an o n log log n time algorithm for triangulating a simple polygon given a simple n vertex polygon the triangulation problem is to partition the interior of the polygon into n 2 triangles by adding n 3 nonintersecting diagonals we propose an o n log log n time algorithm for this problem improving on the previously best bound of o n log n and showing that triangulation is not as hard as sorting improved algorithms for several other computational geometry problems including testing whether a polygon is simple follow from our result

fast methods for the eikonal and related hamilton jacobi equations on unstructured meshes the fast marching method is a numerical algorithm for solving the eikonal equation on a rectangular orthogonal mesh in o m log m steps where m is the total number of grid points the scheme relies on an upwind finite difference approximation to the gradient and a resulting causality relationship that lends itself to a dijkstra like programming approach in this paper we discuss several extensions to this technique including higher order versions on unstructured meshes in rn and on manifolds and connections to more general static hamilton jacobi equations

quantum 2 spheres and big q jacobi polynomials orthogonal bases for the algebras of functions of podles quantum 2 spheres are explicitly determined in terms of bigq jacobi polynomials this gives a group theoretic interpretation of the symmetric bigq jacobi polynomials and the symmetricq hahn polynomials

finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple lie algebra let g be a complex simple lie algebra we show that whent is not a root of 1 all finite dimensional representations of the quantum analogu t g are completely reducible and we classify the irreducible ones in terms of highest weights in particular they can be seen as deformations of the representations of the classical u g

cortical surface based analysis ii inflation flattening and a surface based coordinate system the surface of the human cerebral cortex is a highly folded sheet with the majority of its surface area buried within folds as such it is a difficult domain for computational as well as visualization purposes we have therefore designed a set of procedures for modifying the representation of the cortical surface to i inflate it so that activity buried inside sulci may be visualized ii cut and flatten an entire hemisphere and iii transform a hemisphere into a simple parameterizable surface such as a sphere for the purpose of establishing a surface based coordinate system r 1999 academic press

spherical and hyperbolic embeddings of data many computer vision and pattern recognition problems may be posed as the analysis of a set of dissimilarities between objects for many types of data these dissimilarities are not euclidean i e they do not represent the distances between points in a euclidean space and therefore cannot be isometrically embedded in a euclidean space examples include shape dissimilarities graph distances and mesh geodesic distances in this paper we provide a means of embedding such non euclidean data onto surfaces of constant curvature we aim to embed the data on a space whose radius of curvature is determined by the dissimilarity data the space can be either of positive curvature spherical or of negative curvature hyperbolic we give an efficient method for solving the spherical and hyperbolic embedding problems on symmetric dissimilarity data our approach gives the radius of curvature and a method for approximating the objects as points on a hyperspherical manifold without optimisation for objects which do not reside exactly on the manifold we develop a optimisation based procedure for approximate embedding on a hyperspherical manifold we use the exponential map between the manifold and its local tangent space to solve the optimisation problem locally in the euclidean tangent space this process is efficient enough to allow us to embed data sets of several thousand objects we apply our method to a variety of data including time warping functions shape similarities graph similarity and gesture similarity data in each case the embedding maintains the local structure of the data while placing the points in a metric space

invariant operators for the n dimensional super poincare algebra and the decomposition of the scalar superfield an analysis of supersymmetric kaluza klein theories is begun by obtaining the casimir operators for the super poincare algebra in any number of dimensions the knowledge of these operators is used to decompose the general scalar superfield in 11 dimensions into its irreducible parts the irreducible superfields are expressed as products of grassmann hermite functions and grassmann bargmann wigner multispinor fields some lagrangians for these superfields are written down the formulation is off shell but global

quantitative evaluation of three cortical surface flattening methods during the past decade several computational approaches have been proposed for the task of mapping highly convoluted surfaces of the human brain to simpler geometric objects such as a sphere or a topological disc we report the results of a quantitative comparison of freesurfer circlepack and lscm with respect to measurements of geometric distortion and computational speed our results indicate that freesurfer performs best with respect to a global measurement of metric distortion whereas lscm performs best with respect to angular distortion and best in all but one case with a local measurement of metric distortion freesurfer provides more homogeneous distribution of metric distortion across the whole cortex than circlepack and lscm lscm is the most computationally efficient algorithm for generating spherical maps while circlepack is extremely fast for generating planar maps from patches

stochastic jump diffusion process for computing medial axes in markov random fields proposes a statistical framework for computing medial axes of 2d shapes in the paper the computation of medial axes is posed as a statistical inference problem not as a mathematical transform the paper contributes to three aspects in computing medial axes 1 prior knowledge is adopted for axes and junctions so that axes around junctions are regularized 2 multiple interpretations of axes are possible each being assigned a probability 3 a stochastic jump diffusion process is proposed for estimating both axes and junctions in markov random fields we argue that the stochastic algorithm for computing medial axes is compatible with existing algorithms for image segmentation such as region growing snake and region competition thus our method provides a new direction for computing medial axes from texture images experiments are demonstrated on both synthetic and real 2d shapes

the weighted region problem finding shortest paths through a weighted planar subdivision the problem of determining shortest paths through a weighted planar polygonal subdivision with n vertices is considered distances are measured according to a weighted euclidean metric the length of a path is defined to be the weighted sum of euclidean lengths of the subpaths within each region an algorithm that constructs a restricted shortest path map with respect to a given source point is presented the output is a partitioning of each edge of the subdivion into intervals of e optimality allowing an e optimal path to be traced from the source to any query point along any edge the algorithm runs in worst case time o es and requires o e space where e is the number of events in our algorithm and s is the time it takes to run a numerical search procedure in the worst case e is bounded above by o n 4 and we give an o n 4 lower bound but it is likeky that e will be much smaller in practice we also show that s is bounded by o n 4 l where l is the precision of the problem instance including the number of bits in the user specified tolerance e again the value of s should be smaller in practice the algorithm applies the continuous dijkstra paradigm and exploits the fact that shortest paths obey snell s law of refraction at region boundaries a local optimaly property of shortest paths that is well known from the analogous optics model the algorithm generalizes to the multi source case to compute voronoi diagrams

grassmannian legendre functions the grassmannian analogues of the legendre polonomials are obtained from the corresponding grassmannian hermite multinomials the generating function recurrence relations and differential equation are given in contrast to the hermitian case there is no berezin weight function that orthogonalizes these functions generalizations to the tchebyscheff and gegenbauer cases are also possible

constructing approximate shortest path maps in three dimensions we present a new technique for constructing a data structure that approximates shortest path maps in re d by applying this technique we get the following two results on approximate shortest path maps in re 3 i given a polyhedral surface or a convex polytope p with n edges in re 3 a source point s on p and a real parameter 0 eps leq 1 we present an algorithm that computes a subdivision of p of size o n eps log 1 eps which can be used to answer efficiently approximate shortest path queries namely given any point t on p one can compute in o log n eps time a distance delta p s t such that d p s t leq delta p s t leq 1 eps d p s t where d p s t is the length of a shortest path between s and t on p the map can be computed in o n 2 log n n eps log 1 eps log n eps time for the case of a polyhedral surface and in o n eps 3 log 1 eps n eps 1 5 log 1 eps log n time if p is a convex polytope ii given a set of polyhedral obstacles o with a total of n edges in re 3 a source point it s in re 3 rm setminus inter cup scriptstyle o in o o and a real parameter 0 eps leq 1 we present an algorithm that computes a subdivision of re 3 which can be used to answer efficiently approximate shortest path queries that is for any point t in re 3 one can compute in o log n eps time a distance delta o s t that eps approximates the length of a shortest path from it s to it t that avoids the interiors of the obstacles this subdivision can be computed in roughly o n 4 eps 6 time

dynamic programming generation of curves on brain surfaces dynamic programming algorithms are presented for automated generation of length minimizing geodesics and curves of extremal curvature on the neocortex of the macaque and the visible human probabilistic models of curve variation are constructed in terms of the variability in speed curvature and torsion in the frenet representation

determining approximate shortest paths on weighted polyhedral surfaces in this article we present an approximation algorithm for solving the single source shortest paths problem on weighted polyhedral surfaces we consider a polyhedral surface p as consisting of n triangular faces where each face has an associated positive weight the cost of travel through a face is the euclidean distance traveled multiplied by the face s weight for a given parameter e 0 e 1 the cost of the computed paths is at most 1 p e times the cost of corresponding shortest paths our algorithm is based on a novel way of discretizing polyhedral surfaces and utilizes a generic greedy approach for computing shortest paths in geometric graphs obtained by such discretization its running time is o c p n se log n e log 1 e time where c p captures geometric parameters and the weights of the faces of p

approximating shortest path for the skew lines problem in time doubly logarithmic in 1 epsilon we consider two three dimensional situations when a polytime algorithm for approximating a shortest path can be constructed the main part of the paper treats a well known problem of constructing a shortest path touching lines in r3 given a list of straight lines l l1 ln in r3 and two points s and t find a shortest path that starting from s touches the lines li in the given order and ends at t we remark that such a shortest path is unique we show that it can be length position e approximated i e both its length and its position can be found approximately in time rn d 16 o n2 log log 1 e where d is the minimal distance between consecutive lines of l is the minimum of sines of angles between consecutive lines and r is the radius of a ball where the initial approximation can be placed such a radius can be easily computed from the initial data as computational model we take real ram extended by square and cubic roots extraction this problem of constructing a shortest path touching lines is known for quite some time to be a challenging problem the existing methods for approximating shortest paths based on adding steiner points which form a grid and subsequently applying dijkstra s algorithm for finding a shortest path in the grid provide a complexity bound which depends polynomially on 1 e while our algorithm for the problem under consideration has complexity linear in log log 1 e our algorithm is motivated by the observation that the shortest path in question is a geodesic in a certain length space of non positive curvature in the sense of a d alexandrov and it relies on the elementary theory of cat 0 spaces in the second part of the paper we analyze very simple grid approximations we assume that a parameter a 0 describing separability of obstacles is given and the part of a grid with mesh size a outside the obstacles is built for semi algebraic obstacles all these precalculations are polytime we show that there is an algorithm of time complexity o 1 a 6 which given a separated obstacles in a unit cube finds a path between given vertices s and t of the grid whose length is bounded from above by 84 96a where is the length of a shortest path on the other hand as we show by an example one cannot approximate the length of a shortest path better than 7 if one uses only grid polygons constructed only from grid edges for semi algebraic obstacles our computational model is bitwise for a general type of obstacles the model is bitwise modulo constructing the part of the grid admissible for our paths observe that the existing methods for approximating shortest paths are not directly applicable for semi algebraic obstacles since they usually place the steiner points forming a grid on the edges of polyhedral obstacles

computational surface flattening a voxel based approach a voxel based method for flattening a surface in 3d space into 2d while best preserving distances is presented triangulation or polyhedral approximation of the voxel data are not required the problem is divided into two main parts voxel based calculation of the minimal geodesic distances between points on the surface and finding a configuration of points in 2d that has euclidean distances as close as possible to these distances the method suggested combines an efficient voxel based hybrid distance estimation method that takes the continuity of the underlying surface into account with classical multidimensional scaling mds for finding the 2d point configuration the proposed algorithm is efficient simple and can be applied to surfaces that are not functions experimental results are shown

a boundary focused quadrilateral mesh generation algorithm for multi material structures this paper describes a new boundary focused mesh generation algorithm the algorithm produces quadrilateral meshes from images of multi material structures the challenge of meshing such images is to create high quality elements which are aligned with complex material boundaries the approach proposed in this paper uses the following steps 1 extract boundaries according to pixel colours 2 smooth the jagged boundaries 3 generate geodesic isolines aligned with smoothed boundaries 4 generate a mesh from the boundaries and isolines 5 optimize the mesh using the pisces pattern which is introduced in this paper application examples are presented to contrast the reliability and effectiveness of the new algorithm with existing approaches

approximating generalized distance functions on weighted triangulated surfaces with applications given p a simple connected possibly non convex polyhedral surface composed of positively weighted triangular faces we consider paths from generalized sources points segments polygonal chains or polygonal regions to points on p that stay on p and avoid obstacles segments polygonal chains or polygonal regions the distance function defined by a generalized source is a function that assigns to each point of p the cost of the shortest path from the source to the point in this paper we present an algorithm for computing approximate generalized distance functions we also provide an algorithm that computes a discrete representation of the approximate distance function and as applications algorithms for computing discrete order k voronoi diagrams and for approximately solving facility location problems finally we present experimental results obtained with our implementation of the provided algorithms

irreducible so 2 extended superfields all the irreducible so 2 extended scalar superfields are obtained in so 2 invariant form these superfields can be labelled by the eigenvalues of three casimir operators of the enlarged algebra containing chiral transformation they may be still more simply classified by introducing two invariant simple superspaces lagrangians and supersymmetry transformation rules are obtained from the explicit form of the irreducible superfields the highest superspin multiplet is identified as the multiplet for the linearized so 2 weyl supergravity the linearized so 2 poincare supergravity is obtained by coupling this multiplet with the chiral superfields

a grassmann integral equation the present study introduces and investigates a new type of equation which is called grassmann integral equation in analogy to integral equations studied in real analysis a grassmann integral equation is an equation which involves grassmann berezin integrations and which is to be obeyed by an unknown function over a finite dimensional grassmann algebra gm i e a sought after element of the grassmann algebra gm a particular type of grassmann integral equations is explicitly studied for certain low dimensional grassmann algebras the choice of the equation under investigation is motivated by the effective action formalism of lattice quantum field theory in a very general setting for the grassmann algebras g2n n 2 3 4 the finite dimensional analogues of the generating functionals of the green functions are worked out explicitly by solving a coupled system of nonlinear matrix equations finally by imposing the condition g g0 const 0 r k k k 1 n are the generat

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

connectivity of discrete planes studying connectivity of discrete objects is a major issue in discrete geometry and topology in the present work we deal with connectivity of discrete planes in the framework of reveilles analytical definition these d etat universite louis pasteur strasbourg france 1991 accordingly a discrete plane is a set p a b c of integer points x y z satisfying the diophantine inequalities 0 ax by cz the parameter z estimates the plane intercept while n is the plane thickness given three integers plane coefficients a b and c with 0 a b c one can seek the value of beyond which the discrete plane p a b c is always connected we call this remarkable topological invariable the connectivity number of p a b c and denote it a b c despite several attempts over the last 10 years to determine the connectivity number this is still an open question in the present paper we propose a solution to the problem for this we first investigate some combinatorial properties of discrete planes these structural results facilitate the deeper understanding of the discrete plane structure on this basis we obtain a series of results in particular we provide an explicit solution to the problem under certain conditions we also obtain exact upper and lower bounds on a b c and design an o alogb algorithm for its computation

a single exponential upper bound for finding shortest paths in three dimensions we derive a single exponential time upper bound for finding the shortest path between two points in 3 dimensional euclidean space with nonnecessarily convex polyhedral obstacles prior to this work the best known algorithm required double exponential time given that the problem is known to be pspace hard the bound we present is essentially the best in the worst case sense that can reasonably be expected

casimir operators for massless representations of the super poincare algebra and the reduction of the ten dimensional massless scalar superfield invariant operators for the massless little algebra of spd super poincare algebra in d dimensions are given they are used to decompose the scalar massless superfield in ten dimensions explicit expressions for the irreducible pieces are obtained after exploiting a relevant cartan subalgebra

on shortest paths in polyhedral spaces we consider the problem of computing the shortest path between two points in two or three dimensional space bounded by polyhedral surfaces in the 2 d case the problem is easily solved in time o n 2 log n in the general 3 d case the problem is quite hard to solve and is not even discrete we present a doubly exponential procedure for solving the discrete subproblem of determining the sequence of boundary edges through which the shortest path passes finally we consider a favorable special case of the 3 d shortest path problem namely that of finding the shortest path between two points along the surface of a convex polyhedron and solve it in time o n 3 log n

brain surface conformal parameterization with the ricci flow in brain mapping research parameterized 3 d surface models are of great interest for statistical comparisons of anatomy surface based registration and signal processing here we introduce the theories of continuous and discrete surface ricci flow which can create riemannian metrics on surfaces with arbitrary topologies with user defined gaussian curvatures the resulting conformal parameterizations have no singularities and they are intrinsic and stable first we convert a cortical surface model into a multiple boundary surface by cutting along selected anatomical landmark curves secondly we conformally parameterize each cortical surface to a parameter domain with a user designed gaussian curvature arrangement in the parameter domain a shape index based on conformal invariants is computed and inter subject cortical surface matching is performed by solving a constrained harmonic map we illustrate various target curvature arrangements and demonstrate the stability of the method using longitudinal data to map statistical differences in cortical morphometry we studied brain asymmetry in 14 healthy control subjects we used a manifold version of hotelling s t2 test applied to the jacobian matrices of the surface parameterizations a permutation test along with the cumulative distribution of p values were used to estimate the overall statistical significance of differences the results show our algorithm s power to detect subtle group differences in cortical surfaces

semi continuity of skeletons in two manifold and discrete voronoi approximation the skeleton of a 2d shape is an important geometric structure in pattern analysis and computer vision in this paper we study the skeleton of a 2d shape in a two manifold mathcal m based on a geodesic metric we present a formal definition of the skeleton s omega for a shape omega in mathcal m and show several properties that make s omega distinct from its euclidean counterpart in mathbb r 2 we further prove that for a shape sequence lbrace omega i rbrace that converge to a shape omega in mathcal m the mapping omega rightarrow overline s omega is lower semi continuous a direct application of this result is that we can use a set p of sample points to approximate the boundary of a 2d shape omega in mathcal m and the voronoi diagram of p inside omega subset mathcal m gives a good approximation to the skeleton s omega examples of skeleton computation in topography and brain morphometry are illustrated

shortest paths without a map we study several versions of the shortest path problem when the map is not known in advanced but is specified dynamically we are seeking dynamic decision rules that optimize the worst case ratio of the distance covered to the length of the statically optimal path we describe optimal decision rules for two cases layered graphs of bounded width and two dimensional scenes with unit square obstacles the optimal rules turn out to be intuitive common sense heuristics for slightly more general graphs and scenes we show that no bounded ratio is possible we also show that the computational problem of devising a strategy that achieves a given worst case ratio to the optimum path in a graph is a universal two person game and thus pspace complete whereas optimizing the expected ratio is p hard

estimating shortest paths and minimal distances on digitized three dimensional surfaces abstract the estimation of minimal distances and shortest paths between points on a continuous three dimensional 3d surface given in digitized form is considered the suggested approach combines recently developed length estimators for digitized 3d curves with well known algorithms for computing shortest paths in sparse graphs to estimate the minimal distances and the corresponding shortest paths in a systematic computationally efficient way in comparison to differential geometry techniques for solving the continuous geodesic problem and to computational geometry algorithms for solving the discrete geodesic probelm the suggested hybrid continuous discrete approach is well suited to applications in which continuous surfaces are given in digitized form computation time must be relatively short and an approximate solution is acceptable in particular the suggested method can be used to efficiently obtain a good initial approximation as an input to other algorithms thus improving their speed of convergence and reducing their tendency to converge to insignificant local minima some confusion has recently arisen with respect to the relation between the design of length estimators and the optimization of local distances for distance transformations since 3d length estimation is a central concept in the suggested approach an effort is made to resolve discrepancies and to highlight the similarities and differences between these problems it is shown that while in two dimensions the design of length estimators and the design of local weights for distance transformations are rather similar in three dimensions digital geometry dictates that these problems are different the contexts in which each of the two approaches should be preferred are identified

about thin arithmetic discrete planes arithmetic discrete planes are sets of integer points located within a fixed bounded distance called thickness of a euclidean plane we focus here on a class of thin arithmetic discrete planes i e on a class of arithmetic discrete planes whose thickness is smaller than the usual one namely the so called standard one these thin arithmetic discrete planes have holes but we consider a thickness large enough for these holes to be bounded by applying methods issued from the study of tilings and quasicrystals derived from cut and project schemes we first consider configurations that occur in thin arithmetic discrete planes we then discuss substitution rules acting on thin discrete planes with these geometric rules mapping faces of unit cubes to unions of such faces

near isometric flattening of brain surfaces flattened representations of brain surfaces are often used to visualize and analyze spatial patterns of structural organization and functional activity here we present a set of rigorous criteria and accompanying test cases with which to evaluate flattening algorithms that attempt to preserve shortest path distances on the original surface we also introduce a novel flattening algorithm that is the first to satisfy all of these criteria and demonstrate its ability to produce accurate flat maps of human and macaque visual cortex using this algorithm we have recently obtained results showing a remarkable unexpected degree of consistency in the shape and topographic structure of visual cortical areas within humans and macaques as well as between these two species

thin discrete triangular meshes abstract in this paper we present an approach to describe polyhedra by meshes of discrete triangles the study is based on the theory of arithmetic discrete geometry j p reveilles geometrie discrete calcul en nombres entiers et algorithmique these d etat universite louis pasteur strasbourg december 1991 as distinct from the previous investigations on this topic the triangles we introduce are parts of the thinnest possible discrete 6 tunnel free planes i e those that are usually used in practice given a plane p in the space we define a 6 tunnel free discrete plane called a regular plane which appears to be the best approximation to p given a mesh of triangles we propose a method to approximate any triangle by a discrete triangular patch a portion of a regular plane and we prove that the resulting triangular mesh is 6 tunnel free the properties of the approximation obtained make the suggested approach convenient for practical applications

surface preserving robust watermarking of 3 d shapes this paper describes a new statistical approach for watermarking mesh representations of 3 d graphical objects a robust digital watermarking method has to mitigate among the requirements of watermark invisibility robustness embedding capacity and key security the proposed method employs a mesh propagation distance metric procedure called the fast marching method fmm which defines regions of equal geodesic distance width calculated with respect to a reference location on the mesh each of these regions is used for embedding a single bit the embedding is performed by changing the normalized distribution of local geodesic distances from within each region two different embedding methods are used by changing the mean or the variance of geodesic distance distributions geodesic distances are slightly modified statistically by displacing the vertices in their existing triangle planes the vertex displacements performed according to the fmm ensure a minimal surface distortion while embedding the watermark code robustness to a variety of attacks is shown according to experimental results

untangling the blum medial axis transform for over 30 years blum s medial axis transform mat has proven to be an intriguing tool for analyzing and computing with form but it is one that is notoriously difficult to apply in a robust and stable way it is well documented how a tiny change to an object s boundary can cause a large change in its mat there has also been great difficulty in using the mat to decompose an object into a hierarchy of parts reflecting the natural parts hierarchy that we perceive this paper argues that the underlying cause of these problems is that medial representations embody both the substance of each part of an object and the connections between adjacent parts a small change in an object s boundary corresponds to a small change in its substance but may involve a large change in its connection information the problems with blum s mat are generated because it does not explicitly represent this dichotomy of information to use the blum mat to it s full potential this paper presents a method for separating the substance and connection information of an object this provides a natural parts hierarchy while eliminating instabilities due to small boundary changes the method also allows for graded fuzzy classifications of object parts to match the ambiguity in human perception of many objects

quantum deformations of certain simple modules over enveloping algebras soit f un corps soit f 0 un sous corps de f soit u l algebre enveloppante universelle sur f 0 de l algebre de lie de kac moody correspondant a la matrice a d entiers on montre qu un module integrable de poids le plus eleve v de u admet une deformation quantique

q deformations of the oscillator generating functions that interpolate between the generating functions for hermitian multilinears and the corresponding grassmann multilinears are considered the new functions that arise in this way may be related to state functions of q deformed harmonic oscillators

decomposition of the scalar superfield in ten dimensions explicit expressions for the irreducible superfields included in the ten dimensional scalar superfield are derived by using a method based in a cartan subalgebra

determining the geometry of boundaries of objects from medial data we consider a region in r2 or r3 with generic smooth boundary b and blum medial axis m on which is defined a multivalued radial vector field u from points x on m to the points of tangency of the sphere at x with b we introduce a radial shape operator srad and an edge shape operator se which measure how u bends along m these are not traditional differential geometric shape operators nonetheless we derive all local differential geometric invariants of b from these operators this allows us to define from m u a geometric medial map on m which corresponds via a radial map from m to b to the differential geometric properties of b the geometric medial map also includes a description of the relative geometry of b this is defined using the relative critical set of the radius function r on m this set consists of a network of curves on m which describe where b is thickest and thinnest it is computed using the covariant derivative of the tangential component of the unit radial vector field we further determine how these invariants are related to the differential geometric invariants of m and how these invariants change under deforming diffeomorphisms of m

multivariate tensor based morphometry on surfaces application to mapping ventricular abnormalities in hiv aids here we developed a new method called multivariate tensor based surface morphometry tbm and applied it to study lateral ventricular surface differences associated with hiv aids using concepts from differential geometry and the theory of differential forms we created mathematical structures known as holomorphic one forms to obtain an efficient and accurate conformal parameterization of the lateral ventricular surfaces in the brain the new meshing approach also provides a natural way to register anatomical surfaces across subjects and improves on prior methods as it handles surfaces that branch and join at complex 3d junctions to analyze anatomical differences we computed new statistics from the riemannian surface metrics these retain multivariate information on local surface geometry we applied this framework to analyze lateral ventricular surface morphometry in 3d mri data from 11 subjects with hiv aids and 8 healthy controls our method detected a 3d profile of surface abnormalities even in this small sample multivariate statistics on the local tensors gave better effect sizes for detecting group differences relative to other tbm based methods including analysis of the jacobian determinant the largest and smallest eigenvalues of the surface metric and the pair of eigenvalues of the jacobian matrix the resulting analysis pipeline may improve the power of surface based morphometry studies of the brain

on the intrinsic reconstruction of shape from its symmetries the main question we address is what is the minimal information required to generate closed nonintersecting planar boundaries for this paper we restrict shape to this meaning more precisely we examine whether the medial axis together with dynamics can serve as a language to design shapes and to effect shape changes we represent the medial axis together with a direction of flow along the axis as the shock graph and examine the reconstruction of shape along each of the three types of medial axis points a sub 1 sup 2 a sub 1 sup 3 a sub 3 and the associated six types of shock points first we show that the tangent and curvature of the medial axis and the speed and acceleration of the shock with respect to time of propagation are sufficient to determine the boundary tangent and curvature at corresponding points of the boundary this implies that a rather coarse sampling of the symmetry axis its tangent curvature speed and acceleration is sufficient to regenerate accurately a local neighborhood of shape at regular axis points a sub 1 sup 2 second we examine the reconstruction of shape at branch points a sub 1 sup 3 where three regular branches are joined we show that the three pairs of geometry that is curvature and dynamics that is acceleration must satisfy certain constraints finally we derive similar results for the end points of shock branches a sub 3 points these formulas completely specify the local reconstruction of a shape from its shock graph or medial axis and the conditions required to form a coherent shape from the medial axis

an optimal algorithm for euclidean shortest paths in the plane we propose an optimal time algorithm for a classical problem in plane computational geometry computing a shortest path between two points in the presence of polygonal obstacles our algorithm runs in worst case time o n log n and requires o n log n space where n is the total number of vertices in the obstacle polygons the algorithm is based on an efficient implementation of wavefront propagation among polygonal obstacles and it actually computes a planar map encoding shortest paths from a fixed source point to all other points of the plane the map can be used to answer single source shortest path queries in o log n time the time complexity of our algorithm is a significant improvement over all previously published results on the shortest path problem finally we also discuss extensions to more general shortest path problems involving nonpoint and multiple sources

short note fast geodesics computation with the phase flow method this paper introduces a novel approach for rapidly computing a very large number of geodesics on a smooth surface the idea is to apply the recently developed phase flow method l ying e j candes the phase flow method j comput phys to appear an efficient and accurate technique for constructing phase maps for nonlinear ordinary differential equations on invariant manifolds which are here the unit tangent bundles of the surfaces under study we show how to rapidly construct the whole geodesic flow map which then allows computing any geodesic by straightforward local interpolation an operation with constant complexity a few numerical experiments complement our study and demonstrate the effectiveness of our approach

lower bounds for algebraic computation trees with integer inputs let w subseteq r n be scale invariant and rationally dispersed i e tilde x in w implies lambda tilde x in w for all real lambda 0 and the rationale are dense in both w and r n w it is shown that in the algebraic computation tree model the problem of deciding whether an input tilde x in r n with integer coordinates is a member of w has complexity omega log 2 hat beta w 2n where hat beta w is the number of connected components of w that are not of measure 0 this theorem can be used to prove tight lower bounds for the integral constrained form of many basic problems such as element distinctness set disjointness finding convex hull etc through further transformations it leads to lower bounds for problems such as integer max gap and closest pair of a simple polygon the proof involves a nontrivial extension of the milnor thorn techniques for finding upper bounds on the betti numbers of algebraic varieties

optimal search in planar subdivisions a planar subdivision is any partition of the plane into possibly unbounded polygonal regions the subdivision search problem is the following given a subdivision s with n line segments and a query point p determine which region of s contains p we present a practical algorithm for subdivision search that achieves the same optimal worst case complexity bounds as the significantly more complex algorithm of lipton and tarjan namely o log n search time with o n storage our subdivision search structure can be constructed in linear time from the subdivision representation used in many applications

star unfolding of a polytope with applications we introduce the notion of a star unfolding of the surface cal p of a three dimensional convex polytope with n vertices and use it to solve several problems related to shortest paths on cal p the first algorithm computes the edge sequences traversed by shortest paths on cal p in time o n 6 beta n log n where beta n is an extremely slowly growing function a much simpler o n 6 time algorithm that finds a small superset of all such edge sequences is also sketched the second algorithm is an o n 8 log n time procedure for computing the geodesic diameter of cal p the maximum possible separation of two points on cal p with the distance measured along cal p finally we describe an algorithm that preprocesses cal p into a data structure that can efficiently answer the queries of the following form given two points what is the length of the shortest path connecting them given a parameter 1 le m le n 2 it can preprocess cal p in time o n 6 m 1 delta for any delta 0 into a data structure of size o n 6m 1 delta so that a query can be answered in time o sqrt n m 1 4 log n if one query point always lies on an edge of cal p the algorithm can be improved to use o n 5 m 1 delta preprocessing time and storage and guarantee o n m 1 3 log n query time for any choice of m between 1 and n

from prima quadraginta octant to lattice sphere through primitive integer operations we present here the first integer based algorithm for constructing a well defined lattice sphere specified by integer radius and integer center the algorithm evolves from a unique correspondence between the lattice points comprising the sphere and the distribution of sum of three square numbers in integer intervals we characterize these intervals to derive a useful set of recurrences which in turn aids in efficient computation each point of the lattice sphere is determined by resorting to only a few primitive operations in the integer domain the symmetry of its quadraginta octants provides an added advantage by confining the computation to its prima quadraginta octant detailed theoretical analysis and experimental results have been furnished to demonstrate its simplicity and elegance

approximating shortest paths on a nonconvex polyhedron we present an approximation algorithm that given the boundary p of a simple nonconvex polyhedron in mathbb r 3 and two points s and t on p constructs a path on p between s and t whose length is at most 7 1 varepsilon dp s t where dp s t is the length of the shortest path between s and t on p and varepsilon 0 is an arbitrarily small positive constant the algorithm runs in o n5 3 log5 3 n time where n is the number of vertices in p we also present a slightly faster algorithm that runs in o n8 5 log8 5 n time and returns a path whose length is at most 15 1 varepsilon d p s t

on the polyhedral complexity of the integer points in a hyperball let b n be a hyperball in r n n 2 and denote b z n b email protected z n define polyhedral facet complexity of b z n as fc b z n min p f n 1 p where p is an enclosing polyhedron for b z n i e p email protected z n b z n and f n 1 p is the number of the n 1 facets of p analogously define polyhedral vertex complexity of b z n as vc b z n min p f 0 p where p is an enclosing polyhedron for b z n and f 0 p is the number of the 0 facets vertices of p upper bounds for fc b z n follow from a well known bound for the number of facets and vertices of the convex hull of b z n i barany d g larman the convex hull of the integer points in a large ball math ann 312 1998 167 181 in this note we provide the first nontrivial lower bounds on fc b z n and vc b z n

on different topological classes of spherical geodesic paths and circles in z 3 a discrete spherical geodesic path dsgp between two voxels s and t lying on a discrete sphere is a the shortest path from s to t comprising voxels of the discrete sphere intersected by the discrete geodesic plane passing through s t and the center of the sphere we consider two classes of discretization namely naive and standard for both the sphere and the geodesic plane which gives rise to four distinct topological classes of dsgp we show that the naive naive class does not guarantee the existence of a dsgp whereas the other three classes do we derive the upper bounds of the distance of a dsgp belonging to each class from the real sphere and the real plane for different neighborhood conditions we propose an efficient integer based algorithm to compute the dsgp for any class and neighborhood combination novel number theoretic characterization of discrete sphere has been used for searching the voxels comprising a dsgp the algorithm is output sensitive having its time and space complexities both linear in the length of the dsgp it can also be extended for constructing discrete 3d circles of arbitrary orientations specified by a few appropriate input parameters experimental results and related analysis demonstrate its efficiency and versatility

exact geodesics and shortest paths on polyhedral surfaces we present two algorithms for computing distances along convex and non convex polyhedral surfaces the first algorithm computes exact minimal geodesic distances and the second algorithm combines these distances to compute exact shortest path distances along the surface both algorithms have been extended to compute the exact minimal geodesic paths and shortest paths these algorithms have been implemented and validated on surfaces for which the correct solutions are known in order to verify the accuracy and to measure the run time performance which is cubic or less for each algorithm the exact distance computations carried out by these algorithms are feasible for large scale surfaces containing tens of thousands of vertices and are a necessary component of near isometric surface flattening methods that accurately transform curved manifolds into flat representations

efficient shape modeling entropy adaptive coding and boundary curves vs blum s medial axis we propose efficiency of representation as a criterion for evaluating shape models then apply this criterion to compare the boundary curve representation with the medial axis we estimate the entropy of two compact classes of curves we then construct two adaptive encodings for non compact classes of shapes one using the boundary curve and the other using the medial axis and determine precise conditions for when the medial axis is more efficient finally we apply our results to databases of naturally occurring shapes determining whether the boundary or medial axis is more efficient along the way we construct explicit near optimal boundary based approximations for compact classes of shapes construct an explicit compression scheme for non compact classes of shapes based on the medial axis and derive some new results about the medial axis

invariant feature extraction for 3d model retrieval an adaptive approach using euclidean and topological metrics with the fast development of 3d model construction and widespread popularity of 3d graphic engines more applications employ 3d geometric models to provide an interactive environment as the number of 3d models increases some 3d model retrieval systems have been proposed for indexing and matching these models an important issue in a retrieval system is feature extraction an efficient and invariant feature is a global shape distribution that collects some geometric properties of a model the d2 shape descriptor by osada et al is a one dimensional histogram of euclidean distances between two random points although the d2 is effective for some cases it changes when the model deforms we propose two shape descriptors in this paper gd which is the topological metric and asf which combines both euclidean and topological metrics the topological metric is an invariant deformation factor the two features are also robust against common geometric processing including scaling rotation resampling compression and remeshing in experiments we implement these methods and confirm their feasibility

computing generalized higher order voronoi diagrams on triangulated surfaces we present an algorithm for computing exact shortest paths and consequently distance functions from a generalized source point segment polygonal chain or polygonal region on a possibly non convex triangulated polyhedral surface the algorithm is generalized to the case when a set of generalized sites is considered providing their distance field that implicitly represents the voronoi diagram of the sites next we present an algorithm to compute a discrete representation of the distance function and the distance field then by using the discrete distance field we obtain the voronoi diagram of a set of generalized sites points segments polygonal chains or polygons and visualize it on the triangulated surface we also provide algorithms that by using the discrete distance functions provide the closest furthest and k order voronoi diagrams and an approximate 1 center and 1 median

multiple shooting approach for computing approximately shortest paths on convex polytopes abstract in this paper we use a multiple shooting approach in solving boundary value problems for ode to introduce a novel iterative algorithm for computing an approximate shortest path between two points on the surface of a convex polytope in 3d namely the polytope is partitioned into subpolytopes shooting points and a straightness condition are established the algorithm specifies how to combine shortest paths between shooting points in subpolytopes to become the required shortest path by the straightness condition in particular the algorithm does not rely on steiner points and graph tools on the entire polytope it is implemented in c and a comparison with agarwal har peled and karia s algorithm on the accurate construction of the shortest path is presented

time minimal paths amidst moving obstacles in three dimensions a path planning problem is considered in the presence of moving polygonal obstacles in three dimensions a particle is to be moved from a given initial position to a destination position amidst polygonal disjoint barriers moving along known linear trajectories the particle can move in any direction in space with a single constraint that it cannot move faster than a given speed bound all obstacles are slowly moving i e their speeds are strictly slower than the maximum speed of the particle the destination point is also permitted to move along a known trajectory and is assumed to be collision free at all times three properties are stated and proved for a time minimal path amidst moving polygonal barriers a few extensions are considered including piecewise linear motions of the obstacle

the convex hull of the integer points in a large ball where fk p denotes the number of k dimensional faces of the polytope p the limit as r of the average of r 2 3f0 pr on an interval r r h is determined by balog and deshoullier bd and turns out to be 3 453 h must be large our main result extends 1 1 to any d 2 and to any fk pr with k 0 d 1 theorem 1 for every d 2 there are constants c1 d and c2 d such that for all k 0 d 1

approximating shortest paths on a convex polytope in three dimensions given a convex polytope p with n faces in r3 points s t 6 p and a parameter 0 e 1 we present an algorithm that constructs a path on 6 p from s to t whose length is at most 1 e d p s t where d p s t is the length of the shortest path between s and t on 6 p the algorithm runs in o n log 1 e 1 e 3 time and is relatively simple the running time is o n 1 e 3 if we only want the approximate shortest pathdistance and not the path itself we also present an extension of thealgorithm that computes approximate shortest path distances from a givensource point on 6 p to all vertices of p

analysis of planar shapes using geodesic paths on shape spaces for analyzing shapes of planar closed curves we propose differential geometric representations of curves using their direction functions and curvature functions shapes are represented as elements of infinite dimensional spaces and their pairwise differences are quantified using the lengths of geodesics connecting them on these spaces we use a fourier basis to represent tangents to the shape spaces and then use a gradient based shooting method to solve for the tangent that connects any two shapes via a geodesic using the surrey fish database we demonstrate some applications of this approach 1 interpolation and extrapolations of shape changes 2 clustering of objects according to their shapes 3 statistics on shape spaces and 4 bayesian extraction of shapes in low quality images

n 4 irreducible scalar superfields the n 4 extended supersymmetry in four dimensional space time is considered from the viewpoint of the two included n 2 extended supersymmetries using the results for the structure of the irreducible superfields in n 2 superspace obtained earlier the task of explicit construction of n 4 irreducible superfields with su 2 su 2 internal symmetry in n 2 2 superspace is solved the decomposition of the n 4 scalar superfield into n 4 irreducible ones is described and the rules for collecting the irreducible scalar n 2 2 superfields with su 2 su 2 internal symmetry into n 4 irreducible scalar superfields with su 4 internal symmetry are set to illustrate this method a simple consistent way of constructing the linearized off shell n 4 conformal supergravity in n 4 superspace is shown

a new approach to planar point location given a planar straight line graph g with n vertices and a point p 0 locating p 0 means to find the region of the planar subdivision induced by g which contains p 0 recently lipton and tarjan presented a brilliant but extremely complex point location algorithm which runs in time o log n on a data structure using o n storage this paper presents a practical algorithm which runs in less than 6 lceil log 2 n rceil comparisons on a data structure which uses o n log n storage in the worst case the method rests crucially on a simple partition of each edge of g into o log n segments

construction of iso contours bisectors and voronoi diagrams on triangulated surfaces in the research of computer vision and machine perception 3d objects are usually represented by 2 manifold triangular meshes m in this paper we present practical and efficient algorithms to construct iso contours bisectors and voronoi diagrams of point sites on m based on an exact geodesic metric compared to euclidean metric spaces the voronoi diagrams on m exhibit many special properties that fail all of the existing euclidean voronoi algorithms to provide practical algorithms for constructing geodesic metric based voronoi diagrams on m this paper studies the analytic structure of iso contours bisectors and voronoi diagrams on m after a necessary preprocessing of model m practical algorithms are proposed for quickly obtaining full information about iso contours bisectors and voronoi diagrams on m the complexity of the construction algorithms is also analyzed finally three interesting applications surface sampling and reconstruction 3d skeleton extraction and point pattern analysis are presented that show the potential power of the proposed algorithms in pattern analysis

the discrete geodesic problem we present an algorithm for determining the shortest path between a source and a destination on an arbitrary possibly nonconvex polyhedral surface the path is constrained to lie on the surface and distances are measured according to the euclidean metric our algorithm runs in time o n log n and requires o n2 space where n is the number ofedges ofthe surface afterwe run our algorithm the distance from the source to any other destination may be determined using standard techniques in time o log n by locating the destination in the subdivision created by the algorithm the actual shortest path from the source to a destination can be reported in time o k log n where k is the number of faces crossed by the path the algorithm generalizes to the case of multiple source points to build the voronoi diagram on the surface where n is now the maximum of the number of vertices and the number of sources

palindromic language of thin discrete planes we work on the rveills hyperplane p v 0 with normal vectorvrd shift 0 and thicknessr such a hyperplane is connected as soon as is greater than some value v 0 called the connecting thickness of v with null shift in the case where v satisfies the so called kraaikamp and meester criterion at the connecting thickness the hyperplane has very specific properties first of all the adjacency graph of the voxels forms a tree this tree appeared in many works both in discrete geometry and in discrete dynamical systems in addition it is well known that for a finite coding of length n of discrete lines the number of palindromes in the language is exactly n 1 we extend this notion of language to labeled trees and we compute the number of distinct palindromes in fact for our voxel adjacency trees with n letters we show that the number of palindromes in the language is also n 1 this result establishes a first link between combinatorics on words palindromic languages voxel adjacency trees and connecting thickness of rveills hyperplanes it also provides a better understanding of the combinatorial structure of discrete planes