outlier detection with the kernelized spatial depth function statistical depth functions provide from the deepest point a center outward ordering of multidimensional data in this sense depth functions can measure the extremeness or outlyingness of a data point with respect to a given data set hence they can detect outliers observations that appear extreme relative to the rest of the observations of the various statistical depths the spatial depth is especially appealing because of its computational efficiency and mathematical tractability in this article we propose a novel statistical depth the kernelized spatial depth ksd which generalizes the spatial depth via positive definite kernels by choosing a proper kernel the ksd can capture the local structure of a data set while the spatial depth fails we demonstrate this by the half moon data and the ring shaped data based on the ksd we propose a novel outlier detection algorithm by which an observation with a depth value less than a threshold is declared as an outlier the proposed algorithm is simple in structure the threshold is the only one parameter for a given kernel it applies to a one class learning setting in which normal observations are given as the training data as well as to a missing label scenario where the training set consists of a mixture of normal observations and outliers with unknown labels we give upper bounds on the false alarm probability of a depth based detector these upper bounds can be used to determine the threshold we perform extensive experiments on synthetic data and data sets from real applications the proposed outlier detector is compared with existing methods the ksd outlier detector demonstrates a competitive performance

the approximation algorithms for a class of multiple choice problem abstract given a set of n points which is a unit set of m colored sets we study the minimum circle to cover one of each colored set at least in this paper we study some problems of optimizing some properties of color spanning set firstly we show a way to find a color spanning set for each point and constitute a union of color spanning sets without fcvd the farthest color voronoi diagram the approach to find each color spanning set is based on the nearest neighbor points which have different colors for every color spanning set we can find a enclosing circle to cover all points of each color spanning set which is a good approximate for mdcs the minimum diameter color spanning set problem we propose an approximation algorithm for the secsc the smallest color spanning circle problem with 2 factor approximation result for p n and q m the worst running time of our algorithm is o n 2 although these results are not as good as previous results with fcvd the performance of our algorithms in r d is much better than others moreover an approximation algorithm can solve problem of minimum perimeter of the color spanning set faster with time of o n 2 n m log n and ratio 6 this result improved the ratio with only a little cost of time complexity at last we give an example for the 2 center secsc problem a fast computation of 2 center enclosing circle is proposed with time of o n 2 but the ratio depends on the gap between the nearest distinct colored distance

optimality approximation and complexity in set membership h sub spl infin identification investigates the set membership identification of time invariant discrete time exponentially stable possibly infinite dimensional linear systems from time or frequency domain data corrupted by deterministic noise the aim is to deliver not a single model but a set of models whose size in h sub spl infin norm measures the uncertainty in the identification the main focus of the note is on the optimality properties for finite data and on the tradeoff between optimality and complexity of approximated low order model sets a method is given for evaluating convergent and computationally efficient inner and outer approximations of the value set for a given frequency such approximations allow one to compute within any desired accuracy the identification error of any identified model and to evaluate an optimal model at any given number of frequencies by suitably approximating these values model sets with nominal models in rh sub spl infin are then derived whose order is selected by trading off between model set complexity and identification accuracy degradation this degradation is evaluated by computing the optimality level defined as the ratio between the reduced model identification error and the optimal one a numerical example demonstrates the effectiveness of the presented results

simultaneous uniqueness of infinite clusters in stationary random labeled graphs in processes such as invasion percolation and certain models of continuum percolation in which a possibly random labelf b is attached to each bondb of a possibly random graph percolation models for various values of a parameterr are naturally coupled one can define a bondb to be occupied at levelr iff b r if the labeled graph is stationary then under the mild additional assumption of positive finite energy a result of gandolfi keane and newman ensures that in lattice models for each fixedr at which percolation occurs the infinite cluster is unique a s analogous results exist for certain continuum models a unifying framework is given for such fixed r results and it is shown that if the site density is finite and the labeled graph has positive finite energy then with probability one uniqueness holds simultaneously for all values ofr an example is given to show that when the site density is infinite positive finite energy does not ensure uniqueness even for fixedr in addition with finite site density but without positive finite energy one can have fixed r uniqueness a s for eachr yet not have simultaneous uniqueness

partitions of a simplex leading to accurate spectral finite volume reconstruction in this paper we present several systematic techniques based on the voronoi diagram and its variants to partition a one and two dimensional simplex the fekete points are used as input to generate the voronoi diagram as they concentrate near the edges and are almost optimal for polynomial interpolation in a simplex spectral finite volume reconstructions on the resultant partitions have small lebesgue constants when using the dubiner basis the reconstruction matrix is well conditioned moreover the total number of edges of the partitions the total work when being used in spectral volume methods is shown to be at most twice the minimum number of edges of all partitions for reconstructions of the same order accuracy these suggest that the obtained partitions are well suited for spectral volume methods and other numerical methods which rely on reconstructions from cell averages

on point covers of c oriented polygons let s be any family of n c oriented polygons of the two dimensional euclidean plane e2 i e bounded intersection of halfplanes whose normal directions of edges belong to a fixed collection of c distinct directions let s denote the packing number of s that is the maximum number of pairwise disjoint objects of s let s be the transversal number of s that is the minimum number of points required so that each object contains at least one of those points we provide linear time algorithms t n c t nc for a fat c oriented polytopes translates or homothets of ed proving that g 2 c o a d g 2 c dd and g 2 c 3d3 2 d respectively

space time tradeoffs for emptiness queries we develop the first nontrivial lower bounds on the complexity of online hyperplane and halfspace emptiness queries our lower bounds apply to a general class of geometric range query data structures called partition graphs informally a partition graph is a directed acyclic graph that describes a recursive decomposition of space we show that any partition graph that supports hyperplane emptiness queries implicitly defines a halfspace range query data structure in the fredman yao semigroup arithmetic model with the same asymptotic space and time bounds thus results of bronnimann chazelle and pach imply that any partition graph of size s that supports hyperplane emptiness queries in time t satisfies the inequality st d omega n log n d d 1 d 1 using different techniques we improve previous lower bounds for hopcroft s problem given a set of points and hyperplanes does any hyperplane contain a point in dimensions four and higher using this offline result we show that for online hyperplane emptiness queries omega n d mbox polylog n space is required to achieve polylogarithmic query time and omega n d 1 d mbox polylog n query time is required if only o n polylog n space is available these two lower bounds are optimal up to polylogarithmic factors for two dimensional queries we obtain an optimal continuous tradeoff st 2 omega n 2 between these two extremes finally using a lifting argument we show that the same lower bounds hold for both offline and online halfspace emptiness queries in mathbb r d d 3 2

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

integral grothendieck riemann roch theorem we show that in characteristic zero the obvious integral version of the grothendieck riemann roch formula obtained by clearing the denominators of the todd and chern characters is true without having to divide the chow groups by their torsion subgroups the proof introduces an alternative to grothendieck s strategy we use resolution of singularities and the weak factorization theorem for birational maps

reweighting to achieve elliptically contoured covariates in regression abstract we investigate a method of constructing weights to induce elliptically contoured covariates in regression analyses much recent work in regression has identified various data analytic and model robustness advantages associated with such covariates in particular new estimation methods like sir sirii save and phd have been built around the assumption of elliptically contoured covariates finite samples of regression covariates may deviate from this ideal in practice and the method developed here termed voronoi weighting can be used to induce elliptical symmetry in such samples in a number of examples we show that reweighting cases by the voronoi method can substantially enhance various procedures for covariates that deviate from elliptical symmetry we show that voronoi weighting in conjunction with some trimming via the minimum volume ellipsoid method can be effective

approximations of one dimensional digital signals under the l infty norm approximation of digital signals by means of continuous time functions is often required in many tasks of digital to analog conversion signal processing and coding in many cases the approximation is performed based on an l2 optimality criterion in this paper we study approximations of one dimensional signals under the linfin norm we first introduce approximations in linear spaces for which linear programming methods are known for the particular case of linear approximations i e first order polynomials we propose a geometric solution that is shown to be computationally more efficient than the linear programming approach then we study the problem of piecewise approximations i e dividing the domain into intervals and approximating the signal in linear spaces within every segment independently so as to reach an optimal noncontinuous approximation given an error bound delta we establish a strategy to determine the minimum number k of segments for which the approximation is guaranteed to produce an error within delta we then show how to find the optimal partition that gives the piecewise linfin optimal solution with k segments the computational complexity of the algorithms is studied showing that in many practical situations the number of operations is o n with n being the number of samples

computing connectedness disconnectedness and discreteness we consider finite point set approximations of a manifold or fractal with the goal of determining topological properties of the underlying set we use the minimal spanning tree of the finite set of points to compute the number and size of its connected components by extrapolating the limiting behavior of these quantities as 0 we can say whether the underlying set appears to be connected totally disconnected or perfect we demonstrate the effectiveness of our techniques for a number of examples including a family of fractals related to the sierpinski triangle cantor subsets of the plane the henon attractor and cantori from four dimensional symplectic sawtooth maps for zero measure cantor sets we conjecture that the growth rate of the number of components as 0 is equivalent to the box counting dimension 2000 elsevier science b v all rights reserved

discrete voronoi diagrams and the skiz operator a dynamic algorithm the voronoi diagram vd is a popular tool for partitioning the support of an image an algorithm is presented for constructing vd when the seed set which determines the voronoi regions can be modified by adding and removing seeds the number of pixels and seeds revisited for updating the diagram and the neighbor relationships among seeds is minimized a result on cocircular seeds is presented the adjacency or dual graph of the vd is readily obtained the use of the algorithm for constructing skeletons by influence zones skiz is demonstrated

polygon based contact resolution for superquadrics the representation of discrete objects in the discrete element modelling is a fundamental issue which has a direct impact on the efficiency of discrete element implementation and the dynamic behaviour of particulate systems disks and spheres are the most commonly used geometric shapes due to their geometric simplicity and computational efficiency but they are unable to provide resistance to rolling motion for this reason some non circular spherical objects such as polygons polyhedrons superquadrics or the clustering of disks spheres to form irregular shapes are introduced when superquadrics are used as discrete elements the bottleneck of contact resolution is associated with the searching for intersections of two non linear functions which is a very expensive operation and may sometimes fail in finding the solution in this work an efficient and robust algorithm is proposed for contact resolution of 2d superquadrics in which any superquadric is approximated with a convex polygon through adaptive sampling then by clipping two polygons an efficient linear algorithm is performed to search for intersections and overlap area of the polygons the contact forces and directions are determined by employing a newly established corner corner contact model it is important to highlight that the proposed methodology can also be extended to general non circular discrete object cases the performance of the algorithm is demonstrated via numerical examples copyright 2005 john wiley sons ltd

generators extremals and bases of max cones abstract max cones are max algebraic analogs of convex cones in the present paper we develop a theory of generating sets and extremals of max cones in r n this theory is based on the observation that extremals are minimal elements of max cones under suitable scalings of vectors we give new proofs of existing results suitably generalizing restating and refining them of these it is important that any set of generators may be partitioned into the set of extremals and the set of redundant elements we include results on properties of open and closed cones on properties of totally dependent sets and on computational bounds for the problem of finding the essentially unique basis of a finitely generated cone

optimal point location in a monotone subdivision point location often known in graphics as hit detection is one of the fundamental problems of computational geometry in a point location query we want to identify which of a given collection of geometric objects contains a particular point let mathcal s denote a subdivision of the euclidean plane into monotone regions by a straight line graph of m edges in this paper we exhibit a substantial refinement of the technique of lee and preparata siam j comput 6 1977 pp 594 606 for locating a point in mathcal s based on separating chains the new data structure called a layered dag can be built in o m time uses o m storage and makes possible point location in o log m time unlike previous structures that attain these optimal bounds the layered dag can be implemented in a simple and practical way and is extensible to subdivisions with edges more general than straight line segments

linearization and completeness results for terminating transitive closure queries on spatial databases we study queries to spatial databases where spatial data are modeled as semi algebraic sets using the relational calculus with polynomial inequalities as a basic query language we work with the extension of the relational calculus with terminating transitive closures the main result is that this language can express the linearization of semialgebraic databases we also show that the sublanguage with linear inequalities only can express all computable queries on semilinear databases as a consequence of these results we obtain a completeness result for topological queries on semialgebraic databases

an optimal algorithm for intersecting line segments in the plane the main contribution of this work is an o n log n k time algorithm for computing all k intersections among n line segments in the plane this time complexity is easily shown to be optimal within the same asymptotic cost our algorithm can also construct the subdivision of the plane defined by the segments and compute which segment if any lies right above or below each intersection and each endpoint the algorithm has been implemented and performs very well the storage requirement is on the order of n k in the worst case but it is considerably lower in practice to analyze the complexity of the algorithm an amortization argument based on a new combinatorial theorem on line arrangements is used

tomographic image reconstruction based on a content adaptive mesh model in this paper we explore the use of a content adaptive mesh model camm for tomographic image reconstruction in the proposed framework the image to be reconstructed is first represented by a mesh model an efficient image description based on nonuniform sampling in the camm image samples represented as mesh nodes are placed most densely in image regions having fine detail tomographic image reconstruction in the mesh domain is performed by maximum likelihood ml or maximum a posteriori map estimation of the nodal values from the measured data a camm greatly reduces the number of unknown parameters to be determined leading to improved image quality and reduced computation time we demonstrated the method in our experiments using simulated gated single photon emission computed tomography spect cardiac perfusion images a channelized hotelling observer cho was used to evaluate the detectability of perfusion defects in the reconstructed images a task based measure of image quality a minimum description length mdl criterion was also used to evaluate the effect of the representation size in our application both mdl and cho suggested that the optimal number of mesh nodes is roughly five to seven times smaller than the number of projection bins when compared to several commonly used methods for image reconstruction the proposed approach achieved the best performance in terms of defect detection and computation time the research described in this paper establishes a foundation for future development of a four dimensional space time reconstruction framework for image sequences in which a built in deformable mesh model is used to track the image motion

unramified cohomology of degree 3 and noether s problem let g be a finite group and w be a faithful representation of g over c the group g acts on the field of rational functions c w the question whether the field of invariant functions c w g is purely transcendental over c goes back to emmy noether using the unramified cohomology group of degree 2 of this field as an invariant saltman gave the first examples for which c w g is not rational over c around 1986 bogomolov gave a formula which expresses this cohomology group in terms of the cohomology of the group g

riemann roch for tensor powers mapping a locally free module on a scheme to its l th tensor power gives rise to a natural map from the grothendieck group of all locally free modules to the grothendieck group of all locally free representations of the l th symmetric group in this paper we prove some formulas of riemann roch type for the behavior of this tensor power operation with respect to the push forward homomorphism associated with a projective morphism between schemes we furthermore establish analogous formulas for higher k groups

a unified algorithm for finding maximum and minimum object enclosing rectangles and cuboids abstract given a set of n points in r 2 bounded within a rectangular floor f and a rectangular plate p of specified size we consider the following two problems find an isothetic position of p such that it encloses i maximum and ii minimum number of points keeping p totally contained within f for both of these problems a new algorithm based on interval tree data structure is presented which runs in o nlogn time and consumes o n space if polygonal objects of arbitrary size and shape are distributed in r 2 the proposed algorithm can be tailored for locating the position of the plate to enclose maximum or minimum number of objects with the same time and space complexity finally the algorithm is extended for identifying a cuboid i e a rectangular parallelepiped that encloses maximum number of polyhedral objects in r 3 thus the proposed technique serves as a unified paradigm for solving a general class of enclosure problems encountered in computational geometry and pattern recognition

constructing conformal maps of triangulated surfaces abstract we describe a means of computing the uniformizing conformal map from a triangulated surface whose triangles are realized as euclidean triangles in r 3 onto a fundamental domain in the unit disc d plane c or sphere s 2 mapping such triangulated surfaces arises in a number of applications such as conformal brain flattening we use the circle packing technique of bowers hurdal stephenson et al to first create a quasiconformal approximation to the conformal map then we apply a discrete form of conformal welding to reduce the distortion and converge to the conformal map in the limit

classes de stiefel whitney de formes quadratiques et de representations galoisiennes reelles i u n peu de t o p o l o g i e a l g6b r ique 226 1 1 le fonc t eu r 226 1 2 le t rans fe r t mu l t i p l i c a t i f 228 1 3 t rans f e r t et c lasses de s t ie fe l whi tney 233 1 4 le cas d u n g r o u p e d iscre t t r a d u c t i o n s 238

an optimal parallel algorithm for the visibility of a simple polygon from a point we present a parallel algorithm for computing the visible portion of a simple polygonal chain with 7i vertices from a point in the plane the algorithm runs in o logn time using 0 nf log n processors in the crew pram computational model and hence is asymptotically optimal

time discretized variational formulation of non smooth frictional contact the present work extends the non smooth contact class of algorithms introduced by kane et al to the case of friction the formulation specifically addresses contact geometries e g involving multiple collisions between tightly packed non smooth bodies for which neither normals nor gap functions can be properly defined a key aspect of the approach is that the incremental displacements follow from a minimum principle the objective function comprises terms which account for inertia strain energy contact friction and external forcing the euler lagrange equations corresponding to this extended variational principle are shown to be consistent with the equations of motion of solids in frictional contact in addition to its value as a basis for formulating numerical algorithms the variational framework offers theoretical advantages as regards the selection of trajectories in cases of non uniqueness we present numerical and analytical examples which demonstrate the good momentum and energy conservation characteristics of the numerical algorithms as well as the ability of the approach to account for stick and slip conditions

esflib a library to compute the element free galerkin shape functions we present a library for the computation of the shape functions for the element free galerkin efg method while the efg method is in many respects strikingly similar to the nite element method the construction of the shape functions for the efg method is much more complicated than in the fem thus the question arises whether it is possible to encapsulate the complexity of the shape function construction process while providing a simple interface to the shape function data we present a shape function library design which proved unrestrictive it can be used with almost any programming language without any special requirements on application program data structures highly congurable and robust

intersecting and trimming parametric meshes on finite element shells luiz cristovao g coelho marcelo gattass department of computer science puc rio rio de janeiro brazilluiz henrique de figueiredolncc laborato rio nacional de computac a o cient i ca p etro polis brazilabstractwe present an algorithm for intersecting nite element meshes de ned on parametric surface patches the inter section curves are modeled precisely and both meshes are adjusted to the newly formed borders without unwantedreparametrizations the algorithm is part of an interactive shell modeling program that has been used in the designof large offshore oil structures to achieve good interactive response we represent meshes with a topological datastructure that stores its entities in spatial indexing trees instead of linear lists these trees speed up the intersectioncomputations required to determine points of the trimming curves moreover when combined with the topologicalinformation they allow remeshing using only local queries

stiefel whitney classes of real representations of finite groups abstract we prove the existence and uniqueness of stiefel whitney classes of real representations subject to the hirzebruch axioms within the category of finite groups we also give a method of constructing and computing with these classes using multiplicative transfer and induced representations and apply our formulae in certain simple cases

on line signature verification based on logarithmic spectrum in this paper an on line signature verification scheme based on similarity measurement of logarithmic spectrum is proposed the principal components of the logarithmic spectrum of each signature are extracted we then compute the similarity of logarithmic spectrum between input signature and the reference template by comparing the similarity of logarithmic spectrum with the verification threshold we can determine the authenticity of the input signature based on the experimentation the rates of false rejection errors and false acceptance errors are as low as 1 4 and 2 8 respectively this demonstrates the effectiveness of the proposed scheme

quality estimates and stretched meshes based on delaunay triangulations any other vertex is exploited for the automation of the grid generation process moreover delaunay grids have many interesting properties an extensive documentation of which can be found in ref 10 in the following we will first describe the different elements of frod the derivation of the background mesh as well as the vertex generation for the viscous and inviscid regions the latter half of this paper presents the quality measures that can be derived for the isotropic vertex generation part

acuity matching resolution degradation through wavelet coefficient scaling a wavelet based multiresolution image representation method is developed matching human visual system hvs spatial acuity within multiple regions of interest rois rois are maintained at high original resolution while peripheral areas are gracefully degraded variable resolution images are generated by selectively scaling wavelet detail coefficients prior to reconstruction the technique is equivalent to linear interpolation mip mapping which involves smooth subsampling decomposition prior to texture mapping reconstruction multiple roi degradation is achieved through wavelet coefficient scaling following voronoi partitioning of the image plane

the hamiltonian properties of supergrid graphs in this paper we first introduce a novel class of graphs namely supergrid supergrid graphs include grid graphs and triangular grid graphs as their subgraphs the hamiltonian cycle and path problems for grid graphs and triangular grid graphs were known to be np complete however they are unknown for supergrid graphs the hamiltonian cycle path problem on supergrid graphs can be applied to control the stitching traces of computerized sewing machines in this paper we will prove that the hamiltonian cycle problem for supergrid graphs is np complete it is easily derived from the hamiltonian cycle result that the hamiltonian path problem on supergrid graphs is also np complete we then show that two subclasses of supergrid graphs including rectangular parallelism and alphabet always contain hamiltonian cycles

computing farthest neighbors on a convex polytope let n be a set of n points in convex position in r3 the farthest point voronoi diagram of n partitions r3 into n convex cells we consider the intersection g n of the diagram with the boundary of the convex hull of n we give an algorithm that computes an implicit representation of g n in expected o n log2 n time more precisely we compute the combinatorial structure of g n the coordinates of its vertices and the equation of the plane defining each edge of g n the algorithm allows us to solve the all pairs farthest neighbor problem for n in expected time o n log2 n and to perform farthest neighbor queries on n in o log2 n time with high probability

analysis of a part design procedure in this paper we analyze and illustrate a new ab initio part design procedure in which given a cost function which reflects performance materials and manufacturing considerations the topology and the geometry of the part are automatically produced the analysis is based on demonstration of first the compactness of the metric space over which the cost function is defined and second lower semi continuity of the cost function examples include beams and elastic supports

icp registration using invariant features investigates the use of euclidean invariant features in a generalization of iterative closest point icp registration of range images pointwise correspondences are chosen as the closest point with respect to a weighted linear combination of positional and feature distances it is shown that under ideal noise free conditions correspondences formed using this distance function are correct more often than correspondences formed using the positional distance alone in addition monotonic convergence to at least a local minimum is shown to hold for this method when noise is present a method that automatically sets the optimal relative contribution of features and positions is described this method trades off the error in feature values due to noise against the error in positions due to misalignment experimental results suggest that using invariant features decreases the probability of being trapped in a local minimum and may be an effective solution for difficult range image registration problems where the scene is very small compared to the model

structure of a simple scheduling polyhedron in a one machine nonpreemptive scheduling problem the feasible schedules may be defined by the vector of the corresponding job completion times for given positive processing times the associated simple scheduling polyhedronp is the convex hull of these feasible completion time vectors the main result of this paper is a complete description of the minimal linear system definingp we also give a complete combinatorial description of the face lattice ofp and a simple o n logn separation algorithm this algorithm has potential usefulness in cutting plane type algorithms for more difficult scheduling problems

optimal separable algorithms to compute the reverse euclidean distance transformation and discrete medial axis in arbitrary dimension in binary images the distance transformation dt and the geometrical skeleton extraction are classic tools for shape analysis in this paper we present time optimal algorithms to solve the reverse euclidean distance transformation and the reversible medial axis extraction problems for d dimensional images we also present a d dimensional medial axis filtering process that allows us to control the quality of the reconstructed shape

voronoi diagram and medial axis algorithm for planar domains with curved boundaries i theoretical foundations in this first installment of a two part paper the underlying theory for an algorithm that computes the voronoi diagram and medial axis of a planar domain bounded by free form polynomial or rational curve segments is presented an incremental approach to computing the voronoi diagram is used wherein a single boundary segment is added to an existing boundary segment set at each step the introduction of each new segment entails modifying the voronoi regions of the existing boundary segments and constructing the voronoi region of the new segment we accomplish this by i computing the bisector of the new segment with each of the current boundary segments ii updating the voronoi regions of the current boundary segments by partitioning them with these bisectors and iii constructing the voronoi region of the new segment as a union of regions obtained from the partitioning in ii when all boundary segments are included and their voronoi regions have been constructed the voronoi diagram of the boundary is obtained as the union of the voronoi polygons for each boundary segment to construct the medial axis of a planar domain we first compute the voronoi diagram of its boundary the medial axis is then obtained from the voronoi diagram by i removing certain edges of the voronoi diagram that do not belong to the medial axis and ii adding certain edges that do belong to the medial axis but are absent from the voronoi diagram unambiguous characterizations for edges in both these categories are given details of algorithms based on this theory are deferred to the second installment of this two part paper

the c convergence of hexagonal disk packings to the riemann map let omega mathop subset limits ne c be a simply connected domain the rodin sullivan theorem states that a sequence of disk packings in the unit disk u converges in a well defined sense to a conformal map from to u moreover it is known that the first and second derivatives converge as well here it is proven that for hexagonal disk packings the convergence is c this is done by studying mobius invariants of the disk packings that are discrete analogs of the schwarzian derivative

surface bounded growth modeling applied to human mandibles from a set of longitudinal three dimensional scans of the same anatomical structure the authors have accurately modeled the temporal shape and size changes using a linear shape model on a total of 31 computed tomography scans of the mandible from six patients 14 851 semilandmarks are found automatically using shape features and a new algorithm called geometry constrained diffusion the semilandmarks are mapped into procrustes space principal component analysis extracts a one dimensional subspace which is used to construct a linear growth model the worst case mean modeling error in a cross validation study is 3 7 mm

computational geometry an introduction from the reviews this book offers a coherent treatment at the graduate textbook level of the field that has come to be known in the last decade or so as computational geometry the book is well organized and lucidly written a timely contribution by two founders of the field it clearly demonstrates that computational geometry in the plane is now a fairly well understood branch of computer science and mathematics it also points the way to the solution of the more challenging problems in dimensions higher than two

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

geometric classification of triangulations and their enumeration in a convex polygon abstract triangulation of polygons is a classical problem in computational geometry for an arbitrary polygon the triangulation may depend on its shape in this paper we describe a new geometric classification of triangulations of a convex polygon and then derive expressions for counting the number of nonisomorphic triangulations in each class

the multipoint morisita index for the analysis of spatial patterns abstract in many fields the spatial clustering of sampled data points has significant consequences therefore several indices have been proposed to assess the degree of clustering affecting datasets e g the morisita index ripley s k function and renyi s information the classical morisita index measures how many times it is more likely to randomly select two sampled points from the same quadrat the dataset is covered by a regular grid of changing size than it would be in the case of a random distribution generated from a poisson process the multipoint version takes into account m points with m 2 the present research deals with a new development of the multipoint morisita index m morisita which is directly related to multifractality this relationship to multifractality is first demonstrated and highlighted on a mathematical multifractal set then the new version of the m morisita index is adapted to the characterization of environmental monitoring network clustering and finally an additional extension the functional m morisita index is presented for the detection of structures in monitored phenomena

on the convergence of circle packings to the riemann map rodin and sullivan 1987 proved thurston s conjecture that a scheme based on the circle packing theorem converges to the riemann mapping thereby providing a refreshing geometric view of riemann s mapping theorem we now present a new proof of the rodin sullivan theorem this proof is based on the argument principle and has the following virtues 1 it applies to more general packings the rodin sullivan paper deals with packings based on the hexagonal combinatorics later quantitative estimates were found which also worked for bounded valence packings here the bounded valence assumption is unnecessary and irrelevant 2 our method is rather elementary and accessible to non experts in particular quasiconformal maps are not needed consequently this gives an independent proof of riemann s conformal mapping theorem the rodin sullivan proof uses results that rely on riemann s mapping theorem 3 our approach gives the convergence of the first and second derivatives without significant additional difficulties while previous work has established the convergence of the first two derivatives for bounded valence packings now the bounded valence assumption is unnecessary

reduction of bivariate polynomials from convex dense to dense with application to factorizations in this article we present a new algorithm for reducing the usual sparse bivariate factorization problems to the dense case this reduction simply consists in computing an invertible monomial transformation that produces a polynomial with a dense size of the same order of magnitude as the size of the integral convex hull of the support of the input polynomial this approach turns out to be very efficient in practice as demonstrated with our implementation

on pointers versus addresses summary what is the cost of random access to memory we address this fnndamental problem by studying the simulation of random addressing by a machine which lacks it a pointer machine the model we define allows the use of a data type of our choice a ram program of time i and space s can be simulated in o t log a time using a tree however this is not an obvious lower bound since a high level data type may allow us to encode the data in a more economic way our major contribution is the formalization of incompressibility for general data types the definition extends a similar property 01 strings that underlies the theory of kolmogorov complexity the main theorem states that for all incompressible data types an q t log s lower bound holds incompressibility trivially holds for strings but is harder to prove for a powerlul data type we prove incompressibility for the real numbers with a set of primitives which includes all functions which are continuously differentiable except on a countable closed set this may be the richest set of operations considered in a lower bound proof the prooi relies on the implicit functions theorem and baire s category theorem we also show that the integers with arithmetic x and lz zj any boolean operations and left shift are incompressible the inclusion of right shift reverses the situation and we obtain an o tq s upper bound where s is a functional inverse of ackermann s function

optimal expected case planar point location point location is the problem of preprocessing a planar polygonal subdivision s of size n into a data structure in order to determine efficiently the cell of the subdivision that contains a given query point we consider this problem from the perspective of expected query time we are given the probabilities p z that the query point lies within each cell z in s the entropy h of the resulting discrete probability distribution is the dominant term in the lower bound on the expected case query time we show that it is possible to achieve query time h o sqrt h 1 with space o n which is optimal up to lower order terms in the query time we extend this result to subdivisions with convex cells assuming a uniform query distribution within each cell in order to achieve space efficiency we introduce the concept of entropy preserving cuttings

close point spatial tests and their application to random number generators we study statistical tests of uniformity based on thel p distances between them nearest pairs of points forn points generated uniformly over thek dimensional unit hypercube or unit torus the number of distinct pairs at distance no more thant fort 0 is a stochastic process whose initial part after an appropriate transformation and asn 8 is asymptotically a poisson process with unit rate convergence to this asymptotic is slow in the hypercube as soon ask exceeds 2 or 3 due to edge effects but is reasonably fast in the torus we look at the quality of approximation of the exact distributions of the tests statistics by their asymptotic distributions discuss computational issues and apply the tests to random number generators linear congruential generators fail decisively certain variants of the tests as soon asn approaches the square root of the period length

a simple algorithm for nearest neighbor search in high dimensions the problem of finding the closest point in high dimensional spaces is common in pattern recognition unfortunately the complexity of most existing search algorithms such as k d tree and r tree grows exponentially with dimension making them impractical for dimensionality above 15 in nearly all applications the closest point is of interest only if it lies within a user specified distance spl epsiv we present a simple and practical algorithm to efficiently search for the nearest neighbor within euclidean distance spl epsiv the use of projection search combined with a novel data structure dramatically improves performance in high dimensions a complexity analysis is presented which helps to automatically determine spl epsiv in structured problems a comprehensive set of benchmarks clearly shows the superiority of the proposed algorithm for a variety of structured and unstructured search problems object recognition is demonstrated as an example application the simplicity of the algorithm makes it possible to construct an inexpensive hardware search engine which can be 100 times faster than its software equivalent a c implementation of our algorithm is available

partial match queries in random k d trees we solve the open problem of characterizing the leading constant in the asymptotic approximation to the expected cost used for random partial match queries in random k d trees our approach is new and of some generality in particular it is applicable to many problems involving differential equations or difference equations with polynomial coefficients

a fuzzy approach to visibility maps creation over digital terrains the construction of inter visibility models for a terrain is an important tool for environmental and animal studies efficient crisp algorithms are known to address this problem directly on a raster digital elevation map dem of the study area randomness plays an important role several factors like vegetation height of the observer presence of unmapped human artifacts etc cannot be modeled in advance into a traditional dem model we propose the use of fuzzy terrain models to incorporate uncertainty and unpredictable variability of the landscape accordingly we propose a suitable definition of visibility to obtain intervisibility models an algorithm that uses approximation of fuzzy numbers with a discrete family of intervals is proposed the proposed model produces sound results and can be extended to model complex situations like triangulating a radio source using two directional receivers all of these situations are difficult to manage with a precise stochastic model and are a frequent occurrence in animal behavior studies under field conditions the proposed techniques have been implemented inside a user friendly software package that may easily exchange data with the most common giss we report also some experimental results obtained using over artificial terrain under controlled noise and under sampling conditions

branched circle packings and discrete blaschke products in this paper we introduce the notion of discrete blaschke products via circle packing we first establish necessary and sufficient conditions for the existence of finite branched circle packings next discrete blaschke products are defined as circle packing maps from univalent circle packings that properly fill d z izi 1 to the corresponding branched circle packings that properly cover d it is verified that such maps have all geometric properties of their classical counterparts finally we show that any classical finite blaschke product can be approximated uniformly on compacta of d by discrete ones

a local convergence proof for the minvar algorithm for computing continuous piecewise linear approximations the class of continuous piecewise linear pl functions represents a useful family of approximants because invertibility can be readily imposed and if a pl function is invertible then it can be inverted in closed form many applications arising for example in control systems and robotics involve the simultaneous construction of a forward and inverse system model from data most approximation techniques require that separate forward and inverse models be trained whereas an invertible continuous pl affords simultaneously the forward and inverse system model in a single representation the minvar algorithm computes a continuous pl approximation to data local convergence of minvar is proven for the case when the data generating function is itself a pl function and available directly rather than through data

multidimensional search trees that provide new types of memory reductions an orthogonal query that asks to aggregate the set of records in k dimensional box regions is studied and it is shown that space o n log n log log n k 1 makes possible a combined time complexity o log k n for retrievals insertions and deletions

poisson polyhedra in high dimensions abstract the zero cell of a parametric class of random hyperplane tessellations depending on a distance exponent and an intensity parameter is investigated as the space dimension tends to infinity the model includes the zero cell of stationary and isotropic poisson hyperplane tessellations as well as the typical cell of a stationary poisson voronoi tessellation as special cases it is shown that asymptotically in the space dimension with overwhelming probability these cells satisfy the hyperplane conjecture if the distance exponent and the intensity parameter are suitably chosen dimension dependent functions also the high dimensional limits of the mean number of faces are explored and the asymptotic behaviour of an isoperimetric ratio is analysed in the background are new identities linking the f vector of the zero cell to certain dual intrinsic volumes

optimal sensor positioning a probability perspective study we propose a computational method to optimally position a sensor system each sensor has limited range and viewing angle and it may fail with a certain failure rate the goal is to find the optimal locations as well as the viewing directions of all the sensors and achieve the maximal surveillance of the known environment we set up the problem using the level set framework both the environment and the viewing range of the sensors are represented by level set functions then we solve a system of ordinary differential equations odes to find the optimal viewing directions and locations of all sensors together furthermore we use the intermittent diffusion which converts the odes into stochastic differential equations to find the global maximum of the total surveillance area the numerical examples include various failure rates of sensors different rates of importance of the surveillance region and 3 dimensional setups they show the effectiveness of the proposed method

searching on a line we revisit the problem of searching for a target at an unknown location on a line when given upper and lower bounds on the distance d that separates the initial position of the searcher from the target prior to this work only asymptotic bounds were known for the optimal competitive ratio achievable by any search strategy in the worst case we present the first tight bounds on the exact optimal competitive ratio achievable parameterized in terms of the given bounds on d along with an optimal search strategy that achieves this competitive ratio we prove that this optimal strategy is unique we characterize the conditions under which an optimal strategy can be computed exactly and when it cannot we explain how numerical methods can be used efficiently in addition we answer several related open questions including the maximal reach problem and we discuss how to generalize these results to m rays for any m 2

a fast planar partition algorithm ii randomized optimal algorithms to find a partition of the plane induced by a set of algebraic segments of a bounded degree and a set of linear chains of a bounded degree are given this paper also provides a new technique for clipping called virtual clipping whose overhead per window w depends logarithmically on the number if intersections between the borders of w and the input segments in contrast the overhead of the conventional clipping technique depends linearly on this number of intersections as an application of virtual clipping a new simple and efficient algorithm for plannar point location is given

rigidity of circle domains whose boundary hasσ finite linear measure let be a circle domain in the riemann sphere whose boundary has finite linear measure we show that is rigid in the sense that any conformal homeomorphism of onto any other circle domain is equal to the restriction of a mobius transformation previously kaufman and bishop have independently found examples of non rigid circle domains whose boundary is a cantor set of hausdorff dimension one

the union of convex polyhedra in three dimensions we show that the number of vertices edges and faces of the union of k convex polyhedra in 3 space having a total of n faces is o k3 kn log k this bound is almost tight in the worst case as there exist collections of polyhedra with omega k 3 kn alpha k union complexity we also describe a rather simple randomized incremental algorithm for computing the boundary of the union in o k3 kn log k log n expected time

whitney stratified chains and cochains this paper contains the technical constructions necessary for a geometric cycle definition of cohomology and homology in the context of whitney stratifications cup and cap products are interpreted as the transverse intersection of geometric cocycles and cycles