Pyramids of n-Dimensional Generalized Maps

International audienceGraph pyramids are often used for representing irregular pyramids. Combinatorial pyramids have been recently defined for this purpose. We define here pyramids of n-dimensional generalized maps. This is the main contribution of this work: a generic definition in any dimension which extend and generalize the previous works. Moreover, such pyramids explicitly represent more topological information than graph pyramids. A pyramid can be implemented in several ways, and three representations are discussed in this paper

Removal and Contraction for n-Dimensional Generalized Maps

International audienceRemoval and contraction are basic operations for several methods conceived in order to handle irregular image pyramids, for multi-level image analysis for instance. Such methods are often based upon graph-like representations which do not maintain all topological information, even for 2-dimensional images. We study the definitions of removal and contraction operations in the generalized maps framework. These combinatorial structures enable us to unambiguously represent the topology of a well-known class of subdivisions of n-dimensional (discrete) spaces. The results of this study make a basis for a further work about irregular pyramids of n-dimensional images

Generalized Max Pooling

State-of-the-art patch-based image representations involve a pooling operation that aggregates statistics computed from local descriptors. Standard pooling operations include sum- and max-pooling. Sum-pooling lacks discriminability because the resulting representation is strongly influenced by frequent yet often uninformative descriptors, but only weakly influenced by rare yet potentially highly-informative ones. Max-pooling equalizes the influence of frequent and rare descriptors but is only applicable to representations that rely on count statistics, such as the bag-of-visual-words (BOV) and its soft- and sparse-coding extensions. We propose a novel pooling mechanism that achieves the same effect as max-pooling but is applicable beyond the BOV and especially to the state-of-the-art Fisher Vector -- hence the name Generalized Max Pooling (GMP). It involves equalizing the similarity between each patch and the pooled representation, which is shown to be equivalent to re-weighting the per-patch statistics. We show on five public image classification benchmarks that the proposed GMP can lead to significant performance gains with respect to heuristic alternatives.Comment: (to appear) CVPR 2014 - IEEE Conference on Computer Vision & Pattern Recognition (2014

Serendipity and Tensor Product Affine Pyramid Finite Elements

Using the language of finite element exterior calculus, we define two families of $H^1$-conforming finite element spaces over pyramids with a parallelogram base. The first family has matching polynomial traces with tensor product elements on the base while the second has matching polynomial traces with serendipity elements on the base. The second family is new to the literature and provides a robust approach for linking between Lagrange elements on tetrahedra and serendipity elements on affinely-mapped cubes while preserving continuity and approximation properties. We define shape functions and degrees of freedom for each family and prove unisolvence and polynomial reproduction results.Comment: Accepted to SMAI Journal of Computational Mathematic

Geometric and combinatorial realizations of crystal graphs

For irreducible integrable highest weight modules of the finite and affine Lie algebras of type A and D, we define an isomorphism between the geometric realization of the crystal graphs in terms of irreducible components of Nakajima quiver varieties and the combinatorial realizations in terms of Young tableaux and Young walls. For affine type A, we extend the Young wall construction to arbitrary level, describing a combinatorial realization of the crystals in terms of new objects which we call Young pyramids.Comment: 34 pages, 17 figures; v2: minor typos corrected; v3: corrections to section 8; v4: minor typos correcte

Completely empty pyramids on integer lattices and two-dimensional faces of multidimensional continued fractions

In this paper we develop an integer-affine classification of three-dimensional multistory completely empty convex marked pyramids. We apply it to obtain the complete lists of compact two-dimensional faces of multidimensional continued fractions lying in planes with integer distances to the origin equal 2, 3, 4 ... The faces are considered up to the action of the group of integer-linear transformations. In conclusion we formulate some actual unsolved problems associated with the generalizations for n-dimensional faces and more complicated face configurations.Comment: Minor change

Euclidean decompositions of hyperbolic manifolds and their duals

Epstein and Penner constructed in [EP88] the Euclidean decomposition of a non-compact hyperbolic n-manifold of finite volume for a choice of cusps, n >= 2. The manifold is cut along geodesic hyperplanes into hyperbolic ideal convex polyhedra. The intersection of the cusps with the Euclidean decomposition determined by them turns out to be rather simple as stated in Theorem 2.2. A dual decomposition resulting from the expansion of the cusps was already mentioned in [EP88]. These two dual hyperbolic decompositions of the manifold induce two dual decompositions in the Euclidean structure of the cusp sections. This observation leads in Theorems 5.1 and 5.2 to easily computable, necessary conditions for an arbitrary ideal polyhedral decomposition of the manifold to be a Euclidean decomposition