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Persistent Triangulations
 UNDER CONSIDERATION FOR PUBLICATION IN J. FUNCTIONAL PROGRAMMING
, 2002
"... Triangulations of a surface are of fundamental importance in computational geometry, computer graphics, and engineering and scientific simulations. Triangulations are ordinarily represented as mutable graph structures for which both adding and traversing edges take constant time per operation. These ..."
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Triangulations of a surface are of fundamental importance in computational geometry, computer graphics, and engineering and scientific simulations. Triangulations are ordinarily represented as mutable graph structures for which both adding and traversing edges take constant time per operation. These representations of triangulations make it difficult to support persistence, including “multiple futures”, the ability to use a data structure in several unrelated ways in a given computation; “time travel”, the ability to move freely among versions of a data structure; or parallel computation, the ability to operate concurrently on a data structure without interference. We present a purely functional interface and representation of triangulated surfaces, and more generally of simplicial complexes in higher dimensions. In addition to being persistent in the strongest sense, the interface more closely matches the mathematical definition of triangulations (simplicial complexes) than do interfaces based on mutable representations. The representation, however, comes at the cost of requiring O(lg n) time for traversing or adding triangles (simplices), where n is the number of triangles in the surface. We show both analytically and experimentally that for certain important cases, this extra cost does not seriously affect endtoend running time. Analytically, we present a new randomized algorithm for 3dimensional Convex Hull based on our representations for which the running time matches the Ω(n lg n) lowerbound for the problem. This is achieved by using only O(n) traversals of the surface. Experimentally, we present results for both an implementation of the 3dimensional Convex Hull and for a terrain modeling algorithm, which demonstrate that, although there is some cost to persistence, it seems to be a small constant factor.
A LusternikSchnirelmann theorem for graphs. http://arxiv.org/abs/1211.0750
, 2012
"... Abstract. We prove the discrete LusternikSchnirelmann theorem tcat(G) ≤ crit(G) for general simple graphs G = (V, E). It relates tcat(G), the minimal number of in G contractible graphs covering G, with crit(G), the minimal number of critical points which an injective function f: V → R can have. Als ..."
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Abstract. We prove the discrete LusternikSchnirelmann theorem tcat(G) ≤ crit(G) for general simple graphs G = (V, E). It relates tcat(G), the minimal number of in G contractible graphs covering G, with crit(G), the minimal number of critical points which an injective function f: V → R can have. Also the cup length estimate cup(G) ≤ tcat(G) is valid for any finite simple graph. Let cat(G) be the minimal tcat(H) among all H homotopic to G and let cri(G) be the minimal crit(H) among all graphs H homotopic to G, then cup(G) ≤ cat(G) ≤ cri(G) relates three homotopy invariants for graphs: the algebraic cup(G), the topological cat(G) and the analytic cri(G). 1.
On critical kernels
, 2005
"... We propose a method for collapsing simplicial complexes in a symmetric manner. For that purpose, we introduce the notions of a simple cell, of an essential face, and the one of a core of a cell. Then, we define the critical kernel of a complex. Our main result is that the critical kernel of a given ..."
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We propose a method for collapsing simplicial complexes in a symmetric manner. For that purpose, we introduce the notions of a simple cell, of an essential face, and the one of a core of a cell. Then, we define the critical kernel of a complex. Our main result is that the critical kernel of a given complex X is a collapse of X. We extend this result by giving a necessary and sufficient condition which characterizes a certain class of subcomplexes of X which contain the critical kernel of X. In particular, any complex which belongs to this class is homotopy equivalent to X. 1
Some properties of topological greyscale watersheds
 ROCS. SPIE VISION GEOMETRY XII, FRANCE
, 2004
"... In this paper, we investigate topological watersheds. 1 For that purpose we introduce a notion of “separation between two points ” of an image. One of our main results is a necessary and sufficient condition for a map G to be a watershed of a map F, this condition is based on the notion of separatio ..."
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In this paper, we investigate topological watersheds. 1 For that purpose we introduce a notion of “separation between two points ” of an image. One of our main results is a necessary and sufficient condition for a map G to be a watershed of a map F, this condition is based on the notion of separation. A consequence of the theorem is that there exists a (greedy) polynomial time algorithm to decide whether a map G is a watershed of a map F or not. We also show that, given an arbitrary total order on the minima of a map, it is possible to define a notion of “degree of separation of a minimum ” relative to this order. This leads to another necessary and sufficient condition for a map G to be a watershed of a map F. At last we derive, from our framework, a new definition for the dynamics of a minimum.
Branching and Circular Features in High Dimensional Data
, 2011
"... Large observations and simulations in scientific research give rise to highdimensional data sets that present many challenges and opportunities in data analysis and visualization. Researchers in the application domains such as engineering, computational biology, climate study, imaging and motion ca ..."
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Large observations and simulations in scientific research give rise to highdimensional data sets that present many challenges and opportunities in data analysis and visualization. Researchers in the application domains such as engineering, computational biology, climate study, imaging and motion capture are faced with the problem of how to discover compact representations of highdimensional data while preserving their intrinsic structure. In many applications, the original data is projected onto lowdimensional space via dimensionality reduction techniques prior to modeling. One problem with this approach is that the projection step in the process can fail to preserve structure in the data that is only apparent in high dimensions. Conversely, such techniques may create structural illusions in the projection, implying structure not present in the original highdimensional data. Our solution is to utilize topological techniques to recover important structures in highdimensional data that contains nontrivial topology. Specifically, we are interested in two types of features in high dimensions: local branching structures and global circular structures. We construct local and global circlevalued coordinate functions to represent such features. Subsequently, we perform dimensionality reduction on the data while ensuring such structures are visually preserved. Our results reveal neverbeforeseen structures on realworld data sets from a variety of applications. Branching and Circular Features in High Dimensional Data
Correctness Criteria based on a Homology of Proof Structures in Multiplicative Linear Logic
, 1996
"... Contents 1 Introduction 1 2 Multiplicative linear logic 5 2.1 The calculus : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.2 Cut elimination : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 2.3 Strong normalization : : : : : : : : : : : : : : : ..."
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Contents 1 Introduction 1 2 Multiplicative linear logic 5 2.1 The calculus : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.2 Cut elimination : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 2.3 Strong normalization : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 3 Graphs 15 3.1 Paired directed multigraphs : : : : : : : : : : : : : : : : : : : : : : : : : : : : 15 3.2 Constructions on pdm's : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 17 3.3 Proof nets : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 20 4 Homology 27 4.1 General theory : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 27 4.2 Application on pdm's : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 31
Feasibility of a Skeletal Modeler for Conceptual Mechanical Design
, 2000
"... vii List of Symbols xi Chapter 1 The Need For A Conceptual Modeler 1 1.1 Current practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Process of Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . ..."
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vii List of Symbols xi Chapter 1 The Need For A Conceptual Modeler 1 1.1 Current practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Process of Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Chapter 2 Partitioning Space 12 2.1 Voronoi diagrams and Delaunay triangulations . . . . . . . . . . . . 12 2.2 Power diagrams and regular triangulations . . . . . . . . . . . . . . . 14 2.3 Representing a regular triangulation . . . . . . . . . . . . . . . . . . 17 2.3.1 Vertex insertion algorithm . . . . . . . . . . . . . . . . . . . . 25 2.4 Topology of finite triangulations . . . . . . . . . . . . . . . . . . . . 26 2.4.1 A compact geometric notation . . . . . . . . . . . . . . . . . 27 2.4.2 Division ain't what it used to be . . . . . . . . . . . . . . . . 31 2.4.3 Homology groups . . . . . . ....
Topological properties of thinning in 2D pseudomanifolds
, 2008
"... Preserving topological properties of objects during thinning procedures is an important issue in the field of image analysis. In the case of 2D digital images (i.e. images defined onZ 2) such procedures are usually based on the notion of simple point. By opposition to the case of spaces of higher d ..."
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Preserving topological properties of objects during thinning procedures is an important issue in the field of image analysis. In the case of 2D digital images (i.e. images defined onZ 2) such procedures are usually based on the notion of simple point. By opposition to the case of spaces of higher dimensions (i.e.Z n, n ≥ 3), it was proved in the 80’s that the exclusive use of simple points inZ 2 was indeed sufficient to develop thinning procedures providing an output that is minimal with respect to the topological characteristics of the object. Based on the recently introduced notion of minimal simple set (generalising the notion of simple point), we establish new properties related to topologypreserving thinning in 2D spaces which extend, in particular, this classical result to more general spaces (the 2D pseudomanifolds) and more general objects (the 2D cubical complexes).
Mapping Complicated Surfaces onto a Sphere
"... A method for mapping a closed, triangulated surface of genus 0 onto the unit sphere was developed using a parameterization based on spherical coordinates analogous to latitude and longitude combined with the Mercator scaling. The algorithm was tested on 77 protein and DNA surfaces and produces corre ..."
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A method for mapping a closed, triangulated surface of genus 0 onto the unit sphere was developed using a parameterization based on spherical coordinates analogous to latitude and longitude combined with the Mercator scaling. The algorithm was tested on 77 protein and DNA surfaces and produces correct (bijective) mappings. The mappings produce relatively uniform distortion, as judged by angular criteria, and by distribution of octant areas, thus facilitating the graphical analysis of these surfaces. 1.