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102
The Topological Structure of Asynchronous Computability
 JOURNAL OF THE ACM
, 1996
"... We give necessary and sufficient combinatorial conditions characterizing the tasks that can be solved by asynchronous processes, of which all but one can fail, that communicate by reading and writing a shared memory. We introduce a new formalism for tasks, based on notions from classical algebra ..."
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Cited by 157 (12 self)
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We give necessary and sufficient combinatorial conditions characterizing the tasks that can be solved by asynchronous processes, of which all but one can fail, that communicate by reading and writing a shared memory. We introduce a new formalism for tasks, based on notions from classical algebraic and combinatorial topology, in which a task's possible input and output values are each associated with highdimensional geometric structures called simplicial complexes. We characterize computability in terms of the topological properties of these complexes. This characterization has a surprising geometric interpretation: a task is solvable if and only if the complex representing the task's allowable inputs can be mapped to the complex representing the task's allowable outputs by a function satisfying certain simple regularity properties. Our formalism thus replaces the "operational" notion of a waitfree decision task, expressed in terms of interleaved computations unfolding ...
Multiresolution modeling: Survey & future opportunities
 Proc. of the Eurographics ’99 – State of the Art Reports
, 1999
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Perceptual Completion of Occluded Surfaces
, 1994
"... Researchers in computer vision have primarily studied the problem of visual reconstruction of environmental structure that is plainly visible. In this thesis, the conventional goals of visual reconstruction are generalized to include both visible and occluded forward facing surfaces. This larger f ..."
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Cited by 40 (5 self)
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Researchers in computer vision have primarily studied the problem of visual reconstruction of environmental structure that is plainly visible. In this thesis, the conventional goals of visual reconstruction are generalized to include both visible and occluded forward facing surfaces. This larger fraction of the environment is termed the anterior surfaces. Because multiple anterior surface neighborhoods project onto a single image neighborhood wherever surfaces overlap, surface neighborhoods and image neighborhoods are not guaranteed to be in onetoone correspondence, as conventional "shapefrom" methods assume. The result is that the topology of threedimensional scene structure can no longer be taken for granted, but must be inferred from evidence...
The topological structures of membrane computing
 Fundamenta Informaticae
, 2002
"... In its initial presentation, the P system formalism describes the topology of the membranes as a set of nested regions. This description is too rough and presents several shortcommings: only the nesting of membranes is taken into account, not their adjacency and there is an artificial distinction b ..."
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Cited by 30 (19 self)
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In its initial presentation, the P system formalism describes the topology of the membranes as a set of nested regions. This description is too rough and presents several shortcommings: only the nesting of membranes is taken into account, not their adjacency and there is an artificial distinction between a membrane and its enclosed region. To answer these problems, we shown that most of the notions used to describe P systems find a natural setting and a smooth extension in the framework provided by topological notions developed in the field of homology theory. Notions like membrane structures, adjacency relationships between membranes, local computations, moves between adjacent membranes, etc., can be specified on top of the notion of chain complex. Using an appropriate abstract setting, this technical device enables us to reformulate also the computation within a membrane and proposes a unified view on several computational mechanisms initially inspired by biological processes, namely: Gamma and the CHAM, P systems, L systems and cellular automata. These models can be rephrased as the iteration of simple transformations on a topological collection, the difference coming
Delaunay Conforming Isosurface, Skeleton Extraction and Noise Removal
"... Isosurfaces are routinely used for the visualization of volumetric structures. Further processing (such as quantitative analysis, morphometric measurements, shape description) requires volume representations. The skeleton representation matches these requirements by providing a concise description ..."
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Cited by 27 (2 self)
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Isosurfaces are routinely used for the visualization of volumetric structures. Further processing (such as quantitative analysis, morphometric measurements, shape description) requires volume representations. The skeleton representation matches these requirements by providing a concise description of the object. This paper has two parts. First, we exhibit an algorithm which locally builds an isosurface with two significant properties: it is a 2manifold and the surface is a subcomplex of the Delaunay tetrahedrization of its vertices. Secondly, because of the latter property, the skeleton can in turn be computed from the dual of the Delaunay tetrahedrization of the isosurface vertices. The skeleton representation, although informative, is very sensitive to noise. This is why we associate a graph to each skeleton for two purposes: (i) the amount of noise can be identified and quantified on the graph and (ii) the selection of the graph subpart that does not correspond to noise induces a filtering on the skeleton. Finally, we show some results on synthetic and medical images. An application, measuring the thickness of objects (heart ventricles, bone samples) is also presented.
The Combinatorial Structure of Waitfree Solvable Tasks (Extended Abstract)
, 1996
"... This paper presents a selfcontained study of waitfree solvable tasks. A new necessary and sufficient condition for waitfree solvability is proved, providing a characterization of the waitfree solvable tasks. The necessary condition is used to prove tight bounds on renaming and kset consensus. ..."
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Cited by 25 (14 self)
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This paper presents a selfcontained study of waitfree solvable tasks. A new necessary and sufficient condition for waitfree solvability is proved, providing a characterization of the waitfree solvable tasks. The necessary condition is used to prove tight bounds on renaming and kset consensus. The framework is based on topology, but uses only elementary combinatorics, and does not rely on algebraic or geometric arguments.
Widening the boundary between decidable and undecidable hybrid systems
, 2002
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The Fluid Dynamics applied to Mobile Robot Motion: the Stream Field Method
 Proceedings of the IEEE Int. Conference on Robotics and Automation
, 1994
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Topologydriven surface mappings with robust feature alignment
 In IEEE Visualization
, 2005
"... Figure 1: Surface mapping between horse and lizard. The colorcoding shows the mapping of each region, guided by eight userspecified feature curves. Our topologydriven method provides mappings of different homotopy type between the two surfaces as shown in (c) and (d). We show feature curves in re ..."
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Cited by 23 (11 self)
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Figure 1: Surface mapping between horse and lizard. The colorcoding shows the mapping of each region, guided by eight userspecified feature curves. Our topologydriven method provides mappings of different homotopy type between the two surfaces as shown in (c) and (d). We show feature curves in red. Topological concepts and techniques have been broadly applied in computer graphics and geometric modeling. However, the homotopy type of a mapping between two surfaces has not been addressed before. In this paper, we present a novel solution to the problem of computing continuous maps with different homotopy types between two arbitrary triangle meshes with the same topology. Inspired by the rich theory of topology as well as the existing body of work on surface mapping, our newlydeveloped mapping techniques are both fundamental and unique, offering many attractive advantages. First, our method allows the user to change the homotopy type or global structure of the mapping with minimal intervention. Moreover, to locally affect shape correspondence, we articulate a new technique that robustly satisfies hard feature constraints, without the use of heuristics to ensure validity. In addition to acting as a useful tool for computer graphics applications, our method can be used as a rigorous and practical mechanism for the visualization of abstract topological concepts such as homotopy type of surface mappings, homology basis, fundamental domain, and universal covering space. At the core of our algorithm is a procedure for computing the canonical homology basis and using it as a common cut graph for any surface with the same topology. We demonstrate our results by applying our algorithm to shape morphing in this paper. 1
A quasilinear algorithm to compute the tree of shapes of nD images
"... Abstract. To compute the morphological selfdual representation of images, namely the tree of shapes, the stateoftheart algorithms do not have a satisfactory time complexity. Furthermore the proposed algorithms are only effective for 2D images and they are far from being simple to implement. That ..."
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Cited by 21 (12 self)
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Abstract. To compute the morphological selfdual representation of images, namely the tree of shapes, the stateoftheart algorithms do not have a satisfactory time complexity. Furthermore the proposed algorithms are only effective for 2D images and they are far from being simple to implement. That is really penalizing since a selfdual representation of images is a structure that gives rise to many powerful operators and applications, and that could be very useful for 3D images. In this paper we propose a simpletowrite algorithm to compute the tree of shapes; it works for nD images and has a quasilinear complexity when data quantization is low, typically 12 bits or less. To get that result, this paper introduces a novel representation of images that has some amazing properties of continuity, while remaining discrete. 1