) the simplex of consistent probabilities. Each of them is in turn associated with a different probability measure that depends on the original b.f. We focus in particular on the geometry and properties of the orthogonal projection of a b.f. onto P and its intersection probability, provide their interpretations in terms of degrees of belief, and discuss their behavior with respect to affine combination.

In this paper we adopt the geometric approach to the theory of evidence to
study the geometric counterparts of the plausibility functions, or upper probabilities.
The computation of the coordinate change between the two natural
reference frames in the belief space allows us to introduce the dual notion
of basic plausibility assignment and understand its relation with the classical
basic probability assignment. The convex shape of the plausibility space P is
recovered in analogy to what was done for the belief space, and the pointwise geometric relation between a belief function and the corresponding plausibility vector is discussed. The orthogonal projection of an arbitrary belief
function s onto the probabilistic subspace is computed and compared with
other significant entities, such as the relative plausibility and mean probability
vectors.

In this paper the geometric structure of the space S_Theta of the belief functions defined over a discrete set Theta (belief space) is analyzed. Using the Moebius inversion lemma we prove the recursive bundle structure of the belief space and show how an arbitrary belief function can be uniquely represented as a convex combination of certain elements of the fibers, giving $\mathcal{S}$ the form of a simplex. The commutativity of orthogonal sum and convex closure operator is proved and used to depict the geometric structure of conditional subspaces, i.e. sets of belief functions conditioned by a given function s.
Future applications of this geometric methods to classical problems like probabilistic approximation and canonical decomposition are outlined.

Human identification from gait is a challenging task in realistic surveillance scenarios in which people walking along arbitrary directions are imaged by a single camera.
In this paper, motivated by the view-invariance issue in the human ID from gait problem, we address the general problem of classifying the “content” of human motions of unknown “style”. Given a dataset of sequences in which different people walking normally are seen from several well separated views, we propose a three-layer scheme based on bilinear models, in which image sequences are mapped to observation vectors of fixed dimension using Markov modeling. We test our approach on the CMU Mobo database,
showing how bilinear separation outperforms other approaches, opening the way to view- and action-invariant identity recognition, as well as subject- and view-invariant action recognition.

In this paper, we analyze Shafer’s belief functions (BFs) as geometric entities, focusing in particular on the geometric behavior of Dempster’s rule of combination in the belief space, i.e., the set of all the admissible BFs defined over a given finite domain Theta. The study of the orthogonal sums of affine subspaces allows us to unveil a convex decomposition of Dempster’s rule of combination in terms of Bayes’ rule of conditioning and prove that under specific conditions orthogonal sum and affine closure commute. A direct consequence of these results is the simplicial shape of the conditional subspaces , i.e., the sets of all the possible combinations of a given BF s. We show how Dempster’s rule exhibits a rather elegant behavior when applied to BFs assigning the same mass to a fixed subset (constant mass loci). The resulting affine spaces have a common intersection that is characteristic of the conditional subspace, called focus. The affine geometry of these foci eventually suggests an interesting geometric construction of the orthogonal sum of two BFs.

In this paper we prove that a recent Bayesian approximation of belief functions, the relative belief of singletons, meets a number of properties with respect to Dempster's rule of combination which mirrors those satisfied by the relative plausibility of singletons. In particular, its operator commutes with Dempster's sum of plausibility functions, while perfectly representing a plausibility function when combined through Dempster's rule. This suggests a classification of all Bayesian approximations into two families according to the operator they relate to.

In this paper we introduce the relative belief of singletons as a novel
Bayesian approximation of a belief function.We discuss its nature in terms of degrees of belief under several different angles, and its applicability to different classes of belief functions.

In this paper we introduce three alternative combinatorial formulations of the theory of evidence (ToE), by proving that both plausibility and commonality functions share the structure of \sum function" with belief functions. We compute their Moebius inverses, which we call basic plausibility and commonality assignments. In the framework of the geometric approach to uncertainty measures the equivalence of the associated formulations of the ToE is mirrored by the geometric congruence of the related simplices. We can then describe the point-wise geometry of these sum functions in terms of rigid transformations mapping them onto each other. Combination rules can be applied to plausibility and commonality functions through their Moebius inverses, leading to interesting applications of such inverses to the probabilistic transformation problem.

Object tracking consists of reconstructing the configuration of an articulated body from a sequence of images provided by one or more cameras. In this paper we present a general method for pose estimation based on the evidential reasoning. The proposed framework integrates different levels of description of the object to improve robustness and precision, overcoming the limitations of approaches using single-feature representations. Several image descriptions extracted from a single-camera view are fused together using the Dempster-Shafer ”theory of evidence”. Feature data are expressed as belief functions over the set of their possibile values. There is no need of any a-priori assumptions about the model of the object. Learned refinement maps between feature spaces and the parameter space Q describing the configuration of the object characterize the relationships among distinct representations of the pose and play the role of the model. During training the object follows a sample trajectory in Q. Each feature space is reduced to a discrete frame of discernment (FOD) and refinements are built by mapping these FODs into subsets of the sample trajectory. During tracking new sensor data are converted to belief functions which are projected and combined in the approximate state space. Resulting degrees of belief indicate the best pose estimate at the current time step. The choice of a sufficiently dense (in a topological sense) sample trajectory is a critical problem.
Experimental results concerning a simple tracking system are shown.

In this paper we extend our geometric approach to the theory of evidence
in order to include other important classes of nite fuzzy measures.
In particular we describe the geometric counterparts of possibility measures or fuzzy sets, represented as consonant belief functions.
The correspondence between chains of subsets and convex sets of consonant functions is studied and its properties analyzed, eventually yielding an elegant representation of the region of consonant belief functions
in terms of the notion of simplicial complex.

In this paper we propose a credal representation of the interval probability associated with a belief function (b.f.), and show how it relates to several classical Bayesian transformations of b.f.s through the notion of “focus” of a pair of simplices.
While a belief function corresponds to a polytope of probabilities consistent with it, the related interval probability is geometrically represented by a pair of upper and lower simplices. Starting from the interpretation of the pignistic function as the center of mass of the credal set of consistent probabilities, we prove that relative belief of singletons, relative plausibility of singletons and intersection probability can all be described as foci of different pairs of simplices in the region of all probability measures. The formulation of frameworks similar to the Transferable Belief Model for such Bayesian transformations appears then at hand.

Automatic gesture recognition is an important and challenging problem in computer vision. In this paper we present an original technique for hand gesture recognition based on a dynamic shape representation by combining size functions and hidden Markov models (HMM). Size functions are objects of topological nature which allow to describe shape with completeness and a remarkable tolerance to noise. HMM allow, instead, for the inclusion of dynamics in the model. Each gesture is described by a different probabilistic finite state machine which models a succession of so called canonical postures of the hand. The state dynamics describe the transition between canonical postures while the observation equations are maps from the set of canonical postures to size functions. Tests on real image sequences are included.