the h infty calculus and sums of closed operators we develop a very general operator valued functional calculus for operators with an h infty calculus we then apply this to the joint functional calculus of two sectorial operators when one has an h infty calculus using this we prove theorem of dore venni type on sums of sectorial operators and apply our results to the problem of l p maximal regularity our main assumption is the r boundedness of certain sets of operators and therefore methods from the geometry of banach spaces are essential here in the final section we exploit the special banach space structure of l 1 spaces and c k spaces to obtain some more detailed results in this setting

the h calculus and sums of closed operators we develop a very general operator valued functional calcu lus for operators with an h 1 calculus we then apply this to the joint functional calculus of two commuting sectorial operators when one has an h 1 calculus using this we prove theorem of dore venni type on sums of commuting sectorial operators and apply our results to the problem of lp maximal regularity our main assumption is the r boundedness of certain sets of operators and therefore methods from the geometry of ba nach spaces are essential here in the final section we exploit the special banach space structure of l1 spaces and c k spaces to obtain some more detailed results in this setting

a finite element method for equal width equation abstract a numerical solution of the equal width ew equation is obtained using space splitting technique and quadratic b spline galerkin finite element method solitary wave motion interaction of two solitary waves wave undulation and wave generation are studied using the proposed method comparisons are made with analytical solutions and with some spline finite element method calculations at selected times accuracy and efficiency are discussed by computing the numerical conserved laws and l 2 l error norms

interpolation of weighted and vector valued hardy spaces real and complex interpolation methods when applied to the couple hpo e0 wo hp el wl give what is expected if eo and e1 are quasi banach lattices of measurable functions satisfying certain mild conditions and if log wi powj l p1 e bmo wo w1 being weights on the unit circle the last condition is in fact necessary it is expected of course that the resulting spaces coincide with the subspaces of analytic functions in the corresponding interpolation spaces for the couple lpo eo wo lp e1 w1

active contours without edges we propose a new model for active contours to detect objects in a given image based on techniques of curve evolution mumford shah 1989 functional for segmentation and level sets our model can detect objects whose boundaries are not necessarily defined by the gradient we minimize an energy which can be seen as a particular case of the minimal partition problem in the level set formulation the problem becomes a mean curvature flow like evolving the active contour which will stop on the desired boundary however the stopping term does not depend on the gradient of the image as in the classical active contour models but is instead related to a particular segmentation of the image we give a numerical algorithm using finite differences finally we present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable also the initial curve can be anywhere in the image and interior contours are automatically detected

global detection of salient convex boundaries as an important geometric property of many structures or structural components convexity plays an important role in computer vision and image understanding in this paper we describe a general approach that can force various edge grouping algorithms to detect only convex structures from a set of boundary fragments the basic idea is to remove some fragments and fragment connections so that on the remaining ones a prototype edge grouping algorithm that detects closed boundaries without the convexity constraint can only produce convex closed boundaries we show that this approach takes polynomial time and preserves the grouping optimality by not excluding any valid convex boundary from the search space choosing the recently developed ratio contour algorithm as the prototype grouping algorithm we develop a new convex grouping algorithm which can detect convex salient boundaries with good continuity and proximity in a globally optimal fashion to facilitate the application of this convex grouping algorithm we develop a new fragment connection method based on four point bezier curves we demonstrate the performance of this convex grouping algorithm by conducting experiments on both synthetic and real images in addition we provide a comparison with some prior edge grouping algorithms finally we show that the proposed convex grouping algorithm can be further extended to detect convex open boundaries derive region based image hierarchies and incorporate some simple human computer interactions

holomorphic martingales and interpolation between hardy spaces the complex method a probabilistic proof is given to identify the complex interpolation space of h1 t and h t as hp t introduction in this note a soft propabilistic proof of p w jones s theorem on the complex interpolation space between h1 and hoc is given we shall work with n th varopoulos s space of holomorhpic martingales the observation presented in this article is that the use of a stopping time decomposition simplifies constructions of serguei v kislyakov and quanhua xu to obtain the following theorem the complex interpolation space th t h t 0 o 0 e 1 coincides with hp t provided that 1 1 03 p as is well known this result has been obtained by p w jones using l estimates to the a problem see j his work also contains the description of the real interpolation spaces for the couple hl h at about the same time jean bourgain obtained a marcinkiewicz type decomposition using completely different techniques see b1 recently in a series of papers jean bourgain b2 serguei kislyakov kl k2 k x gilles pisier p and quanhua xu xl x2 obtained deep results concerning real and complex interpolation methods between vector valued weighted hardy spaces of analytic functions s kislyakov s paper kl contains the following idea to approximate the characteristic function 1 1f1 2 by analytic functions on the circle t he starts a max l itl and then considers 1 a iha where h denotes the hilbert transform on t received by the editors august 11 1992 and in revised form february 1 1994 originally communicated to the proceedings of the ams by dale alspach 1991 mathematics subject classification primary 60g46 42b30

martingale transforms and complex uniform convexity les transformees de martingale et les operateurs integraux singuliers de calderon zygmund sont bornees comme des operateurs de l 2 l 1 dans l 2 l q quand 0 q 1 si y est un sous espace reflexif de l 1 alors l 1 y peut se renormer pour etre 2 complexe uniformement convexe on donne une nouvelle demonstration de la propriete de cotype 2 de l 1 h

operators with an absolute functional calculus we study sectorial operators with a special type of functional calculus which we term an absolute functional calculus a typical example of such an operator is an invertible operator a defined on a banach space x considered on the real inter polation space dom a x p where 0 1 and 1 p in general the absolute functional calculus can be characterized in terms of real interpolation spaces we show that operators of this type have a strong form of the h calculus and behave very well with respect to the joint functional calculus we give applications of these results to recent work of arendt batty and bu on the existence of holder continuous solutions for the abstract cauchy problem

a spectral method for the equal width equation a spectral discretization of the equal width equation ewe is presented the method is shown to be convergent and nonlinearly stable time stepping is performed with high order adams methods the spectral accuracy of the scheme reveals some features of the ewe that the methods previously used could not bare out properly for instance we may now study the changes in amplitude and velocity of solitary waves after collisions

integral operators mapping into the space of bounded analytic functions abstract we address the problem of studying the boundedness compactness and weak compactness of the integral operators t g f z 0 z f g d acting from a banach space x into h we obtain a collection of general results which are appropriately applied and mixed with specific techniques in order to solve the posed questions to particular choices of x

fast and robust 2d shape extraction using discrete point sampling and centerline grouping in complex images this paper initially develops the discrete point sampling operator s concept model and parameters that we have previously proposed and makes its belt shaped regions in a discrete point sampling map more salient and appropriate for centerline extraction the cross sectional features of these belt shaped regions are then analyzed and seven types of feature points are defined to facilitate descriptions of such features based on these feature points a three level detection system is proposed including feature points line segments and centerlines to extract centerlines from the belt shaped regions eight basic types of centerlines and five types of relationships among the centerlines are defined by computational geometry algorithms and gestalt laws are used to cluster them into groupings if some prior information about a desired shape is available retrieval grouping may be carried out by a discrete point sampling map the purpose of which is to find centerlines by best matching with prior information discrete point sampling effectually overcomes the influences of interference from noise textures and uneven illumination and greatly reduces the difficulty of centerline extraction centerline clustered groupings and retrieval grouping can offer a strong anti interference ability with nonlinear deformations such as articulation and occlusion this method can extract large scale complex shapes combined of lines and planes from complex images the wheel location results of noise test and other shape extraction experiments show that our method has a strong capability to persist with nonlinear deformations

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

extensions of multilinear operators and banach space properties a new characterization of the dunford pettis property in terms of the extensions of multilinear operators to the biduals is obtained for the first time multilinear characterizations of the reciprocal dunford pettis property and pelczynski s property v are also found polynomial and holomorphic versions of these properties are given as well

theorems of hardy and paley for vector valued analytic functions and related classes of banach spaces we investigate the classes of banach spaces where analogues of the classical hardy inequality and the paley gap theorem hold for vector valued functions we show that the vector valued paley theorem is valid for a large class of banach spaces necessarily of cotype 2 which includes all banach lattices of cotype 2 all banach spaces whose dual is of type 2 and also the preduals of c algebras for the trace class s1 and the dual of the algebra of all bounded operators on a hilbert space a stronger result holds namely the vector valued analogue of the fefferman theorem on multipliers from h into in particular for the latter spaces the vector valued hardy inequality holds this inequality is also true for every banach space of type 1 bourgain 0 introduction if f z oajeiit is an analytic trigonometric polynomial then j o j elajl j 1 1 c1 r if t i dt

rbf ps scheme for solving the equal width equation the equal width equation is solved by a rbf ps scheme this technique is meshless and there is no need to linearize the nonlinear terms the radial kernels are used to transform the pde into a system of odes a higher order ode solver is used to solve the ode system the method is tested for single solitary wave interaction of two and three solitary waves the development of an undular bore and the interaction of two solitary waves the results are compared with earlier work and other methods

solitary waves of the regularized long wave equation abstract a finite element solution of the regularised long wave equation based on galerkin s method using cubic splines as element shape functions is set up a linear stability analysis shows the scheme to be unconditionally stable test problems including the migration and interaction of solitary waves are used to validate the method which is found to be accurate and efficient the three invariants of the motion are evaluated to determine the conservation properties of the algorithm the temporal evolution of a maxwellian initial pulse is then studied

a comparison of fourier pseudospectral methods for the solution of the korteweg de vries equation abstract various fourier pseudospectral methods are used to approximate the korteweg de vries equation the methods differ in their treatment of the time discretisation the methods are compared for computational efficiency using the 1 and 2 soliton test problems they are also compared from an accuracy viewpoint using the zabusky kruskal recurrence problem

l estimates for the bar partial problem in a half planeproblem in a half plane l approche de dualite pour trouver des solutions de f est suffisante pour plusieurs problemes apparaissant en theorie h p on donne des operateurs solution qui permettent de resoudre des problemes non resolubles par la dualite

arens regularity of banach algebras and the geometry of banach spaces we show that a banach algebra a such that a has the property v of a pelczynski is arens regular a strong uniqueness property of the extension of the product on a unital c algebra a to a is proved the algebraic structure of the bidual of a c algebra can be obtained through the local reflexivity principle we give examples which show that the results are sharp

a comparison of measures for detecting natural shapes in cluttered backgrounds we propose a new measure of perceptual saliency and quantitatively compare its ability to detect natural shapes in cluttered backgrounds to five previously proposed measures as defined in the new measure the saliency of an edge is the fraction of closed random walks which contain that edge the transition probability matrix defining the random walk between edges is based on a distribution of natural shapes modeled by a stochastic motion each of the saliency measures in our comparison is a function of a set of affinity values assigned to pairs of edges although the authors of each measure define the affinity between a pair of edges somewhat differently all incorporate the gestalt principles of good continuation and proximity in some form in order to make the comparison meaningful we use a single definition of affinity and focus instead on the performance of the different functions for combining affinity values the primary performance criterion is accuracy we compute false positive rates in classifying edges as signal or noise for a large set of test figures in almost every case the new measure significantly outperforms previous measures

edge grouping combining boundary and region information this paper introduces a new edge grouping method to detect perceptually salient structures in noisy images specifically we define a new grouping cost function in a ratio form where the numerator measures the boundary proximity of the resulting structure and the denominator measures the area of the resulting structure this area term introduces a preference towards detecting larger size structures and therefore makes the resulting edge grouping more robust to image noise to find the optimal edge grouping with the minimum grouping cost we develop a special graph model with two different kinds of edges and then reduce the grouping problem to finding a special kind of cycle in this graph with a minimum cost in ratio form this optimal cycle finding problem can be solved in polynomial time by a previously developed graph algorithm we implement this edge grouping method test it on both synthetic data and real images and compare its performance against several available edge grouping and edge linking methods furthermore we discuss several extensions of the proposed method including the incorporation of the well known grouping cues of continuity and intensity homogeneity introducing a factor to balance the contributions from the boundary and region information and the prevention of detecting self intersecting boundaries

robust and efficient detection of salient convex groups this paper describes an algorithm that robustly locates salient convex collections of line segments in an image the algorithm is guaranteed to find all convex sets of line segments in which the length of the gaps between segments is smaller than some fixed proportion of the total length of the lines this enables the algorithm to find convex groups whose contours are partially occluded or missing due to noise we give an expected case analysis of the algorithm performance this demonstrates that salient convexity is unlikely to occur at random and hence is a strong clue that grouped line segments reflect underlying structure in the scene we also show that our algorithm run time is o n sup 2 log n nm when we wish to find the m most salient groups in an image with n line segments we support this analysis with experiments on real data and demonstrate the grouping system as part of a complete recognition system

edge linking by sequential search edge detection is mainly a two stage process edge enhancement followed by edge linking in this paper we develop a linking algorithm for the combination of edge elements enhanced by an optimal filter the linking algorithm link is based on sequential search from a starting node transitions are made to the goal nodes based on a maximum likelihood metric results of our search algorithm are compared to the sequential edge linking algorithm sel as well as to two common nonsequential algorithms based on tracing the zero crossings loci in the convolution output of the 2g operators and based on nonmaximal suppression and thresholding of the convolution output of the g operator it is shown that the link algorithm provides comparable results require only local calculations and can accommodate any a priori information in the path metric used to guide the search

hankel operators on bounded analytic functions for u a domain in the complex plane and g a bounded measurable function on u the generalized hankel operator sg on h u is the operator of multiplication by g followed by projection into l h under certain conditions on u we show that either sg is compact or there is an embedded on which sg is bicontinuous we characterize those g s for which sg is compact in the case that u is a behrens roadrunner domain

the complex convexity of quasi normed linear spaces abstract a notion of uniform convexity is defined for quasi normed complex spaces by replacing norms of midpoints of segments in the space by norms of centers of complex discs in the space complex uniformly convex spaces c u c spaces always have cotype in banach lattices possessing cotype and being complex uniformly convex are the same duals of c algebras are c u c and in fact have cotype 2 the behavior of certain martingale difference sequences in c u c spaces is examined this leads to an isomorphic description of the spaces having the property

a joint functional calculus for sectorial operators with commuting resolvents 1 introduction in this paper we are concerned with h 1 functional calculus in the sense of the construction introduced by mcintosh 23 and developed recently by cowling doust mcintosh and yagi 8 given v in 0 p we will use the notation sv fz 2 c n f0g jarg z j vg to denote the open sector of angle 2v around the half line r let x be a complex banach space given a linear operator a on x we denote by d a n a and r a the domain the kernel and the range of a respectively we also denote by j a the spectrum of a and by r a the resolvent set of a

an adaptive contour closure algorithm and its experimental evaluation the potential of edge based complete image segmentation into regions has not gained the due attention in the literature thus far the present paper attempts to explore this potential by proposing an adaptive grouping algorithm to solve the contour closure problem that is the key to a successful edge based complete image segmentation the effectiveness of the proposed algorithm is extensively tested in the range image domain and compared to several region based segmentation methods within a rigorous comparison framework on three range image databases of varying quality acquired by different range scanners it is shown that the proposed approach is able to achieve very appealing performance with respect to both segmentation quality and computation time

globally optimal regions and boundaries as minimum ratio weight cycles we describe a new form of energy functional for the modeling and identification of regions in images the energy is defined on the space of boundaries in the image domain and can incorporate very general combinations of modeling information both from the boundary intensity gradients etc and from the interior of the region texture homogeneity etc we describe two polynomial time digraph algorithms for finding the global minima of this energy one of the algorithms is completely general minimizing the functional for any choice of modeling information it runs in a few seconds on a 256 spl times 256 image the other algorithm applies to a subclass of functionals but has the advantage of being extremely parallelizable neither algorithm requires initialization

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

inferring global perceptual contours from local features we address the problem of contour inference from partial data as obtained from state of the art edge detectors we argue that in order to obtain more pereeptually salient contours it is necessary to impose generic constraints such as continuity and co curvilinearity the implementation is in the form of a convolution with a mask which encodes both the orientation and the strength of the possible continuations we first show how the mask called the extension field is derived then how the contributions from different sites are collected to produce a saliency map we show that the scheme can handle a variety of input data from dot patterns to oriented edgels in a unified manner and demonstrate results on a variety of input stimuli we also present a similar approach to the problem of inferring contours formed by end points in both cases the scheme is non linear non iterative and unified in the sense that all types of input tokens are handled in the same manner

winding number constrained contour detection salient contour detection can benefit from the integration of both contour cues and region cues however this task is difficult due to different nature of region representations and contour representations to solve this problem this paper proposes an energy minimization framework based on winding number constraints in this framework both region cues such as color texture homogeneity and contour cues such as local contrast and continuity are represented in a joint objective function which has both region and contour labels the key problem is how to design constraints that ensure the topological consistency of the two kinds of labels our technique is based on the topological concept of winding number using a fast method for winding number computation a small number of linear constraints are derived to ensure label consistency our method is instantiated by ratio based energy functions by successfully integrating both region and contour cues our method shows advantages over competitive methods our method is extended to incorporate user interaction which leads to further improvements

hahn banach extension of multilinear forms and summability the aim of this paper is to investigate close relations between the validity of hahn banach extension theorems for multilinear forms on banach spaces and summability properties of sequences from these spaces a case of particular importance occurs when we consider banach spaces which have the property that every bilinear form extends to any superspace

optimal image and video closure by superpixel grouping detecting independent objects in images and videos is an important perceptual grouping problem one common perceptual grouping cue that can facilitate this objective is the cue of contour closure reflecting the spatial coherence of objects in the world and their projections as closed boundaries separating figure from background detecting contour closure in images consists of finding a cycle of disconnected contour fragments that separates an object from its background searching the entire space of possible groupings is intractable and previous approaches have adopted powerful perceptual grouping heuristics such as proximity and co curvilinearity to constrain the search we introduce a new formulation of the problem by transforming the problem of finding cycles of contour fragments to finding subsets of superpixels whose collective boundary has strong edge support few gaps in the image our cost function a ratio of a boundary gap measure to area promotes spatially coherent sets of superpixels moreover its properties support a global optimization procedure based on parametric maxflow extending closure detection to videos we introduce the concept of spatiotemporal closure analogous to image closure we formulate our spatiotemporal closure cost over a graph of spatiotemporal superpixels our cost function is a ratio of motion and appearance discontinuity measures on the boundary of the selection to an internal homogeneity measure of the selected spatiotemporal volume the resulting approach automatically recovers coherent components in images and videos corresponding to objects object parts and objects with surrounding context providing a good set of multiscale hypotheses for high level scene analysis we evaluate both our image and video closure frameworks by comparing them to other closure detection approaches and find that they yield improved performance

structure of subspaces of the compact operators having the dunford pettis property the structure of the subspaces m subset k ell p having the dunford pettis property dpp is studied where k ell p is the space of all compact operators on ell p and 1 p infty the following conditions are shown to be equivalent i m has the dpp ii m is isomorphic to a subspace of c 0 iii the sets sx s in b m subset ell p and s x s in b m subset ell p are relatively compact for all x in ell p and x in ell p the equivalence between i and iii was recently proven in the case of arbitrary hilbert spaces by brown and ulger it is also shown that i and ii are equivalent for subspaces m subset k ell p 1 oplus ldots oplus ell p k this result is optimal in the sense that for 1 p q infty there is a dpp subspace m subset k ell q ell p that fails to be isomorphic to a subspace of c 0

a polynomially bounded operator on hilbert space which is not similar to a contraction let e 0 we prove that there exists an operator te e2 2 such that for any polynomial p we have iip te ii not similar to a contraction i e there does not exist an invertible operator s f2 e f2 such that jis 1tesii tributed to halmos after his well known 1970 paper ten problems in hilbert space we also give some related finite dimensional estimates department of mathematics texas a m university college station texas 77843 universite paris vi equipe d7analyse case 186 75252 paris cedex 05 france e mail address gipqccr jussieu fr this content downloaded from 157 55 39 104 on sun 19 jun 2016 06 52 47 utc all use subject to http about jstor org terms

salient closed boundary extraction with ratio contour we present ratio contour a novel graph based method for extracting salient closed boundaries from noisy images this method operates on a set of boundary fragments that are produced by edge detection boundary extraction identifies a subset of these fragments and connects them sequentially to form a closed boundary with the largest saliency we encode the gestalt laws of proximity and continuity in a novel boundary saliency measure based on the relative gap length and average curvature when connecting fragments to form a closed boundary this new measure attempts to remove a possible bias toward short boundaries we present a polynomial time algorithm for finding the most salient closed boundary we also present supplementary preprocessing steps that facilitate the application of ratio contour to real images we compare ratio contour to two closely related methods for extracting closed boundaries elder and zucker s method based on the shortest path algorithm and williams and thornber s method based on spectral analysis and a strongly connected components algorithm this comparison involves both theoretic analysis and experimental evaluation on both synthesized data and real images

rigidity of composition operators on the hardy space h p abstract let be an analytic map taking the unit disk d into itself we establish that the class of composition operators f c f f exhibits a rather strong rigidity of non compact behaviour on the hardy space h p for 1 p and p 2 our main result is the following trichotomy which states that exactly one of the following alternatives holds i c is a compact operator h p h p ii c fixes a linearly isomorphic copy of l p in h p but c does not fix any copies of l 2 in h p iii c fixes a copy of l 2 in h p moreover in case iii the operator c actually fixes a copy of l p 0 1 in h p provided p 1 we reinterpret these results in terms of norm closed ideals of the bounded linear operators on h p which contain the compact operators k h p in particular the class of composition operators on h p does not reflect the quite complicated lattice structure of such ideals

segmentation of multiple salient closed contours from real images using a saliency measure based on the global property of contour closure we have developed a segmentation method which identifies smooth closed contours bounding objects of unknown shape in real images the saliency measure incorporates the gestalt principles of proximity and good continuity that previous methods have also exploited unlike previous methods we incorporate contour closure by finding the eigenvector with the largest positive real eigenvalue of a transition matrix for a markov process where edges from the image serve as states element i j of the transition matrix is the conditional probability that a contour which contains edge j will also contain edge i we show how the saliency measure defined for individual edges can be used to derive a saliency relation defined for pairs of edges and further show that strongly connected components of the graph representing the saliency relation correspond to smooth closed contours in the image finally we report for the first time results on large real images for which segmentation takes an average of about 10 seconds per object on a general purpose workstation

strict singularity of volterra type operators on hardy spaces abstract in this paper we first characterize the boundedness and compactness of volterra type operator s g f z 0 z f g d z d defined on hardy spaces h p 0 p the spectrum of s g is also obtained then we prove that s g fixes an isomorphic copy of l p and an isomorphic copy of l 2 if the operator s g is not compact on h p 1 p in particular this implies that the strict singularity of the operator s g coincides with the compactness of the operator s g on h p at last we post an open question for further study

graph partitioning active contours gpac for image segmentation in this paper we introduce new types of variational segmentation cost functions and associated active contour methods that are based on pairwise similarities or dissimilarities of the pixels as a solution to a minimization problem we introduce a new curve evolution framework the graph partitioning active contours gpac using global features our curve evolution is able to produce results close to the ideal minimization of such cost functions new and efficient implementation techniques are also introduced in this paper our experiments show that gpac solution is effective on natural images and computationally efficient experiments on gray scale color and texture images show promising segmentation results

unsupervised contour closure algorithm for range image edge based segmentation this paper presents an efficient technique for extracting closed contours from range images edge points edge points are assumed to be given as input to the algorithm i e previously computed by an edge based range image segmentation technique the proposed approach consists of three steps initially a partially connected graph is generated from those input points then the minimum spanning tree of that graph is computed finally a postprocessing technique generates a single path through the regions boundaries by removing noisy links and closing open contours the novelty of the proposed approach lies in the fact that by representing edge points as nodes of a partially connected graph it reduces the contour closure problem to a minimum spanning tree partitioning problem plus a cost function minimization stage to generate closed contours experimental results with synthetic and real range images together with comparisons with a previous technique are presented