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

Combinatorial models for topology-based geometric modeling

Many combinatorial (topological) models have been proposed in geometric modeling, computational geometry, image processing or analysis, for representing subdivided geometric objects, i.e. partitionned into cells of different dimensions: vertices, edges, faces, volumes, etc. We can distinguish among models according to the type of cells (regular or not regular ones), the type of assembly ("manifold" or "non manifold"), the type of representation (incidence graphs or ordered models), etc

Conversion between chains of maps and chains of surfaces; application to the computation of incidence graphs homology

Many combinatorial cellular structures have been defined in order to represent the topology of subdivided geometric objects. Two main classes can be distinguished. According to the terminology of [8], one is related to incidence graphs and the other to ordered models. Both classes have their own specificities and their use is relevant in different contexts. It is thus important to create bridges between them. So we define here chains of surfaces (a subclass of incidence graphs) and chains of maps without multi-incidence (a subclass of ordered models), which are able to represent the topology of subdivided objects, whose cells have " manifold-like " properties. We show their equivalence by providing conversion operations. As a consequence, it is hence possible to directly apply on each model results obtained on the other. We extend here classical results related to homology computation obtained for incidence graphs corresponding to regular CW −complexes and recent results about combinatorial cell complexes where cells are not necessarily homeomorphic to balls

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

Homology of Cellular Structures Allowing Multi-incidence

International audienceThis paper focuses on homology computation over ‘cellular’ structures that allow multi-incidence between cells. We deal here with combinatorial maps, more precisely chains of maps and subclasses such as maps and generalized maps. Homology computation on such structures is usually achieved by computing simplicial homology on a simplicial analog. But such an approach is computationally expensive because it requires computing this simplicial analog and performing the homology computation on a structure containing many more cells (simplices) than the initial one. Our work aims at providing a way to compute homologies directly on a cellular structure. This is done through the computation of incidence numbers. Roughly speaking, if two cells are incident, then their incidence number characterizes how they are attached. Having these numbers naturally leads to the definition of a boundary operator, which induces a homology. Hence, we propose a boundary operator for chains of maps and provide optimization for maps and generalized maps. It is proved that, under specific conditions, the homology of a combinatorial map as defined in the paper is equivalent to the homology of its simplicial analogue

Incremental Computation of the Homology of Generalized Maps: An Application of Effective Homology Results

This paper deals with the incremental computation of the homology of " cellular " combinatorial structures, namely combinatorial maps and incidence graphs. " Incremental " is related to the operations which are applied to construct such structures: basic operations, i.e. the creation of cells and the identification of cells, are considered in the paper. Such incremental computation is done by applying results of effective homology [RS06]: a correspondence between the chain complex associated with a given combinatorial structure is maintained with a " smaller " chain complex , from which the homology groups and homology generators can be more efficiently computed

A Boundary Operator for Computing the Homology of Cellular Structures

71 pagesThe paper focuses on homology computation over cellular structures through the computation of incidence numbers. Roughly speaking, if two cells are incident, then their incidence number characterizes how they are attached. Having these numbers naturally leads to the definition of a boundary operator, which induces a cellular homology. More precisely, the two main families of cellular structures (incidence graphs and ordered models) are studied through various models. A boundary operator is then proposed for the most general structure, and is optimized for the other structures. It is proved that, under specific conditions, the cellular boundary operator proposed in this paper defines a cellular homology equivalent to the simplicial one

Extracting cell complexes from digital images

In this paper, we define a method for constructing cell complexes from 4-dimensional binary digital images on a dual grid. First, we revisit a method similar to Kenmochi et al. method [6], [7], [8] for treating with images of dimension 3. Then, we extend this method to 4-dimensional images. The idea consists in considering the black 4-xels of the image as 0-cells of a cell complex. The cells of higher dimension of the complex are constructed by deforming the 4-cubes of the dual grid. Finally, the resulting complex can be simplified, for instance, by merging adjacent 4-cells which share a common 3-cell. More concretely, 0,1,2,3-cells non-incident to 4-cells are stored, together with 3-cells (and their boundary) incident to exactly one 4-cell

Altered Cortisol Metabolism Increases Nocturnal Cortisol Bioavailability in Prepubertal Children With Type 1 Diabetes Mellitus

ObjectiveDisturbances in the activity of the hypothalamus-pituitary-adrenal axis could lead to functional alterations in the brain of diabetes patients. In a later perspective of investigating the link between the activity of the hypothalamus-pituitary-adrenal axis and the developing brain in children with diabetes, we assessed here nocturnal cortisol metabolism in prepubertal children with type 1 diabetes mellitus (T1DM).MethodsPrepubertal patients (aged 6–12 years) diagnosed with T1DM at least 1 year previously were recruited, along with matched controls. Nocturnal urine samples were collected, with saliva samples taken at awakening and 30 minutes after awakening. All samples were collected at home over 5 consecutive days with no detectable nocturnal hypoglycaemia. The State-Trait Anxiety Inventory (trait scale only) and Child Depression Inventory were also completed. Glucocorticoid metabolites in the urine, salivary cortisol (sF) and cortisone (sE) were measured by liquid chromatography–tandem mass spectrometry. Metabolic data were analysed by logistic regression, adjusting for sex, age, BMI and trait anxiety score.ResultsUrine glucocorticoid metabolites were significantly lower in T1DM patients compared to controls. 11β-hydroxysteroid dehydrogenase type 1 activity was significantly higher, while 11β-hydroxysteroid dehydrogenase type 2, 5(α+β)-reductase and 5α-reductase levels were all lower, in T1DM patients compared to controls. There was a significant group difference in delta sE level but not in delta sF level between the time of awakening and 30 minutes thereafter.ConclusionsOur findings suggest that altered nocturnal cortisol metabolism and morning HPA axis hyperactivity in children with T1DM leads to greater cortisol bioavailability and lower cortisol production as a compensatory effect. This altered nocturnal glucocorticoid metabolism when cortisol production is physiologically reduced and this HPA axis hyperactivity question their impact on brain functioning