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

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

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

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

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

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

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

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

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

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

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

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

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

a mechanical counterexample to kam theory with low regularity abstract we give a mechanical example concerning the fact that some regularity is necessary in kam theory we consider the model given by the vertical bouncing motion of a ball on a periodically moving plate denoting with f the motion of the plate some variants of moser invariant curve theorem apply if f is small in norm c 5 and every motion has bounded velocity this is not possible if the function f is only c 1 indeed we construct a function f c 1 with arbitrary small derivative in norm c 0 for which a motion with unbounded velocity exists

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

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

a simple unified approach to some convergence theorems of l simon abstract we give a new and easier proof for the convergences result of l simon 15 concerning global and bounded solutions of a heat equation with analytic nonlinearity by constructing a new liapunov functional for the elliptic equation we are able to prove the second convergence theorem of 15 in the same way as the first theorem the results are more general and can be applied to other problems such as fourth order equation and vector valued systems of gradient type

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