an efficient algorithm for decomposing a polygon into star shaped polygons abstract in this paper we show how a theorem in plane geometry can be converted into a o n log n algorithm for decomposing a polygon into star shaped subsets the computational efficiency of this new decomposition contrasts with the heavy computational burden of existing methods

morphological decomposition of 2 d binary shapes into conditionally maximal convex polygons mathematical morphology is an approach to image analysis based on the geometric concept of shape and size naturally it provides a very effective tool for shape analysis in this paper we present a morphological shape segmentation algorithm that decomposes a 2 d two dimensional binary shape into a collection of restricted convex polygons we use simple morphological operations to extract shape information from different skeleton subsets to form a group of conditionally maximal convex polygonal shape components then a set of more meaningful and natural convex polygonal components are constructed from these maximal components the resulting algorithm is very simple and the decomposition is always unique the components produced have well defined mathematical characterizations and they seem to be in good agreement with the nature structures of the given shapes the shape segments produced can be used to construct structural shape description and for other shape analysis purposes

segmenting dot patterns by voronoi diagram concavity this correspondence defines a signed distance called internal concavity on paths of the voronoi diagram of a dot pattern an algorithm using internal concavity to segment dot patterns is described the segmentation algorithm produces subsets of the dirichlet tessellation delaunay triangulation of the dot pattern

classification of invariant image representations using a neural network a neural network nn based approach for classification of images represented by translation scale and rotation invariant features is presented the utilized network is a multilayer perceptron mlp classifier with one hidden layer back propagation learning is used for its training two types of features are used moment invariants derived from geometrical moments of the image and features based on zernlike moments which are the mapping of the image onto a set of complex orthogonal polynomials the performance of the mlp is compared to the bayes nearest neighbor and minimum mean distance statistical classifiers through extensive experimentation with noiseless as well as noisy binary images of all english characters 26 classes the following conclusions are reached 1 the mlp outperforms the other three classifiers especially when noise is present 2 the nearest neighbor classifier performs about the same as the nn for the noiseless case 3 the nn can do well even with a very small number of training samples 4 the nn has a good degree of fault tolerance and 5 the zernlike moment based features possess strong class separability power and are more powerful than moment invariants

scale invariant distribution and multifractality of volatility multipliers in stock markets abstract the statistical properties of the multipliers of the absolute returns are investigated using 1 min high frequency data of financial time series the multiplier distribution is found to be independent of the box size s when s is larger than some crossover scale providing direct evidence of the existence of scale invariance in financial data the multipliers with base a 2 are well approximated by a normal distribution and the most probable multiplier scales as a power law with respect to the base a we unravel that the volatility multipliers possess multifractal nature which is independent of construction of the multipliers that is the values of s and a

the searchlight scheduling problem the problem of searching for a mobile robber in a simple polygon by a number of searchlights is considered a searchlight is a stationary point which emits a single ray that cannot penetrate the boundary of the polygon the direction of the ray can be changed continuously and a point is detected by a searchlight at a given time if and only if it is on the ray a robber is a point that can move continuously with unbounded speed first it is shown that the problem of obtaining a search schedule for an instance having at least one searchlight on the polygon boundary can be reduced to that for instances having no searchlight on the polygon boundary the reduction is achieved by a recursive search strategy called the one way sweep strategy then various sufficient conditions for the existence of a search schedule are presented by using the concept of a searchlight visibility graph finally a simple necessary and sufficient condition for the existence of a search schedule for instances having exactly two sea

computing conforming partitions of orthogonal polygons with minimum stabbing number abstract let p be an orthogonal polygon with n vertices a partition of p into rectangles is called conforming if it results from cutting p along a set of interior disjoint line segments each having both endpoints on the boundary of p the stabbing number of a partition of p into rectangles is the maximum number of rectangles stabbed by any orthogonal line segment inside p in this paper we consider the problem of finding a conforming partition of p with minimum stabbing number we first give an o n log n time algorithm to solve the problem when p is a histogram for an arbitrary orthogonal polygon even with holes we give an integer programming formulation of the problem and show that a simple rounding results in a 2 approximation algorithm for the problem finally we show that the problem is np hard if p is allowed to have holes

the korteweg de vries equation and its symmetry preserving discretization the korteweg de vries equation is one of the most important nonlinear evolution equations in the mathematical sciences in this article invariant discretization schemes are constructed for this equation both in the lagrangian and in the eulerian form we also propose invariant schemes that preserve the momentum numerical tests are carried out for all invariant discretization schemes and related to standard numerical schemes we find that the invariant discretization schemes give generally the same level of accuracy as the standard schemes with the added benefit of preserving galilean transformations which is demonstrated numerically as well

lie group classification of second order ordinary difference equations a group classification of invariant difference models i e difference equations and meshes is presented in the continuous limit the results go over into lie s classification of second order ordinary differential equations the discrete model is a three point one and we show that it can be invariant under lie groups of dimension 0 n 6

invariant pattern recognition a review in this document we review and compare some of the classical and modern techniques for solving the problem of invariant pattern recognition such techniques include integral transforms construction of algebraic moments and the use of structured neural networks in all cases we assume that the nature of the invariance group is known a priori many of the methods described apply to specific geometrical transformation groups however some of the techniques are highly general and applicable to large classes of invariance groups we also review some results regarding the existence and structure of invariants under certain kinds of groups

extraction of parametric descriptions of circular gcs from a pair of contours for 3 d shape recognition this paper addresses the issue of recognizing 3 d three dimensional shapes of objects from their 2 d contour pairs a shape model that is represented by a family of spheres of varying radii with their centers on its axis is developed for the shape class of circular generalized cones gcs with a space curved axis the quasi invariant property of the shape model is defined and is used to recover the 3 d shapes of the object class by solving the problem of stereo matching between their contour pairs the recovered shapes are represented by a set of their parametric primitives and spatial relations between them so that they may be used for recognition even under occluded environment

efficient morphological shape representation by varying overlapping levels among representative disks abstract the morphological skeleton transform the morphological shape decomposition and the overlapped morphological shape decomposition are three basic morphological shape representation schemes in this paper we propose a new way of generalizing these basic representation algorithms to improve representational efficiency in all three basic algorithms a fixed overlapping policy is used to control the overlapping relationships among representative disks of different sizes in our new algorithm different overlapping policies are used to generate shape components that have different overlapping relationships among themselves the overlapping policy is selected dynamically according to local shape features experiments show that compared to the three basic algorithms our algorithm produces more efficient representations with lower numbers of representative points

shape description and recognition using the high order morphological pattern spectrum abstract shape analysis and recognition is an important issue in modern image processing and computer vision it needs to solve three problems possibility sensitivity and robustness to deal with the third one this paper proposes the high order morphological pattern hp spectrum which relies upon the difference between two different sized alternating sequential morphological transformations on the original image the hp spectrum is characterized by the translating invariance and rotating invariance for the isotropic structuring element it is also proved that the higher order components of hp spectrum for a noisy image are exactly identical to these for the noise free image under certain conditions also the results of the experiments conducted have shown that the hp spectrum shows the strongest robustness in a noisy environment

morphological decomposition of 2 d binary shapes into convex polygons a heuristic algorithm in many morphological shape decomposition algorithms either a shape can only be decomposed into shape components of extremely simple forms or a time consuming search process is employed to determine a decomposition in this paper we present a morphological shape decomposition algorithm that decomposes a two dimensional 2 d binary shape into a collection of convex polygonal components a single convex polygonal approximation for a given image is first identified this first component is determined incrementally by selecting a sequence of basic shape primitives these shape primitives are chosen based on shape information extracted from the given shape at different scale levels additional shape components are identified recursively from the difference image between the given image and the first component simple operations are used to repair certain concavities caused by the set difference operation the resulting hierarchical structure provides descriptions for the given shape at different detail levels the experiments show that the decomposition results produced by the algorithm seem to be in good agreement with the natural structures of the given shapes the computational cost of the algorithm is significantly lower than that of an earlier search based convex decomposition algorithm compared to nonconvex decomposition algorithms our algorithm allows accurate approximations for the given shapes at low coding costs

symmetries of discrete dynamical systems differential difference equations of the form un fn t un 1 un un 1 are classified according to their continuous lie point symmetry groups it is shown that for nonlinear equations the symmetry group can be at most seven dimensional the integrable toda lattice is a member of this class and has a four dimensional symmetry group

on fully discrete galerkin methods of second order temporal accuracy for the nonlinear schrodinger equation we approximate the solutions of an initial and boundary value problem for nonlinear schrodinger equations with emphasis on the cubic nonlinearity by two fully discrete finite element schemes based on the standard galerkin method in space and two implicit crank nicolson type second order accurate temporal discretizations for both schemes we study the existence and uniqueness of their solutions and provel 2 error bounds of optimal order of accuracy for one of the schemes we also analyze one step of newton s method for solving the nonlinear systems that arise at every time step we then implement this scheme using an iterative modification of newton s method that at each time stept n requires solving a number of sparse complex linear systems with a matrix that does not change withn the effect of this inner iteration is studied theoretically and numerically

efficient shape matching through model based shape recognition abstract an efficient shape matching method for shape recognition is proposed it first uses a polygonal approximation technique to describe a shape then using the proposed matching algorithm based on some polygonal attributes the dissimilarity measure between an input shape and its reference model is obtained the input shape can be classified into the class model which has the minimum dissimilarity measure with it in addition it can provide the best matching orientation information for the input shape and its model so that further applications such as automated inspection and assembly can be effectively performed the proposed shape matching is invariant to translation rotation and scale changes of a shape experiments using noisy shapes and real images are performed and the recognition results demonstrate that the proposed method is very effective

the discretized boussinesq equation and its conditional symmetry reduction in this article we show that we can carry out the symmetry preserving discretization of the boussinesq equation with respect to three of its more significant conditional symmetries we perform the symmetry reduction of the obtained nonlinear discrete schemes with respect to the conditional symmetries and obtain the reduced discrete equations which unlike in the continuous case are not always reducible to second order difference equations a numerical comparison with the exact continuous solution given by weierstrass elliptic functions is carried out

a morphological signature transform for shape description abstract a novel shape description method based on the multiresolution morphological image processing by non convex multiple structuring elements is presented a binary image which contains the object shape to be described is represented by means of a multiresolution pyramid a set of primary structuring elements along with their rotated versions is used the method is based on the successive morphological erosions of the input image at different resolutions by primary and rotated structuring elements the areas of successively eroded images are computed for each structuring element at each pyramid level the obtained set of numbers is arranged into vectors ordered and used as a shape descriptor results are included to show that the method is robust against noise and invariant to both rotation and scale change

inspection of 2 d objects using pattern matching method abstract a pattern matching scheme is developed for the inspection of objects in industrial environment the inspection includes dimensional verification and shape matching which compares a 2 dimensional image of an object to a pattern image the method proves to be computationally efficient and accurate for real time application

a generalized discrete morphological skeleton transform with multiple structuring elements for the extraction of structural shape components a common problem shared by several leading morphological shape representation algorithms is that there is much overlapping among the representative disks of the same size a shape component represented by a group of connected disk centers sometimes uses many heavily overlapping representative disks to represent a relatively simple shape part a shape component may also contain a large number of representative disks that form a complicated structure we introduce a generalized discrete morphological skeleton transform that uses eight structuring elements to generate skeleton subsets so that no two skeletal points from the same skeleton subset are adjacent to each other each skeletal point represents a shape part that is in general an octagon with four pairs of parallel opposing sides the number of representative points needed to represent a given shape is significantly lower than that in the standard skeleton transform a collection of shape components needed to build a structural representation is easily derived from the generalized skeleton transform each shape component covers a significant area of the given shape and severe overlapping is avoided the given shape can also be accurately approximated using a small number of shape components

convecting reference frames and invariant numerical models abstract in the recent paper by bernardini et al 1 the discrepancy in the performance of finite difference and spectral models for simulations of flows with a preferential direction of propagation was studied in a simplified investigation carried out using the viscous burgers equation the authors attributed the poorer numerical results of finite difference models to a violation of galilean invariance in the discretization and propose to carry out the computations in a reference frame moving with the bulk velocity of the flow here we further discuss this problem and relate it to known results on invariant discretization schemes non invariant and invariant finite difference discretizations of burgers equation are proposed and compared with the discretization using the remedy proposed by bernardini et al

difference schemes with point symmetries and their numerical tests symmetry preserving difference schemes approximating second and third order ordinary differential equations are presented they have the same three or four dimensional symmetry groups as the original differential equations the new difference schemes are tested as numerical methods the obtained numerical solutions are shown to be much more accurate than those obtained by standard methods without an increase in cost for an example involving a solution with a singularity in the integration region the symmetry preserving scheme contrary to standard ones provides solutions valid beyond the singular point

a generalized morphological skeleton transform using both internal and external skeleton points the morphological skeleton transform mst is a leading morphological shape representation scheme in the mst a given shape is represented as the union of all the maximal disks contained in the shape the concept of external skeleton points and external maximal disks has been used for shape description and characterization purposes in this paper we develop a generalized morphological skeleton transform that combines the concepts of internal and external maximal disks into a unified framework in this framework a shape is described in terms of disk components that need to be added as well as disk components that need to be removed the procedures and formulae describing the extraction of the disk components and the reconstruction of the original shape from these components are developed the correctness of the procedures and formulae is established this new framework seems to provide a more powerful and more natural way of modeling the approximation and reconstruction of binary shapes using primitive shape components

singularities and regularities on line pictures via symmetrical trapezoids an algorithm that decomposes a line image into singular and regular regions is presented we define a contour trapezoid as the building block of regular regions and postulate a maximal trapezoid set as the core of regular regions then we describe an algorithm that calculates a maximal trapezoid set of a polygon and show how to use it to find a skeleton of a polygonal approximation of a contour experiments are explained to show the behavior of the new concept on real images as compared to previous algorithms

ordered structural shape matching with primitive extraction by mathematical morphology abstract structural shape matching procedures begin by decomposing the shapes to be matched into primitive parts and extracting the properties of and relationships among these primitives the resultant structural descriptions are matched to structural models usually via a tree search whose time is exponential in the number of primitives in this paper we introduce a new fast method of structural shape matching called ordered structural matching which has polynomial time complexity instead of exponential the approach presented also uses mathematical morphology to rapidly decompose the shapes into primitives to be used in the matching the primitives so produced are pieces of the original shape but not necessarily those pieces that a human would select the method has been tested on the toy task of determining which plastic shapes match the shapes of the holes in a child s six sided toy box it is also directly applicable to the inspection and manipulation of certain airplane parts

reasoning about pictures and similarity retrieval for image information systems based on sk set knowledge representation spatial reasoning and similarity retrieval are two important functions of any image information system good spatial knowledge representation for images is necessary to adequately support these two functions in this paper we propose a new spatial knowledge representation called the sk set based on morphological skeleton theories spatial reasoning algorithms which achieve more accurate results by directly analysing skeletons are described sk set facilitates browsing and progressive visualization we also define four new types of similarity measures and propose a similarity retrieval algorithm for performing image retrieval moreover using sk set as a spatial knowledge representation will reduce the storage space required by an image database significantly

long term memory in stock market prices a test for long term memory that is robust to short range dependence is developed it is a modification of the r s statistic and the relevant asymptotic sampling theory is derived via functional central limit theory contrary to previous findings when applied to daily and monthly stock returns indexes over several time periods this test yields no evidence of long range dependence once short range dependence is accounted for monte carlo experiments show that the modified r s test has power against at least two specific models of long term memory suggesting that models with short range dependence may adequately capture the behavior of historical stock returns copyright 1991 by the econometric society

symmetries of the discrete burgers equation a discrete cole hopf transformation is used to derive a discrete burgers equation that is linearizable to a discrete heat equation a five dimensional symmetry algebra is obtained that reduces to the lie point symmetry algebra of the usual burgers equation in the continuous limit this lie algebra is used to obtain explicit invariant solutions

symmetries and first integrals of ordinary difference equations this paper describes a new symmetry based approach to solving a given ordinary difference equation by studying the local structure of the set of solutions we derive a systematic method for determining one parameter lie groups of symmetries in closed form such groups can be used to achieve successive reductions of order if there are enough symmetries the difference equation can be completely solved several examples are used to illustrate the technique for transitive and intransitive symmetry groups it is also shown that every linear second order ordinary difference equation has a lie algebra of symmetry generators that is isomorphic to sl 3 the paper concludes with a systematic method for constructing first integrals directly which can be used even if no symmetries are known

region extraction from complex shapes an algorithm is described which extracts primitive regions i e convex spiral shaped and biconcave lens from complex shapes the interior region bounded by the shape is decomposed by first slicing it into a set of convex subregions and then rotating and dissolving the various boundaries between subregions until a satisfactory decomposition is obtained the same algorithm also is used to decompose the exterior region between the shape and its convex hull the algorithm has been implemented as an algol w computer program for the univac 90 80 and results of running the program are presented for a wide variety of complex shapes these results compare favorably with the experience reported by previous programs

efficiency and multifractality analysis of csi 300 based on multifractal detrending moving average algorithm abstract we apply the multifractal detrending moving average mfdma to investigate and compare the efficiency and multifractality of 5 min high frequency china securities index 300 csi 300 the results show that the csi 300 market becomes closer to weak form efficiency after the introduction of csi 300 future we find that the csi 300 is featured by multifractality and there are less complexity and risk after the csi 300 index future was introduced with the shuffling surrogating and removing extreme values procedures we unveil that extreme events and fat distribution are the main origin of multifractality besides we discuss the knotting phenomena in multifractality and find that the scaling range and the irregular fluctuations for large scales in the f q s vs s plot can cause a knot

new self similar solutions of the nonlinear schrodinger equation with moving mesh computations we study the blow up self similar solutions of the radially symmetric nonlinear schrodinger equation nls given by iut urr d 1rur u u 2 with dimension d 2 these solutions become infinite in a finite time t by a series of careful numerical computations partly supported by analytic results we demonstrate that there is a countably infinite set of blow up self similar solutions which satisfy a second order complex ordinary differential equation with an integral constraint these solutions are characterised by the number of oscillations in their amplitude when d is close to 2 the solutions are computed as functions of d and their behaviour in the critical limit as d 2 is investigated the stability of these solutions is then studied by solving the nls by using an adaptive numerical method this method uses moving mesh partial differential equations and exploits the scaling invariance properties of the underlying equation we demonstrate that the single humped self similar solution is globally stable whereas the multi humped solutions all appear to be unstable

the jacobi last multiplier for linear partial difference equations we present a discretization of the jacobi last multiplier with some applications to the computation of solutions of linear partial difference equations

analysis of the efficiency of the shanghai stock market a volatility perspective abstract by applying the rolling window method we investigate the efficiency of the shanghai stock market through the dynamic changes of local hurst exponents based on multifractal detrended fluctuation analysis we decompose the realized volatility into continuous sample paths and jump components and analyze their long range correlations of decomposing components our results reveal that the efficiency of the shanghai stock market improved greatly based on the time varying hurst exponents

construction of invariant compact finite difference schemes in this paper we propose a method which is based on equivariant moving frames for development of high order accurate invariant compact finite difference schemes that preserve lie symmetries of underlying partial differential equations in this method variable transformations that are obtained from the extended symmetry groups of partial differential equations pdes are used to transform independent and dependent variables and derivative terms of compact finite difference schemes constructed for these pdes such that the resulting schemes are invariant under the chosen symmetry groups the unknown symmetry parameters that arise from the application of these transformations are determined through selection of convenient moving frames in some cases owing to selection of convenient moving frames numerical representation of invariant or symmetry preserving compact numerical schemes is found to be notably simpler than that of standard noninvariant compact numerical schemes further the accuracy of these invariant compact schemes can be arbitrarily set to a desired order by considering suitable compact finite difference algorithms application of the proposed method is demonstrated through construction of invariant compact finite difference schemes for some common linear and nonlinear pdes including the linear advection diffusion equation in one or two dimensions the inviscid burgers equation in one dimension viscous burgers equation in one or two dimensions spherical burgers equation in one dimension and shallow water equations in two dimensions results obtained from our numerical simulations indicate that invariant compact finite difference schemes not only inherit selected symmetry properties of underlying pdes but are also comparably more accurate than the standard noninvariant base numerical schemes considered here

symplectic integration of hamiltonian wave equations the numerical integration of a wide class of hamiltonian partial differential equations by standard symplectic schemes is discussed with a consistent hamiltonian approach we discretize the hamiltonian and the poisson structure separately then form the the resulting ode s the stability accuracy and dispersion of different explicit splitting methods are analyzed and we give the circumstances under which the best results can be obtained in particular when the hamiltonian can be split into linear and nonlinear terms many different treatments and examples are compared

computational complexity of art gallery problems we study the computational complexity of the art gallery problem originally posed by klee and its variations specifically the problem of determining the minimum number of vertex guards that can see an n wall simply connected art gallery is shown to be np hard the proof can be modified to show that the problems of determining the minimum number of edge guards and the minimum number of point guards in a simply connected polygonal region are also np hard as a byproduct the problem of decomposing a simple polygon into a minimum number of star shaped polygons such that their union is the original polygon is also shown to be np hard

symmetry adapted moving mesh schemes for the nonlinear schrodinger equation in this paper we consider symmetry preserving difference schemes for the nonlinear schrodinger equation where n is the number of space dimensions this equation describes one dimensional waves in n space dimensions in many physical situations including phenomena in plasma physics and nonlinear optics we will consider the nonintegrable case n 2 for which the equation admits solutions that blow up in a finite time and construct discretizations based upon moving mesh schemes that have the same lie group properties and lagrangian structures as the continuous counterpart

symmetry preserving difference schemes for some heat transfer equations lie group analysis of differential equations is a generally recognized method which provides invariant solutions integrability conservation laws etc in this paper we present three characteristic examples of the construction of invariant difference equations and meshes where the original continuous symmetries are preserved in discrete models conservation of symmetries in difference modelling helps to retain qualitative properties of the differential equations in their difference counterparts

morphological decomposition of 2 d binary shapes into modestly overlapped octagonal and disk components one problem with several leading morphological shape representation algorithms is the heavy overlapping among the representative disks of the same size a shape component formed by grouping connected disk centers may use many heavily overlapping disks to represent a simple shape part sometimes these representative disks form complicated structures a generalized skeleton transform was recently introduced which allows a shape to be represented as a collection of modestly overlapped octagonal shape parts however the generalized skeleton transform needs to be applied many times furthermore an octagonal component is not easily matched up with another octagonal component in this paper we describe a octagon fitting algorithm which identifies a special maximal octagon for each image point in a given shape this transform leads to the development of two new shape decomposition algorithms these algorithms are more efficient to implement the octagon fitting algorithm only needs to be applied once the components generated are better characterized mathematically the disk components used in the second decomposition algorithm are more primitive than octagons and easily matched up with other disk components from another shape the experiments show that the new decomposition algorithms produce as efficient representations as the old algorithm for both exact and approximate cases a simple shape matching algorithm using disk components is also demonstrated

comparison of some lie symmetry based integrators lie symmetry based integrators are constructed in order to preserve the local invariance properties of the equations the geometrical methods leading to discretized equations for numerical computations involve many different concepts therefore they give rise to numerical schemes that vary in the accuracy in the computational cost and in the implementation in this paper a comparison is made between some alternative lie symmetry based methods illustrated on the example of the burgers equation the importance of the symmetry preservation is numerically highlighted

invariant discretization schemes for the shallow water equations invariant discretization schemes are derived for the one and two dimensional shallow water equations with periodic boundary conditions while originally designed for constructing invariant finite difference schemes we extend the use of difference invariants to allow construction of invariant finite volume methods as well it is found that the classical invariant schemes converge to the lagrangian formulation of the shallow water equations these schemes require redistribution of the grid points according to the physical fluid velocity i e the mesh cannot remain fixed in the course of the numerical integration invariant eulerian discretization schemes are proposed for the shallow water equations in computational coordinates instead of using the fluid velocity as the grid velocity an invariant moving mesh generator is invoked in order to determine the location of the grid points at the subsequent time level the numerical conservation of energy mass and momentum is evaluated for both the invariant

stochastic models for closed boundary analysis representation and reconstruction the analysis of closed boundaries of arbitrary shapes on a plane is discussed specifically the problems of representation and reconstruction are considered a one to one correspondence between the given closed boundary and a univariate or multivariate sequence of real numbers is set up univariate or multivariate circular autoregressive models are suggested for the representation of the sequence of numbers derived from the closed boundary the stochastic model representing the closed boundary is invariant to transformations like sealing translation choice of starting point and rotation over angles that are multiples of 2 pi n where n is the number of observations methods for estimating the unknown parameters of the model are given and a decision rule for choosing the appropriate number of coefficients is included constraints on the estimates are derived so that the estimates are invariant to the transformations of the boundaries the stochastic model enables the reconstruction of a dosed boundary using fft algorithms results of simulations are included and the application to contour coding is discussed

some np hard polygon decomposition problems the inherent computational complexity of polygon decomposition problems is of theoretical interest to researchers in the field of computational geometry and of practical interest to those working in syntactic pattern recognition three polygon decomposition problems are shown to be np hard and thus unlikely to admit efficient algorithms the problems are to find minimum decompositions of a polygonal region into perhaps overlapping convex star shaped or spiral subsets we permit the polygonal region to contain holes the proofs are by transformation from boolean three satisfiability a known np complete problem several open problems are discussed

on a convex hull algorithm for polygons and its application to triangulation problems abstract a frequently used algorithm for finding the convex hull of a simple polygon in linear running time has been recently shown to fail in some cases due to its simplicity the algorithm is nevertheless attractive in this paper it is shown that the algorithm does in fact work for a family of simple polygons known as weakly externally visible polygons some application areas where such polygons occur are briefly discussed in addition it is shown that with a trivial modification the algorithm can be used to internally and externally triangulate certain classes of polygons in 0 n time

division based analysis of symmetry and its application a computational method das is proposed for symmetry analysis of a planar figure closed by a simply connected curve das determines both the symmetric axis and the symmetric point pairs on the curve consistently based on the duality of two geometric plane divisions the delaunay triangulation and the voronoi diagram

multiscaling and clustering of volatility abstract the dynamics of prices in stock markets has been studied intensively both experimentally data analysis and theoretically models nevertheless while the distribution of returns of the most important indices is known to be a truncated levy the behaviour of volatility correlations is still poorly understood what is well known is that absolute returns have memory on a long time range this phenomenon is known in financial literature as clustering of volatility in this paper we show that volatility correlations are power laws with a non unique scaling exponent this kind of multiscale phenomenology is known to be relevant in fully developed turbulence and in disordered systems and it is pointed out here for the first time for a financial series in our study we consider the new york stock exchange nyse daily index from january 1966 to june 1998 for a total of 8180 working days

shape description by time series time series modeling techniques are adapted to represent or describe two dimensional closed contours both linear and nonlinear models are fitted it is found that to detect small changes in shape nonlinear modeling is necessary even though linear models may be sufficient to differentiate between shapes which differ widely a nonlinear model called the noncausal quadratic volterra model is developed for the purpose implementation is illustrated with shapes of aircraft

auto correlated behavior of wti crude oil volatilities a multiscale perspective abstract in this paper we investigate the long range auto correlated behavior of wti crude oil volatility series employing multifractal detrended fluctuation analysis our findings show that the for small time scales the auto correlations of volatilities were multifractal while for large time scales the auto correlations were nearly monofractal based on multiscale analysis we also investigate the dynamics of auto correlations for different intervals of time scales and find that several shocks could make significant effects on the auto correlated behaviors for small time scales analyzing the dynamics of multifractality degrees of auto correlations for small time scales we find that the stronger auto correlations were always related to the lower degrees of multifractality at last we have discussions on the determination factors of price behavior the predictive implications of scaling behavior in volatilities for oil markets and the reasons why long range auto correlations of volatility were always strong for both small time scales and large time scales our results are very important theoretically and practically

some sequential algorithms for a generalized distance transformation based on minkowski operations a generalized distance transformation gdt of binary images and the related medial axis transformation mat are discussed these transformations are defined in a discrete space of arbitrary dimension and arbitrary grids the gdt is based on successive morphological operations using alternatively n arbitrary structuring elements n is called the period of the gdt the gdt differs from the classical distance transformations based on a point to point distance however the well known chessboard city block and hexagonal distance transformations are special cases of the one period gdt whereas the octagonal distance transformation is a special case of the two period gdt in this paper both one and two period gdts are discussed different sequential algorithms are proposed for computing such gdts these algorithms need a maximum of two scannings of the image the computation of the mat is also discussed

linear combination of power law functions for detecting multiscaling using detrended fluctuation analysis abstract in many instances the fluctuation function obtained from detrended fluctuation analysis dfa cannot be described by a uniform power law function along scales in fact the manifestation of crossover scales may reflect the simultaneous action of different stochastic mechanisms displayed predominantly within certain scale ranges this note proposes the use of a linear combination of power law functions for adjusting dfa data the idea is that each power law function recast the dominance of certain stochastic mechanisms e g the mean reversion and long term trends at specific scale domains different values of the scaling exponents are numerically estimated by means of a nonlinear least squares fitting of power law functions examples of crude oil market and heart rate variability are discussed with some detail for illustrating the advantages of taking a linear combination of power law functions for describing scaling behavior from dfa

a method for shift rotation and scale invariant pattern recognition using the form and arrangement of pattern specific features abstract a new computer based method for visual pattern recognition is described the process extracts shape features from binary edge images represented in either real valued or discrete coordinates both separate and overlapping features are extracted maximizing their completeness and continuity without using templates or mathematical feature descriptions these features and their arrangement are encoded in a shift rotation and scale invariant manner and used to recognize incoming patterns from a set of previously encoded patterns stored in memory new patterns may be automatically classified and stored in memory allowing the system to learn and generalize without operator intervention testing shows the system to be completely shift rotation and scale invariant and insensitive to local image distortion

scaling properties of foreign exchange volatility in this paper we investigate the scaling properties of foreign exchange volatility our methodology is based on a wavelet multi scaling approach which decomposes the variance of a time series and the covariance between two time series on a scale by scale basis through the application of a discrete wavelet transformation it is shown that foreign exchange rate volatilities follow different scaling laws at different horizons particularly there is a smaller degree of persistence in intra day volatility as compared to volatility at one day and higher scales therefore a common practice in the risk management industry to convert risk measures calculated at shorter horizons into longer horizons through a global scaling parameter may not be appropriate this paper also demonstrates that correlation between the foreign exchange volatilities is the lowest at the intra day scales but exhibits a gradual increase up to a daily scale the correlation coefficient stabilizes at scales one day and higher therefore the benefit of currency diversification is the greatest at the intra day scales and diminishes gradually at higher scales lower frequencies the wavelet cross correlation analysis also indicates that the association between two volatilities is stronger at lower frequencies

binary shape recognition using the morphological skeleton transform abstract binary image representation by its morphological skeleton transform has been proposed in the past as an information preserving representation in this paper this skeleton representation is adopted as the start point for the development of a binary shape recognition scheme a skeleton matching algorithm is presented that efficiently characterizes the similarity between two skeletons as a distance measure this skeleton matching algorithm resembles the well known from speech processing elastic matching technique

the numerical study of blowup with application to a nonlinear schrodinger equation abstract we discuss the use of numerical methods in the study of the solutions of evolution problems which exhibit finite time unbounded growth we first examine a naive approach in which the growth rate of the numerical solution is accepted as an approximation of the true growth rate as we shall demonstrate for a radial nonlinear schrodinger equation this approach is inadequate since different discretizations exhibit different growth rates the spurious behaviour of discretizations in the neighbourhood of the singularity is discussed a reliable procedure for the estimation of the blowup parameters is considered which eliminates the discrepancies between different numerical methods

binary object representation and recognition using the hilbert morphological skeleton transform abstract a binary shape representation called the hilbert morphological skeleton transform hmst is introduced this representation combines the morphological skeleton transform mst with the clustering capabilities of the hilbert transform the hmst preserves the skeleton properties including information preservation progressive visualization and compact representation then an object recognition algorithm the hilbert skeleton matching algorithm hsma is introduced this algorithm performs a single sweep over the hmst s and renders the similarity between them as a distance measure testing the hsma against the skeleton matching algorithm sma and invariant moments revealed that the hsma algorithm achieves slightly better object recognition rates while substantially reducing the complexity in an experiment of 14 400 shape matches the hsma achieved a 90 36 recognition rate as opposed to 89 76 for the sma and 89 49 for invariant moments on the other hand the hsma improved the sma processing more than 40

moving mesh methods for problems with blow up in this paper we consider the numerical solution of pdes with blow up for which scaling invariance plays a natural role in describing the underlying solution structures it is a challenging numerical problem to capture the qualitative behaviour in the blow up region and the use of nonuniform meshes is essential we consider moving mesh methods for which the mesh is determined using so called moving mesh partial differential equations mmpdes specifically the underlying pde and the mmpde are solved for the blow up solution and the computational mesh simultaneously motivated by the desire for the mmpde to preserve the scaling invariance of the underlying problem we study the effect of different choices of mmpdes and monitor functions it is shown that for suitable ones the mmpde solution evolves towards a moving mesh which close to the blow up point automatically places the mesh points in such a manner that the ignition kernel which is well known to be a natural coordinate in describing the behaviour of blow up approaches a constant as t to t the blow up time several numerical examples are given to verify the theory for these mmpde methods and to illustrate their efficacy

invariantization of the crank nicolson method for burgers equation abstract in this article a geometric technique to construct numerical schemes for partial differential equations pdes that inherit lie symmetries is proposed the moving frame method enables one to adjust the numerical schemes in a geometric manner and systematically construct proper invariant versions of them to illustrate the method we study invariantization of the crank nicolson scheme for burgers equation with careful choice of normalization equations the invariantized schemes are shown to surpass the standard scheme successfully removing numerical oscillation around sharp transition layers

linear time euclidean distance transform algorithms two linear time and hence asymptotically optimal algorithms for computing the euclidean distance transform of a two dimensional binary image are presented the algorithms are based on the construction and regular sampling of the voronoi diagram whose sites consist of the unit feature pixels in the image the first algorithm which is of primarily theoretical interest constructs the complete voronoi diagram the second more practical algorithm constructs the voronoi diagram where it intersects the horizontal lines passing through the image pixel centers extensions to higher dimensional images and to other distance functions are also discussed

multifractal detrended fluctuation analysis of nonstationary time series abstract we develop a method for the multifractal characterization of nonstationary time series which is based on a generalization of the detrended fluctuation analysis dfa we relate our multifractal dfa method to the standard partition function based multifractal formalism and prove that both approaches are equivalent for stationary signals with compact support by analyzing several examples we show that the new method can reliably determine the multifractal scaling behavior of time series by comparing the multifractal dfa results for original series with those for shuffled series we can distinguish multifractality due to long range correlations from multifractality due to a broad probability density function we also compare our results with the wavelet transform modulus maxima method and show that the results are equivalent

finding contour based abstractions of planar patterns abstract an algorithm is described to detect a number of points on the contour of a planar shape which constitute the vertices of a schematic polygonal representation of the shape itself a set of points initially extracted from the chain coded representation of the contour is iteratively examined while removing some points and inserting new ones the number of selected points decreases in size from iteration to iteration and the selection process converges towards an expected perceptually significant set of points the polygon obtained by linking successive points approximates the contour in an intuitive way it is not constrained within a given tolerance and is likely to locally change from a coarse to a more faithful approximating shape in correspondence with contour regions increasing in details

pattern spectrum and multiscale shape representation the results of a study on multiscale shape description smoothing and representation are reported multiscale nonlinear smoothing filters are first developed using morphological opening and closings g matheron 1975 used openings and closings to obtain probabilistic size distributions of euclidean space sets continuous binary images these distributions are used to develop a concept of pattern spectrum a shape size descriptor a pattern spectrum is introduced for continuous graytone images and arbitrary multilevel signals as well as for discrete images by developing a discrete size family of patterns large jumps in the pattern spectrum at a certain scale indicate the existence of major protruding or intruding substructures of the signal at the scale an entropy like shape size complexity measure is also developed based on the pattern spectrum for shape representation a reduced morphological skeleton transform is introduced for discrete binary and graytone images this transform is a sequence of skeleton components sparse images which represent the original shape at various scales it is shown that the partially reconstructed images from the inverse transform on subsequences of skeleton components are the openings of the image at a scale determined by the number of eliminated components in addition two way correspondences are established among the degree of shape smoothing via multiscale openings or closings the pattern spectrum zero values and the elimination or nonexistence of skeleton components at certain scales

spatial structure of the focusing singularity of the nonlinear schro dinger equation a geometrical analysis the cubic nonlinear schrodinger equation has solutions that become singular in finite time for spatial dimension d geq 2 numerical simulations demonstrate that in three dimensions the blowup is self similar and symmetric with the structure described by a nonlinear nonautonomous profile equation in two dimensions the blowup is again believed to be symmetric but the self similarity is weakly broken with structure described by the profile equation in the limit as d tends to 2 from above this paper gives a proof of the existence of a locally unique solution to the profile equation for d 2 and sufficiently close to 2 satisfying the boundary and global conditions associated with the blowup solution dynamical systems methods are used to transform previously derived asymptotic analysis into constructions of manifolds of solutions satisfying the relevant boundary conditions and to follow these manifolds to show that they have a transverse intersection

on supraconvergence phenomenon for second order centered finite differences on non uniform grids in the present note we consider an example of a boundary value problem for a simple second order ordinary differential equation which may exhibit a boundary layer phenomenon we show that usual central finite differences which are second order accurate on a uniform grid can be substantially upgraded to the fourth order by a suitable choice of the underlying non uniform grid this example is quite pedagogical and may give some ideas for more complex problems

dot pattern processing using voronoi neighborhoods a sound notion of the neighborhood of a point is essential for analyzing dot patterns the past work in this direction has concentrated on identifying pairs of points that are neighbors examples of such methods include those based on a fixed radius k nearest neighbors minimal spanning tree relative neighborhood graph and the gabriel graph this correspondence considers the use of the region enclosed by a point s voronoi polygon as its neighborhood it is argued that the voronoi polygons possess intuitively appealing characteristics as would be expected from the neighborhood of a point geometrical characteristics of the voronoi neighborhood are used as features in dot pattern processing procedures for segmentation matching and perceptual border extraction using the voronoi neighborhood are outlined extensions of the voronoi definition to other domains are discussed

global shape analysis by k syntactic similarity the k syntactic similarity approach is couched in graphical representation terms and its ability to provide global recognition capability while retaining a low time complexity is explored one potential application domain that of composite shape decomposition into approximately convex subshapes is described this is shown to be equivalent to finding cycles within a particular graph the approach yields valid decompositions in many cases of interest and is capable of identifying those cases where additional semantic considerations are necessary for proper analysis the permissible graph structures representing composite shapes given a reasonable set of relations are determined experimental results on nonideal data are given

long term memory and volatility clustering in high frequency price changes abstract we studied the long term memory in diverse stock market indices and foreign exchange rates using detrended fluctuation analysis dfa for all high frequency market data studied no significant long term memory property was detected in the return series while a strong long term memory property was found in the volatility time series the possible causes of the long term memory property were investigated using the return data filtered by the ar 1 model reflecting the short term memory property the garch 1 1 model reflecting the volatility clustering property and the figarch model reflecting the long term memory property of the volatility time series the memory effect in the ar 1 filtered return and volatility time series remained unchanged while the long term memory property diminished significantly in the volatility series of the garch 1 1 filtered data notably there is no long term memory property when we eliminate the long term memory property of volatility by the figarch model for all data used although the hurst exponents of the volatility time series changed considerably over time those of the time series with the volatility clustering effect removed diminish significantly our results imply that the long term memory property of the volatility time series can be attributed to the volatility clustering observed in the financial time series

relative sensitivity of a family of closet point graphs in computer vision applications abstract various graph theoretic definitions of neighborhood have been used in the past in computer vision and clustering applications in this paper we study the properties of a set of four related closest point graphs using monte carlo methods i the delaunay triangulation dt and its dual voronoi tessellation ii the gabriel graph gg iii the relative neighborhood graph rng and iv the minimum spanning tree mst we examine the suitability of these graphs in the detection of structure in noisy images using the hough transform we also explore each graph s structural stability under random positional noise delaunay triangulation is shown to be the least sensitive to such noisy conditions

continuous symmetries of difference equations lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations in this article we review the results of a much more recent program the use of lie groups to study difference equations we show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners one is to use generalized symmetries acting on solutions of difference equations but leaving the lattice invariant the other is to restrict them to point symmetries but to allow them to also transform the lattice

on homoclinic structure and numerically induced chaos for the nonlinear schro dinger equation it has recently been demonstrated that standard discretizations of the cubic nonlinear schrodinger nls equation may lead to spurious numerical behavior in particular the origins of numerically induced chaos and the loss of spatial symmetry are related to the homoclinic structure associated with the nls equation in this paper an analytic description of the homoclinic structure via soliton type solutions is provided and some consequences for numerical computations are demonstrated differences between an integrable discretization and standard discretizations are highlighted

the crude oil price bubbling and universal scaling dynamics of price volatility abstract the main goal of this paper is to reveal the effect of crude oil price bubbling on the price volatility dynamics for this purpose the time series of volatility at different horizons are mapped into a model of kinetic roughening of interface growing in a stochastic environment in this way we found that the volatility dynamics obeys the family viscek dynamic scaling ansatz although during the period from january 2 1986 to july 25 2014 the volatility remains a slightly anti persistent the dynamic exponent is found to be quite different during different regimes of price evolution accordingly we define the intrinsic time of price volatility and metric of volatility horizons this allows us to construct the langevin type equation governing the volatility dynamics during bubble and non bubble periods the data analysis indicates that the bubbling does not affect the intrinsic time of volatility but strongly affect the metric of volatility horizons in this regard numerical data suggest the existence of two universal metrics characterizing the volatility dynamics during the bubble and non bubble regimes of crude oil price evolution respectively the results of this work help us to get a further insight into the dynamics of crude oil price volatility

discretization of partial differential equations preserving their physical symmetries a procedure for obtaining a minimal discretization of a partial differential equation preserving all of its lie point symmetries is presented minimal in this case means that the differential equation is replaced by a partial difference scheme involving n difference equations where n is the number of independent and dependent variables we restrict ourselves to one scalar function of two independent variables as examples invariant discretizations of the heat burgers and korteweg de vries equations are presented some exact solutions of the discrete schemes are obtained

continuous symmetries of lagrangians and exact solutions of discrete equations one of the difficulties encountered when studying physical theories in discrete space time is that of describing the underlying continuous symmetries like lorentz or galilei invariance one of the ways of addressing this difficulty is to consider point transformations acting simultaneously on difference equations and lattices in a previous article we have classified ordinary difference schemes invariant under lie groups of point transformations the present article is devoted to an invariant lagrangian formalism for scalar single variable difference schemes the formalism is used to obtain first integrals and explicit exact solutions of the schemes equations invariant under two and three dimensional groups of lagrangian symmetries are considered

symmetry preserving discretization of ordinary differential equations large symmetry groups and higher order equations ordinary differential equations odes and ordinary difference systems o ss invariant under the actions of the lie groups and of projective transformations of the independent variables x and dependent variables y are constructed the odes are continuous limits of the o ss or conversely the o ss are invariant discretizations of the odes the invariant o ss are used to calculate numerical solutions of the invariant odes of order up to five the solutions of the invariant numerical schemes are compared to numerical solutions obtained by standard runge kutta methods and to exact solutions when available the invariant method performs at least as well as standard ones and much better in the vicinity of singularities of solutions

high order accurate finite difference schemes based on symmetry preservation abstract in this paper we present a mathematical approach that is based on modified equations and the method of equivariant moving frames for construction of high order accurate invariant finite difference schemes that preserve lie symmetry groups of underlying partial differential equations pdes in the proposed approach invariant or symmetry preserving numerical schemes with a desired or fixed order of accuracy are constructed from some non invariant base numerical schemes modified forms of pdes are used to improve the order of accuracy of existing schemes and these modified forms are obtained through addition of defect correction terms to the original forms of pdes these defect correction terms of modified pdes that are noted from truncation error analysis are either completely removed from schemes or their representation is significantly simplified by considering convenient moving frames this feature of the proposed method can especially be useful to avoid cumbersome numerical representations when high order schemes are developed from low order ones via the method of modified equations the proposed method is demonstrated via construction of invariant numerical schemes with fixed and higher order of accuracy for some common linear and nonlinear problems including the linear advection diffusion equation in 1d and 2d inviscid burgers equation and viscous burgers equation and the performance of these invariant numerical schemes is further evaluated our results indicate that such invariant numerical schemes obtained from existing base numerical schemes have the potential to significantly improve the quality of results not only in terms of desired higher order accuracy but also in the context of preservation of appropriate symmetry properties of underlying pdes

construction of partial difference schemes i the clairaut schwarz young theorem on the lattice difference operators in several variables do not always commute their commutation depends on the lattice equations in this paper we study the conditions over a lattice to allow commutativity of these difference operators and their implications for the discretization of partial differential equations