on critical point detection of digital shapes in this paper we present a nonlinear algorithm for critical point detection cpd of 2d digital shapes the algorithm eliminates the problems arising from curvature approximation and gaussian filtering in the existing algorithms based on the definition of critical level we establish a set of criteria for the design of an effective cpd algorithm for the first time by quantifying the critical level to the modified area confined by three consecutive pseudocritical points a simple but very effective algorithm is developed the comparison of our experimental results with those of many other cpd algorithms shows that the proposed algorithm is superior in that it provides a sequence of figures at every detail level and each has a smaller integral error than the others with the same number of critical points the experimental results on shapes with various complexities also show the algorithm is reliable and robust with regard to noise

mean curvature flow orbits moment maps given a compact riemannian manifold together with a group of isometries we discuss mcf of the orbits and some applications e g finding minimal orbits we then specialize to lagrangian orbits in kaehler manifolds in particular in the kaehler einstein case we find a relation between mcf and moment maps which for example proves that the minimal lagrangian orbits are isolated

on some results in fuzzy metric spaces abstract we define a hausdorff topology on a fuzzy metric space introduced by kramosil and michalek kybernetica 11 1975 326 334 and prove some known results of metric spaces including baire s theorem for fuzzy metric spaces

building blocks for designing arbitrarily smooth subdivision schemes with conic precision since subdivision schemes featured by high smoothness and conic precision are strongly required in many application contexts in this work we define the building blocks to obtain new families of non stationary subdivision schemes enjoying such properties to this purpose we firstly derive a non stationary extension of the lane riesenfeld algorithm and we exploit the resulting class of schemes to design a non stationary family of alternating primal dual subdivision schemes all featured by reproduction of 1 x e t x e t x t 0 i r then we focus our attention on interpolatory subdivision schemes with conic precision that can be obtained as a byproduct of the above classes in particular we present a novel construction of a family of non stationary interpolatory 2 n point schemes which generalizes the well known dubuc deslauriers family in such a way the n th n 2 family member reproduces 2 n 3 e t x e t x t 0 i r and keeps the original smoothness of its stationary counterpart unchanged non stationary extension of lane riesenfeld algorithm new family of alternating primal dual subdivision schemes reproducing conics new family of non stationary interpolatory 2 n point schemes reproducing conics explicit formulation and recurrence relations analysis of the main properties of the above families

a family of non stationary subdivision schemes reproducing exponential polynomials we present a family of non stationary subdivision schemes taking the exponential quasi spline as the basic limit function reaching up to interpolatory refinable functions the exponential quasi spline generalizes the exponential b spline the associated non stationary subdivision scheme can reproduce the desired number of exponential polynomials due to the built in parameter permitting a wide range of tension control our scheme can avoid the undesirable artifact generated by usual non stationary interpolatory schemes on the irregularly spaced region of control points under a natural condition on the rate of convergence of the shape parameters all the subdivision schemes in the family have the same smoothness and approximation order as those of the asymptotically equivalent stationary schemes we present numerical examples to verify that our subdivision schemes have high flexibility and versatility in design

reductions of minimal lagrangian submanifolds with symmetries let m be a fano manifold equipped with a kahler form omega in 2 pi c 1 m and k a connected compact lie group acting on m as holomorphic isometries in this paper we show the minimality of a k invariant lagrangian submanifold l in m with respect to a globally conformal kahler metric is equivalent to the minimality of the reduced lagrangian submanifold l 0 l k in a kahler quotient m 0 with respect to the hsiang lawson metric furthermore we give some examples of kahler reductions by using a circle action obtained from a cohomogenenity one action on a kahler einstein manifold of positive ricci curvature applying these results we obtain several examples of minimal lagrangian submanifolds via reductions

corner detection on digital curves based on local symmetry of the shape abstract the detection of corners on digital curves is a problem of important in shape analysis this paper describes a new method for detecting corners based on local symmetry of the shape the proposed method consists of two steps the first is the computation of symmetry measure for symmetry relative to a line at every point on a digital curve the second is the location of reasonable local maxima of the symmetry measure as corners the effectiveness of the new method is illustrated with the results of experiments for a number of different types of digital curves

wavelet based corner detection abstract a non parametric algorithm for detecting and locating corners of planar curves is proposed the algorithm is based on the multiscale wavelet transform of the orientation of the curve which can effectively utilize both the information of local extrema positions and magnitudes of the transform results the corner candidates can then be selected easily based on this information according to the angle preserving concept intrinsic ratios of several corner models have been derived and used to evaluate the corner candidates the corner angles can also be obtained during these processes to make the evaluation process robust a masking algorithm is proposed experiments depict that our detector is more effective than the single scale corner detectors while is more efficient than the multiscale corner detector by rattarangsi and chin proc int conf pattern recognition pp 923 930 1990

convexity preserving interpolatory subdivision with conic precision the paper is concerned with the problem of shape preserving interpolatory subdivision for arbitrarily spaced planar input data an efficient non linear subdivision algorithm reproducing conic sections and respecting the convexity properties of the initial data is here presented significant numerical examples are included to illustrate the effectiveness of the proposed method and the smoothness of the limit curves

locating perceptually salient points on planar curves this paper describes the underlying ideas and algorithmic details of a computer program that performs at a human level of competence for a significant subset of the curve partitioning task it extends and rounds out the technique and philosophical approach originally presented by fischler and bolles 1986 in particular it provides a unified strategy for selecting and dealing with interactions between salient points even when these points are salient at different scales of resolution experimental results are presented involving on the order of 1000 real and synthetically generated images

reproduction of exponential polynomials by multivariate non stationary subdivision schemes with a general dilation matrix we study scalar multivariate non stationary subdivision schemes with a general integer dilation matrix we characterize the capability of such schemes to reproduce exponential polynomials in terms of simple algebraic conditions on their symbols these algebraic conditions provide a useful theoretical tool for checking the reproduction properties of existing schemes and for constructing new schemes with desired reproduction capabilities and other enhanced properties we illustrate our results with several examples

a curve bend function based method to characterize contour shapes in this paper a new curve bend function cbf based method is presented to extract main features of a contour it utilizes two quantities one called a curve bend angle cba and the other known as a type coefficient of curve segment defined on a contour the cba measures the bending degree at each point on the contour while the type coefficient characterizes the properties convexity and concavity of simple curve segments the cbf combines the advantages of the two quantities and provides an overall description of a contour a local maximum or minimum of the cbf corresponds to a critical point of the contour and the peak value is the cosine of the cba hence the problem of locating critical points of a contour is reduced to the problem of finding peak points of the cbf the sign of a peak value indicates the related cba being an inner or outer angle our experimental results and comparisons between the cbf method and rosenfeld johnston s method show that our method is effective and accurate in generating the description of a contour

a family of non uniform subdivision schemes with variable parameters for curve design in this paper we present non uniform subdivision schemes with variable parameter sequences a locally different tension parameter is set at each edge of the initial control polygon to control locally the shape of the resulting curve such that the scheme becomes non uniform due to the variable parameters the scheme can reproduce locally different analytic curves such as conics lissajous trigonometric and catenary curves hence blending curves including such analytic components can be successfully generated we discuss the convergence and smoothness of the proposed non uniform schemes and present some numerical results to demonstrate their advantages in geometric modeling furthermore as an application we propose a chamfering algorithm which can be used in designing automobile and mechanical products

a discrete geometry approach for dominant point detection we propose two fast methods for dominant point detection and polygonal representation of noisy and possibly disconnected curves based on a study of the decomposition of the curve into the sequence of maximal blurred segments 2 starting from results of discrete geometry 3 4 the notion of maximal blurred segment of width n 2 has been proposed well adapted to possibly noisy curves the first method uses a fixed parameter that is the width of considered maximal blurred segments the second method is deduced from the first one based on a multi width approach to obtain a non parametric method that uses no threshold for working with noisy curves comparisons with other methods in the literature prove the efficiency of our approach thanks to a recent result 5 concerning the construction of the sequence of maximal blurred segments the complexity of the proposed methods is o nlogn an application of vectorization is also given in this paper

polynomial based non uniform interpolatory subdivision with features control starting from a well known construction of polynomial based interpolatory 4 point schemes in this paper we present an original affine combination of quadratic polynomial samples that leads to a non uniform 4 point scheme with edge parameters this blending type formulation is then further generalized to provide a powerful subdivision algorithm that combines the fairing curve of a non uniform refinement with the advantages of a shape controlled interpolation method and an arbitrary point insertion rule the result is a non uniform interpolatory 4 point scheme that is unique in combining a number of distinctive properties in fact it generates visually pleasing limit curves where special features ranging from cusps and flat edges to point edge tension effects may be included without creating undesired undulations moreover such a scheme is capable of inserting new points at any positions of existing intervals so that the most convenient parameter values may be chosen as well as the intervals for insertion such a fully flexible curve scheme is a fundamental step towards the construction of high quality interpolatory subdivision surfaces with features control

quantum ergodicity and symmetry reduction abstract we study the spectral and ergodic properties of schrodinger operators on a compact connected riemannian manifold m without boundary in case that the examined system possesses certain symmetries more precisely if m carries an isometric and effective action of a compact connected lie group g we prove a generalized equivariant version of the semiclassical weyl law with an estimate for the remainder using a semiclassical functional calculus for h dependent functions and relying on recent results on singular equivariant asymptotics we then deduce an equivariant quantum ergodicity theorem under the assumption that the symmetry reduced hamiltonian flow on the principal stratum of the singular symplectic reduction of m is ergodic in particular we obtain an equivariant version of the shnirelman zelditch colin de verdiere theorem as well as a representation theoretic equidistribution theorem if m g is an orbifold similar results were recently obtained by kordyukov when g is trivial one recovers the classical results

algebraic conditions on non stationary subdivision symbols for exponential polynomial reproduction we present an accurate investigation of the algebraic conditions that the symbols of a non singular univariate binary non stationary subdivision scheme should fulfill in order to reproduce spaces of exponential polynomials a subdivision scheme is said to possess the property of reproducing exponential polynomials if for any initial data uniformly sampled from some exponential polynomial function the scheme yields the same function in the limit the importance of this property is due to the fact that several curves obtained by combinations of exponential polynomials such as conic sections spirals or special trigonometric and hyperbolic functions are considered of interest in geometric modeling since the space of exponential polynomials trivially includes standard polynomials this work extends the theory on polynomial reproduction to the non stationary context a significant application of the derived algebraic conditions on the subdivision symbols is the construction of new non stationary subdivision schemes with specific reproduction properties

shape controlled interpolatory ternary subdivision ternary subdivision schemes compare favorably with their binary analogues because they are able to generate limit functions with the same or higher smoothness but smaller support in this work we consider the two issues of local tension control and conics reproduction in univariate interpolating ternary refinements we show that both these features can be included in a unique interpolating 4 point subdivision method by means of non stationary insertion rules that do not affect the improved smoothness and locality of ternary schemes this is realized by exploiting local shape parameters associated with the initial polyline edges

gromov hausdorff convergence of non archimedean fuzzy metric spaces we introduce the notion of the gromov hausdorff fuzzy distance between two non archimedean fuzzy metric spaces in the sense of kramosil and michalek basic properties involving convergence and the fuzzy version of the completeness theorem are presented we show that the topological properties induced by the classic gromov hausdorff distance on metric spaces can be deduced from our approach

multiresolution subdivision snakes we present a new family of snakes that satisfy the property of multiresolution by exploiting subdivision schemes we show in a generic way how to construct such snakes based on an admissible subdivision mask we derive the necessary energy formulations and provide the formulas for their efficient computation depending on the choice of the mask such models have the ability to reproduce trigonometric or polynomial curves they can also be designed to be interpolating a property that is useful in user interactive applications we provide explicit examples of subdivision snakes and illustrate their use for the segmentation of bioimages we show that they are robust in the presence of noise and provide a multiresolution algorithm to enlarge their basin of attraction which decreases their dependence on initialization compared to singleresolution snakes we show the advantages of the proposed model in terms of computation and segmentation of structures with different sizes

a parallel algorithm for detecting dominant points on multiple digital curves abstract a new parallel algorithm is proposed for detecting corners of multiple planar curves or shapes this algorithm is based on an analysis of morphological residues and corner characteristics the method is different from traditional chain code based corner detection algorithms which associate with a chain coding and floating point computation for multiple objects in an image the proposed algorithm treats them uniformly and applies simultaneously on curves and shapes of multiple objects the experiments demonstrate that the algorithm can be effectively executed on an simd parallel computer a boundary constrained morphological method is presented for filling closed curves into shapes which are subsequently used for corner detection

hyperbolic forms of ternary non stationary subdivision schemes originated from hyperbolic b splines in this work ternary non stationary subdivision schemes based on hyperbolic b spline basis functions have been presented the proposed hyperbolic ternary three point and four point subdivision schemes give pleasing as well as consistent curves with the control polygons as compared to the existing non stationary subdivision schemes obtained from trigonometric b splines the main speciality of the hyperbolic schemes is that they can reproduce hyperbolas and parabolas quite efficiently which has been demonstrated with the help of curvature plots

unified and extended form of three types of splines the three types refer to polynomial trigonometric and hyperbolic splines in this paper we unify and extend them by a new kind of spline ue spline for short defined over the space cos wt sin wt 1 t t l where l is an arbitrary nonnegative integer w is a frequency sequence w i a i a i r existing splines such as usual polynomial b splines cb splines hb splines nuat splines ah splines fb splines and the third form fb splines etc are all special cases of ue splines ue splines inherit most properties of usual polynomial b splines and enjoy some other advantageous properties for modelling they can exactly represent classical conics the catenary the helix and even the eight curve a kind of snake like curves etc

optimized polygonal approximation by dominant point deletion an algorithm for polygonal approximation based on dominant point dp deletion is presented in this paper the algorithm selects an initial set of dps and starts eliminating them one by one depending upon the error associated with each dp the associated error value is based on global measure a local optimization of few neighboring points is performed after each deletion although the algorithm does not guarantee an optimal solution the combination of local and global optimization is expected to produce optimal results the algorithm is extensively tested on various shapes with varying number of dps and error threshold in general optimal results were observed for about 96 of the times a good comparative study is also presented in this paper

fuzzy uniform structures and continuous t norms we introduce and discuss a notion of fuzzy uniform structure that provides a direct link with the classical theory of uniform spaces more exactly for each continuous t norm we prove that the category of all fuzzy uniform spaces in our sense and fuzzy uniformly continuous mappings is isomorphic to the category of uniform spaces and uniformly continuous mappings by means of a covariant functor moreover we also describe the inverse functor and then we discuss completeness and completion of these fuzzy uniform structures it follows from our results that each fuzzy uniform structure in our sense induces a hutton 0 1 uniformity and conversely each hutton 0 1 uniformity induces a fuzzy uniform structure in our sense

non parametric dominant point detection abstract a new method for detecting dominant points is presented it does not require any input parameter and the dominant points obtained by this method remain relatively the same even when the object curve is scaled or rotated in this method for each boundary point a support region is assigned to the point based on its local properties each point is then smoothed by a gaussian filter with a width proportional to its determined support region a significance measure for each point is then computed dominant points are finally obtained through nonmaximum suppression unlike other dominant point detection algorithms which are sensitive to scaling and rotation of the object curve the new method will overcome this difficulty furthermore it is robust in the presence of noise the proposed new method is compared to the teh chin algorithm see c teh and r t chin ieee trans pattern anal mach intell 11 859 872 1989 in terms of the computational complexity the approximation errors and the number of detected dominant points of an object curve

a robust method for scale independent detection of curvature based criticalities and intersections in line drawings a novel two phase iterative method is proposed to identify curvature based criticalities and intersections in line drawings the method is based on evaluating a field obtained from the image via diffusion using adaptively changing ellipse shaped analysis windows the deviation of the average field strength values within the analysis windows from that of an external reference field is used as a metric to evaluate the criticality of a region the external reference field is computed from an image of a straight line the experimental results depict that the method is effective in detecting curvature based criticalities and intersections even for noisy and disconnected drawings as well as drawings drawn with a variety of brush characteristics such as glass ocean ripple scatter stamp strokes effects our method can be employed as a part of a design grammar interpreter highlightsthe method detects curvature based criticalities and intersections efficiently the method is effective for the noisy and disconnected line drawings the method is effective for line drawings with a variety of brush characteristics the method is utilized to represent shapes in a design grammar interpreter features in varying scales can be detected in digital drawings

reduced resolution and scale space for dominant feature detection in contours abstract analysis of contours or curves at various scales or levels of smoothing scale space is an important tool in curve segmentation and feature detection in this paper we propose the application of the hierarchical discrete correlation algorithm for efficiently calculating and creating a scale space of curves in conjunction with this procedure we also investigate the use of the reduced resolution or gaussian pyramid representation of the set of smoothed curves as the basis for initially localizing and detecting features we also create an approximation to the above algorithms that is computationally less expensive finally we propose a new inter scale method for curve segmentation and feature detection based on the motion of a curve through scale space

from approximating subdivision schemes for exponential splines to high performance interpolating algorithms in this work we construct three novel families of approximating subdivision schemes that generate piecewise exponential polynomials and we show how to convert these into interpolating schemes of great interest in curve design for their ability to reproduce important analytical shapes and to provide highly smooth limit curves with a controllable tension in particular throughout this paper we will focus on the derivation of 6 point interpolating schemes that turn out to be unique in combining vital ingredients like c 2 continuity simplicity of definition ease of implementation user independency tension control and ability to reproduce salient trigonometric and transcendental curves

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

dual univariate interpolatory subdivision of every arity algebraic characterization and construction abstract a new class of univariate stationary interpolatory subdivision schemes of dual type is presented as opposed to classical primal interpolatory schemes these new schemes have masks with an even number of elements and are not step wise interpolants a complete algebraic characterization which covers every arity is given in terms of identities of trigonometric polynomials associated to the schemes this characterization is based on a necessary condition for refinable functions to have prescribed values at the nodes of a uniform lattice as a consequence of the poisson summation formula a strategy for the construction is then showed alongside meaningful examples for applications that have comparable or even superior properties in terms of regularity length of the support and or polynomial reproduction with respect to the primal counterparts

non uniform interpolatory subdivision surface abstract this paper presents an interpolatory subdivision scheme with non uniform parametrization for arbitrary polygon meshes with arbitrary manifold topology this is the first attempt to generalize the non uniform four point interpolatory curve subdivision to surface with extraordinary points the scheme is constructed from the inspiration of the relation between the non uniform four point interpolatory subdivision scheme and the non uniform b spline refinement rule numerical examples and comparisons with the uniform interpolatory subdivision schemes indicate that the quality of the limit surface can be improved by using non uniform parameter values for non uniform initial data

boundary based corner detection using neural networks abstract in this paper we present a novel boundary based corner detection approach using artificial neural networks anns two neural networks are proposed one for detecting corner points with high curvature and the other for detecting tangent points and inflection points that generally have low curvature for a given boundary point p i the first ann uses the normalized coordinates of points on the forward arm neighboring points succeeding p i or on the backward arm neighboring points preceding p i of the point p i as the input vector the output feature of the network is the angle of the forward arm or backward arm with respect to the x axis the boundary point with sufficiently small angle between the forward and backward arms is identified as a corner since the feature points of tangency and inflection have relatively low curvature the signs of curvature rather than the magnitude of curvature for points in the neighborhood of p i are used as the input vector to the second ann the curvature sign at each boundary point is derived from the outputs of the first ann the outputs of the second ann only respond to the sign patterns of tangent points and inflection points by using both anns all features of corners tangent points and inflection points can be extracted from the boundary of any arbitrary shape experimental results have shown that the proposed anns have good detection and localization for objects in random orientations and with moderate scale changes

scale based detection of corners of planar curves a technique for detecting and localizing corners of planar curves is proposed the technique is based on gaussian scale space which consists of the maxima of absolute curvature of the boundary function presented at all scales the scale space of isolated simple and double corners is first analyzed to investigate the behavior of scale space due to smoothing and interactions between two adjacent corners the analysis shows that the resulting scale space contains line patterns that either persist terminate or merge with a neighboring line next the scale space is transformed into a tree that provides simple but concise representation of corners at multiple scales finally a multiple scale corner detection scheme is developed using a coarse to fine tree parsing technique the parsing scheme is based on a stability criterion that states that the presence of a corner must concur with a curvature maximum observable at a majority of scales experiments were performed to show that the scale space corner detector is reliable for objects with multiple size features and noisy boundaries and compares favorably with other corner detectors tested

some properites of fuzzy metric spaces abstract we prove that the topology induced by any complete fuzzy metric space in the sense of george and veeramani is completely metrizable we also show that every separable fuzzy metric space admits a precompact fuzzy metric and that a fuzzy metric space is compact if and only if it is precompact and complete finally we generalize among other classical results the niemytzki tychonoff theorem to fuzzy metric spaces

cubic bezier approximation of a digitized curve in this paper we present an efficient technique for piecewise cubic bezier approximation of digitized curve an adaptive breakpoint detection method divides a digital curve into a number of segments and each segment is approximated by a cubic bezier curve so that the approximation error is minimized initial approximated bezier control points for each of the segments are obtained by interpolation technique i e by the reverse recursion of de castaljau s algorithm two methods two dimensional logarithmic search algorithm tdlsa and an evolutionary search algorithm esa are introduced to find the best fit bezier control points from the approximate interpolated control points esa based refinement is proved to be better experimentally experimental results show that bezier approximation of a digitized curve is much more accurate and uses less number of points compared to other approximation techniques

generalized daubechies wavelet families we present a generalization of the orthonormal daubechies wavelets and of their related biorthogonal flavors cohen daubechies feauveau 9 7 our fundamental constraint is that the scaling functions should reproduce a predefined set of exponential polynomials this allows one to tune the corresponding wavelet transform to a specific class of signals thereby ensuring good approximation and sparsity properties the main difference with the classical construction of daubechies is that the multiresolution spaces are derived from scale dependent generating functions however from an algorithmic standpoint mallat s fast wavelet transform algorithm can still be applied the only adaptation consists in using scale dependent filter banks finite support ensures the same computational efficiency as in the classical case we characterize the scaling and wavelet filters construct them and show several examples of the associated functions we prove that these functions are square integrable and that they converge to their classical counterparts of the corresponding order

the hausdorff fuzzy quasi metric removing the condition of symmetry in the notion of a fuzzy pseudo metric in kramosil and michalek s sense one has the notion of a fuzzy quasi pseudo metric then for each fuzzy quasi pseudo metric on a set x we construct a fuzzy quasi pseudo metric on the collection of all nonempty subsets of x called the hausdorff fuzzy quasi pseudo metric we investigate several properties of this structure and present several illustrative examples as well as an application to the domain of words the notion of hausdorff fuzzy quasi pseudo metric when quasi pseudo metric fuzziness is considered in the sense of george and veeramani is also discussed

on detecting dominant points abstract detecting dominant points is a crucial preprocessing step for shape recognition and point based motion estimation polygonal approximation has been a commonly used approach in detecting dominant points this paper presents two alternatives which detect stable dominant points in the first method we find a set of positive maximum and negative minimum curvature points along the gaussian smoothed boundary followed by a split and merge polygonal approximation algorithm the resulting break points vertices of the approximated polygon are the dominant points experimental results show that dominant points obtained by this method are less sensitive to the orientation of the boundary than other polygonal approximation algorithms in the sense that the number and the location of the dominant points along the contour remain relatively unchanged in the second method we smooth a boundary by a gaussian filter using various widths until the extreme curvature points remain relatively unchanged for a range of filter widths the resulting extreme curvature points which are stable to orientation and a reasonable range of scaling are the dominant points

exponential pseudo splines looking beyond exponential b splines abstract pseudo splines are a rich family of functions that allows the user to meet various demands for balancing polynomial reproduction i e approximation power regularity and support size such a family includes as special members b spline functions universally known for their usefulness in different fields of application when replacing polynomial reproduction by exponential polynomial reproduction a new family of functions is obtained this new family is here constructed and called the family of exponential pseudo splines it is the nonstationary counterpart of polynomial pseudo splines and includes exponential b splines as a special subclass in this work we provide a computational strategy for deriving the explicit expression of the laurent polynomial sequence that identifies the family of exponential pseudo spline nonstationary subdivision schemes for this family we study its symmetry properties and perform its convergence and regularity analysis finally we also show that the family of primal exponential pseudo splines fills in the gap between exponential b splines and interpolatory cardinal functions this extends the analogous property of primal pseudo spline stationary subdivision schemes

polygonal approximation of digital planar curves through break point suppression this paper presents a new algorithm that detects a set of dominant points on the boundary of an eight connected shape to obtain a polygonal approximation of the shape itself the set of dominant points is obtained from the original break points of the initial boundary where the integral square is zero for this goal most of the original break points are deleted by suppressing those whose perpendicular distance to an approximating straight line is lower than a variable threshold value the proposed algorithm iteratively deletes redundant break points until the required approximation which relies on a decrease in the length of the contour and the highest error is achieved a comparative experiment with another commonly used algorithm showed that the proposed method produced efficient and effective polygonal approximations for digital planar curves with features of several sizes

fuzzy stone ccech type compactifications abstract in this paper we investigate the fuzzy sets with lattice precisely fuzzy lattice values some new global properties on the weakly induced spaces and induced spaces are obtained based on these results we take the induced space of real unit interval i as new basic space and establish a reasonable theory of stone cech compactifications via fuzzy imbedding theory 5 and n compactness 10 12 for so called t 3 1 2 and weakly induced spaces its stone cech type compactification always exists we give a counterexample to show that the previous stone cech compactification 6 is not the largest t2 compactification of a given space compared with this we now have a positive results for the new stone cech type compactification moreover although the preorder among the compactifications for a given space does not generally satisfy the anti symmetric law we fortunately establish uniqueness of the largest t2 type compactification among the compactifications for a given space

analysis of asymptotically equivalent binary subdivision schemes non stationary uniform subdivision schemes are presented including some important examples as the up function and exponential splines the multivariate analysis of subdivision schemes is extended to a class of non stationary schemes which are asymptotically equivalent to converging stationary or non stationary schemes

corner point detection using nest moving average abstract corners are rich in high frequency components a low pass filter suppresses corners and results in a smooth curve therefore corner points have a larger amount of shift than other points and can easily be detected by a low pass filter experiments suggest that a moving average can do this job suitably with a low computational load and good locality in spatial domain a moving average with a smaller size window is used to perprocess the digital boundary with quantization error or noise

the hausdorff fuzzy metric on compact sets we propose a method for constructing a hausdorff fuzzy metric on the set of the nonempty compact subsets of a given fuzzy metric space in the sense of george and veeramani we discuss several important properties as completeness completion and precompactness some illustrative examples are given

curve fitting approach for tangent angle and curvature measurements abstract the measures of tangent angle and curvature of a digital curve play an important role in image analysis such as corner detection 1d shape representation and shape signature in the generalized hough transform instead of using the discrete measurement approach the least squares method is employed to fit known digital points to two cubic polynomial functions one with y f x that minimizes the sum of the vertical distances and the other with x g y that minimizes the sum of the horizontal distances from the known points to the approximated curve the tangent angle and curvature are therefore directly computable from the first and second order derivatives of the continuous functions hybrid procedures are also proposed to select the better curve from f and g for accurate evaluation of tangent angle and curvature experimented results on both analytic curves and real world images are included

techniques for assessing polygonal approximations of curves given the enormous number of available methods for finding polygonal approximations to curves techniques are required to assess different algorithms some of the standard approaches are shown to be unsuitable if the approximations contain varying numbers of lines instead we suggest assessing an algorithm s results relative to an optimal polygon and describe a measure which combines the relative fidelity and efficiency of a curve segmentation we use this measure to compare the application of 23 algorithms to a curve first used by teh and chin 1989 their integral square errors ises are assessed relative to the optimal ise in addition using an example of pose estimation it is shown how goal directed evaluation can be used to select an appropriate assessment criterion

on the detection of dominant points on digital curves a parallel algorithm is presented for detecting dominant points on a digital closed curve the procedure requires no input parameter and remains reliable even when features of multiple sizes are present on the digital curve the procedure first determines the region of support for each point based on its local properties then computes measures of relative significance e g curvature of each point and finally detects dominant points by a process of nonmaximum suppression this procedure leads to the observation that the performance of dominant points detection depends not only on the accuracy of the measure of significance but also on the precise determination of the region of support this solves the fundamental problem of scale factor selection encountered in various dominant point detection algorithms the inherent nature of scale space filtering in the procedure is addressed and the performance of the procedure is compared to those of several other dominant point detection algorithms using a number of examples

limits of level and parameter dependent subdivision schemes in this paper we present a new matrix approach for the analysis of subdivision schemes whose non stationarity is due to linear dependency on parameters whose values vary in a compact set indeed we show how to check the convergence in c r s and determine the holder regularity of such level and parameter dependent schemes efficiently via the joint spectral radius approach the efficiency of this method and the important role of the parameter dependency are demonstrated on several examples of subdivision schemes whose properties improve the properties of the corresponding stationary schemes

compactification of lattice valued convergence spaces we define compactness for stratified lattice valued convergence spaces and show that a tychonoff theorem is true further a generalization of the classical richardson compactification is given this compactification has a universal property

a family of smooth and interpolatory basis functions for parametric curve and surface representation interpolatory basis functions are helpful to specify parametric curves or surfaces that can be modified by simple user interaction their main advantage is a characterization of the object by a set of control points that lie on the shape itself i e curve or surface in this paper we characterize a new family of compactly supported piecewise exponential basis functions that are smooth and satisfy the interpolation property they can be seen as a generalization and extension of the keys interpolation kernel using cardinal exponential b splines the proposed interpolators can be designed to reproduce trigonometric hyperbolic and polynomial functions or combinations of them we illustrate the construction and give concrete examples on how to use such functions to construct parametric curves and surfaces

corner detection and interpretation on planar curves using fuzzy reasoning the problem of corner detection on planar curves is examined based on human perception of local graphic features first a set of fuzzy patterns of contour points are established then corner detection is characterized as a fuzzy classification problem that contains three stages evaluation classification and location compared with existing methods the proposed approach is superior in that it explains the curve instead of simple labeling and it performs based on human perception experimental results on shapes of various complexities are presented the performance with respect to noise is also addressed

reliable polygonal approximations of imaged real objects through dominant point detection abstract the problem of dominant point detection is posed taking into account what usually happens in practice the algorithms found in the literature often prove their performance with laboratory contours but the shapes in real images present noise quantization and high inter and intra shape variability these effects are analyzed and solutions to them are proposed we will also focus on the conditions for an efficient few points and precise low error dominant point extraction that preserves the original shape a measurement of the committed error optimization error e 0 that takes into account both aspects is defined for studying this feature

fuzzy quasi pseudometrics on algebraic structures abstract in this work we state a number of theorems about fuzzy quasi pseudometrizable algebraic structures our most useful results are 1 a fuzzy semitopological group whose topology is induced by a left invariant fuzzy quasi pseudometric then it is a fuzzy paratopological topological group 2 if the topology on a semigroup s is induced by an invariant fuzzy quasi pseudometric then s is a fuzzy topological semigroup and 3 the same conclusion is valid for a left invariant fuzzy quasi pseudometric on a monoid g such that the left translations are open and the right translations are continuous at the identity e of g by means of the standard fuzzy quasi pseudometric m d associated to a quasi pseudometric d our results apply in the case of semitopological groups semigroups and monoids in order to obtain new results that allow us to generalize and to strengthen previous outcomes

a shape recognition scheme based on relative distances of feature points from the centroid abstract in this paper a simple but efficient scheme to recognize shapes is proposed in our scheme all of the boundary points are used to calculate the centroid of an object then the distance between each feature point and the centroid is computed we have shown that the time complexity of our shape matching algorithm is o m where m is the number of feature points the most important advantage of our method is its independence to translation rotation and scaling of objects

regularity of non stationary subdivision a matrix approach in this paper we study scalar multivariate non stationary subdivision schemes with integer dilation matrix m and present a unifying general approach for checking their convergence and for determining their holder regularity latter in the case m mi m ge 2 m mi m 2 the combination of the concepts of asymptotic similarity and approximate sum rules allows us to link stationary and non stationary settings and to employ recent advances in methods for exact computation of the joint spectral radius as an application we prove a recent conjecture by dyn et al on the holder regularity of the generalized daubechies wavelets we illustrate our results with several examples

on interpolatory subdivision from approximating subdivision scheme this paper deals with some theoretical analysis of one strategy for deducing interpolatory from approximating subdivision scheme first we derive the explicit formula for one class of interpolatory subdivision schemes deduced from approximating subdivision schemes this explicit formula paves a way for systematic analysis of interpolatory subdivisions derived by the strategy we analyze the polynomial generation property of interpolatory subdivision derived by the strategy moreover the explicit formula also has the merit of providing a method for designing new interpolatory and approximating subdivisions and some new subdivision schemes are constructed

curvature and tangential deflection of discrete arcs a theory based on the commutator of scatter matrix pairs and its application to vertex detection in planar shape data this paper introduces a new theory for the tangential deflection and curvature of plane discrete curves our theory applies to discrete data in either rectangular boundary coordinate or chain coded formats its rationale is drawn from the statistical and geometric properties associated with the eigenvalue eigenvector structure of sample covariance matrices specifically we prove that the nonzero entry of the commutator of a piar of scatter matrices constructed from discrete arcs is related to the angle between their eigenspaces and further we show that this entry is in certain limiting cases also proportional to the analytical curvature of the plane curve from which the discrete data are drawn these results lend a sound theoretical basis to the notions of discrete curvature and tangential deflection and moreover they provide a means for computationally efficient implementation of algorithms which use these ideas in various image processing contexts as a concrete example we develop the commutator vertex detection cvd algorithm which identifies the location of vertices in shape data based on excessive cummulative tangential deflection and we compare its performance to several well established corner detectors that utilize the alternative strategy of finding approximate curvature extrema

standard fuzzy uniform structures based on continuous t norms this paper deals with fuzzy uniform structures previously introduced by the authors fuzzy uniform structures and continuous t norms fuzzy sets syst 161 2009 1011 1021 our approach involves a covariant functor j from the category of fuzzy uniform spaces and fuzzy uniformly continuous mappings in our sense to the category of uniform spaces and uniformly continuous mappings we show that j is well behaved with respect to some significant fuzzy uniform concepts and its behavior provides a method to introduce notions of fine fuzzy uniform structure and stone cech fuzzy compactification in this context our method also applies to obtain fuzzy versions of some classical results on topological algebra and hyperspaces the case of quasi uniform structures is also analyzed

on fuzzy metric groups we introduce and study a notion of fuzzy metric group which permits us to obtain satisfactory extensions of classical theorems on metric groups to the fuzzy setting we also investigate properties of the quotient subgroups of a fuzzy metric group

non parametric planar shape representation based on adaptive curvature functions this paper presents a non parametric method to extract a very short feature vector from the curvature function of a planar shape curvature is adaptively calculated using a new procedure that removes noise from the contour without missing relevant points then its fourier transform is projected onto a set of vectors which have been chosen to be as representative as possible to obtain the similarity between the input object and each vector of the set these similarity values are the elements of the feature vector the proposed method is very fast and classification has proven that the representation is good

the detection of dominant points on digital curves by scale space filtering the detection of dominant points is an important preprocessing step for shape recognition an effective method of scale space filtering with a gaussian kernel is introduced to detect dominant points on digital curves the conventional polygonal approximation algorithms are time consuming and need input parameter tuning for gaussian smoothing the noise and quantization error also they are sensitive to scaling and rotation of the object curve the above dit culty can be overcome by finding out the dominant points at each scale by scale space filtering by tracing back the dominant point contours in the scale space image the stable cardinal curvature points can be detected very accurately this new method requires no input parameters and the resultant dominant points do not change under translation rotation and scaling meanwhile a fast convolution algorithm is proposed to detect the dominant points at each scale dominant points scale space gaussian smoothing curvature

classification of invariant valuations on the quaternionic plane abstract we describe the orbit space of the action of the group sp 2 sp 1 on the real grassmann manifolds gr k h 2 in terms of certain quaternionic matrices of moore rank not larger than 2 we then give a complete classification of valuations on the quaternionic plane h 2 which are invariant under the action of the group sp 2 sp 1

dual univariate m ary subdivision schemes of de rham type in this paper we present an algebraic perspective of the de rham transform of a binary subdivision scheme and propose an elegant strategy for constructing dual m ary approximating subdivision schemes of de rham type starting from two primal schemes of arity m and 2 respectively on the one hand this new strategy allows us to show that several existing dual corner cutting subdivision schemes fit into a unified framework on the other hand the proposed strategy provides a straightforward algorithm for constructing new dual subdivision schemes having higher smoothness and higher polynomial reproduction capabilities with respect to the two given primal schemes

interpolation through an iterative scheme abstract a new method of interpolation is generated through an iterative scheme on dyadic rationals the process is linear local and produces almost twice differentiable functions a fundamental interpolating function is analyzed error formulas are derived the method is applied for curve fitting

a new algorithm for dominant points detection and polygonization of digital curves a new algorithm for detecting dominant points and polygonal approximation of digitized closed curves is presented it uses an optimal criterion for determining the region of support of each boundary point and a new mechanism for selecting the dominant points the algorithm does not require an input parameter and can handle shapes that contain features of multiple sizes efficiently in addition the approximating polygon preserves the symmetry of the shape

a rotationally invariant two phase scheme for corner detection a rotationally invariant two phase scheme is proposed for the detection of corners in the first phase the curvature of each point on the curve is calculated based on the coordinates of five consecutive skipped points and the assumption that thes five points form a b spline curve the points with significant curvatures are collected as candidate corners during the second phase the candidate corners are verified for deviations in the global trend of the curve and the candidate corners due to quantization are removed using a self adjusting dot product window application of the propsed method on the examples given in the literature shows that it is both minimal in computation time and accurate and stable in the detected position of corners with the proposed method recognition of an object can be convenient since the corners detected are rotationally invariant

on completable fuzzy metric spaces in this paper we construct a non completable fuzzy metric space in the sense of george and veeramani which allows to answer an open question related to continuity on the real parameter t in addition the constructed space is not strong non archimedean