a probabilistic fuzzy logic system for modeling and control in this paper a probabilistic fuzzy logic system pfls is proposed for the modeling and control problems similar to the ordinary fuzzy logic system fls the pfls consists of the fuzzification inference engine and defuzzification operation to process the fuzzy information different to the fls it uses the probabilistic modeling method to improve the stochastic modeling capability by using a three dimensional membership function mf the pfls is able to handle the effect of random noise and stochastic uncertainties existing in the process a unique defuzzification method is proposed to simplify the complex operation finally the proposed pfls is applied to a function approximation problem and a robotic system it shows a better performance than an ordinary fls in stochastic circumstance

block diagonal and constraint preconditioners for nonsymmetric indefinite linear systems part i theory we study block diagonal preconditioners and an efficient variant of constraint preconditioners for general two by two block linear systems with zero 2 2 block we derive block diagonal preconditioners from a splitting of the 1 1 block of the matrix from the resulting preconditioned system we derive a smaller so called related system that yields the solution of the original problem solving the related system corresponds to an efficient implementation of constraint preconditioning we analyze the properties of both classes of preconditioned matrices in particular their spectra using analytical results we show that the related system matrix has the more favorable spectrum which in many applications translates into faster convergence for krylov subspace methods we show that fast convergence depends mainly on the quality of the splitting a topic for which a substantial body of theory exists our analysis also provides a number of new relations between block diagonal preconditioners and constraint preconditioners for constrained problems solving the related system produces iterates that satisfy the constraints exactly just as for systems with a constraint preconditioner finally for the lagrange multiplier formulation of a constrained optimization problem we show how scaling nonlinear constraints can dramatically improve the convergence for linear systems in a newton iteration our theoretical results are confirmed by numerical experiments on a constrained optimization problem we consider the general nonsymmetric nonsingular case our only additional requirement is the nonsingularity of the schur complement type matrix derived from the splitting that defines the preconditioners in particular the 1 2 block need not equal the transposed 2 1 block and the 1 1 block might be indefinite or even singular this is the first paper in a two part sequence in the second paper we will study the use of our preconditioners in a variety of applications

a linear time algorithm for computing exact euclidean distance transforms of binary images in arbitrary dimensions a sequential algorithm is presented for computing the exact euclidean distance transform dt of a k dimensional binary image in time linear in the total number of voxels n the algorithm which is based on dimensionality reduction and partial voronoi diagram construction can be used for computing the dt for a wide class of distance functions including the l sub p and chamfer metrics at each dimension level the dt is computed by constructing the intersection of the voronoi diagram whose sites are the feature voxels with each row of the image this construction is performed efficiently by using the dt in the next lower dimension the correctness and linear time complexity are demonstrated analytically and verified experimentally the algorithm may be of practical value since it is relatively simple and easy to implement and it is relatively fast not only does it run in o n time but the time constant is small a simple modification of the algorithm computes the weighted euclidean dt which is useful for images with anisotropic voxel dimensions a parallel version of the algorithm runs in o n p time with p processors

conditional density estimation using probabilistic fuzzy systems we consider conditional density approximation by fuzzy systems fuzzy systems are typically used to approximate deterministic functions in which the stochastic uncertainty is ignored we propose probabilistic fuzzy systems pfss in which the probabilistic nature of uncertainty is taken into account these systems take also fuzzy uncertainty into account by their fuzzy partitioning of input and output spaces we discuss an additive reasoning scheme for pfss that leads to the estimation of conditional probability densities and prove how such fuzzy systems compute the expected value of this conditional density function we show that some of the most commonly used fuzzy systems can compute the same expected output value and we derive how their parameters should be selected in order to achieve this goal the additional information and process understanding provided by the different interpretations of the pfs models are illustrated using a real world example

on the generation of skeletons from discrete euclidean distance maps the skeleton is an important representation for shape analysis a common approach for generating discrete skeletons takes three steps 1 computing the distance map 2 detecting maximal disks from the distance map and 3 linking the centers of maximal disks cmds into a connected skeleton algorithms using approximate distance metrics are abundant and their theory has been well established however the resulting skeletons may be inaccurate and sensitive to rotation in this paper we study methods for generating skeletons based on the exact euclidean metric we first show that no previous algorithms identify the exact set of discrete maximal disks under the euclidean metric we then propose new algorithms and show that they are correct to link cmds into connected skeletons we examine two prevalent approaches connected thinning and steepest ascent we point out that the connected thinning approach does not work properly for euclidean distance maps only the steepest ascent algorithm produces skeletons that are truly medially placed the resulting skeletons have all the desirable properties they have the same simple connectivity as the figure they are well centered they are insensitive to rotation and they allow exact reconstruction the effectiveness of our algorithms is demonstrated with numerous examples

skeletonization based on error reduction the accuracy of a non pixel based skeletonization method is largely dependent on the contour information chosen as input when using a constrained delaunay triangulation to construct an object s skeleton a number of contour pixels must be chosen as a basis for triangulation this paper presents a new method of selecting these contour pixels a new method for measuring skeletonization error is proposed which quantifies the deviation of a skeleton segment from the true medial axis of a stroke in an image the goal of the proposed algorithm is to reduce this error to an acceptable level whilst retaining the superior efficiencies of previous non pixel based techniques experimental results show that the proposed method is adept at following the medial axis of an image and is capable of producing a skeleton that is confirmed by a human s perception of the image it is also computationally efficient and robust against noise

efficient euclidean distance transform algorithm of binary images in arbitrary dimensions in this paper we propose an efficient algorithm i e pbedt for short to compute the exact euclidean distance transform edt of a binary image in arbitrary dimensions the pbedt is based on independent scan and implemented in a recursive way i e the edt of a d dimensional image is able to be computed from the edts of its d 1 dimensional sub images in each recursion all of the rows in the current dimensional direction are processed one by one the points in the current processing row and their closest feature points in d 1 dimensional sub images can be shown in a euclidean plane by using the geometric properties of the perpendicular bisector the closest feature points of d 1 dimensional sub images are easily verified so as to lead to the edt of a d dimensional image after eliminating the invalid points the time complexity of the pbedt algorithm is linear in the amount of both image points and dimensions with a small coefficient compared with the state of the art edt algorithms the pbedt algorithm is much faster and more stable in most cases

a qualified description of extended fuzzy logic abstract the sense of approximate reasoning within fuzzy logic has given rise to speculation as to whether precise reasoning within classical logic can even be considered a feasible way of handling most of the world s problems now another question can be raised can fuzzy logic by itself justify handle the sort of uncertainty found in problems that belong to the class of open world problems this family of problems corresponds to the case of ambiguous mathematical theories mainly due to the fact that there is just imperfect incomplete information about them recently zadeh introduced an extension of fuzzy logic claimed to bridge the gaps in conventional approaches to the open world problems in this paper we present a basic background of extended fuzzy logic fle we address several main concepts such as f transformation and f validity enhance the definition of generalized constraints and propose that fle s rules have two layers one based on the implication model and another based on the conjunction model in addition we explore how fle can handle those conditions described in antonym pairs as well as those occurred without considering rationality in the framework both cases challenge most decision making theories we also suggest fle and second order probability are f complementary rather than f competitive the paper asserts that fle is highly capable of constructing mathematical solutions to problems including computational problems that are stated in natural language

convergence conditions for splitting iteration methods for non hermitian linear systems abstract necessary and sufficient convergence conditions are studied for splitting iteration methods for non hermitian system of linear equations when the coefficient matrix is nonsingular when this theory is specialized to the generalized saddle point problem we obtain convergence theorem for a class of modified accelerated overrelaxation iteration methods which include the uzawa and the inexact uzawa methods as special cases moreover we apply this theory to the two stage iteration methods for non hermitian positive definite linear systems and obtain sufficient conditions for guaranteeing the convergence of these methods

the delta medial axis in this paper we present the delta medial axis dma a quasi linear algorithmic solution addressing several of the main concerns of discrete medial axes ma computation first its sensitivity to small shape perturbations is counterbalanced by a single parameter delta used in a pruning strategy that implicitly takes into account the local topology of the boundary second the discrete nature of images is addressed by approximating the original ma definition in continuous space third to allow real time performances a set of algorithmic optimizations is proposed we compare our approach both qualitatively and quantitatively to recent state of the art solutions and show that the dma presents an excellent choice for a wide range of applications finally to allow fast and efficient use of our algorithm we propose the complete pseudo code highlightsthe delta medial axis a fast and robust skeletonization algorithm is presented it includes a pruning parameter which considers the local topology of the boundary to allow real time performances optimizations are introduced with complete pseudo code a complete comparative study with four recent state of the art algorithms is proposed

on inexact preconditioners for nonsymmetric matrices inexact versions of the block triangular preconditioners for nonsymmetric matrices of block two by two structures in m f murphy g h golub and a j wathen siam j sci comput 21 2000 pp 1969 1972 and i c f ipsen siam j sci comput 23 2001 pp 1050 1051 are presented and the two preconditioners for symmetric block two by two matrices in c durazzi and v ruggiero numer linear algebra appl 10 2003 pp 673 688 are extended to general nonsymmetric matrices moreover we precisely describe the spectral properties of the preconditioned matrices and the finite step termination properties of the preconditioned krylov subspace iteration methods with an optimal or galerkin property with respect to these preconditioners several numerical examples are performed to illustrate the effectiveness of the proposed preconditioners

preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite linear systems for the positive semidefinite system of linear equations of a block two by two structure by making use of the hermitian skew hermitian splitting iteration technique we establish a class of preconditioned hermitian skew hermitian splitting iteration methods theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system moreover the optimal choice of the involved iteration parameter and the corresponding asymptotic convergence rate are computed exactly numerical examples further confirm the correctness of the theory and the effectiveness of the method

detection of 3 d simple points for topology preserving transformations with application to thinning the problems of 3 d digital topology preservation under binary transformations and 3 d object thinning are considered in this correspondence first the authors establish the conditions under which transformation of an object voxel to a non object voxel or its inverse does not affect the image topology an efficient algorithm to detect a simple point has been proposed on the basis of those conditions in this connection some other interesting properties of 3 d digital geometry are also discussed using these properties and the simple point detection algorithm the authors have proposed an algorithm to generate a surface skeleton so that the topology of the original image is preserved the shape of the image is maintained as much as possible and the results are less affected by noise

fuzzy equalization in the construction of fuzzy sets in this note we introduce and study a concept of fuzzy equalization fuzzy equalization concerns a process of building information granules that are semantically and experimentally meaningful the experimental relevance of a given information granule fuzzy set is directly linked with an encapsulation of a certain experimental evidence conveyed by the respective probability density function of available data we establish a detailed equalization algorithm developed for triangular fuzzy sets the study elaborates on the role of the fuzzy equalization in system design

an unified multiscale framework for planar surface and curve skeletonization computing skeletons of 2d shapes and medial surface and curve skeletons of 3d shapes is a challenging task in particular there is no unified framework that detects all types of skeletons using a single model and also produces a multiscale representation which allows to progressively simplify or regularize all skeleton types in this paper we present such a framework we model skeleton detection and regularization by a conservative mass transport process from a shape s boundary to its surface skeleton next to its curve skeleton and finally to the shape center the resulting density field can be thresholded to obtain a multiscale representation of progressively simplified surface or curve skeletons we detail a numerical implementation of our framework which is demonstrably stable and has high computational efficiency we demonstrate our framework on several complex 2d and 3d shapes

automatic correction of ma and sonka s thinning algorithm using p simple points ma and sonka proposed a fully parallel 3d thinning algorithm which does not always preserve topology we propose an algorithm based on p simple points which automatically corrects ma and sonka s algorithm as far as we know our algorithm is the only fully parallel curve thinning algorithm which preserves topology

what are the differences between bayesian classifiers and mutual information classifiers in this paper both bayesian and mutual information classifiers are examined for binary classifications with or without a reject option the general decision rules are derived for bayesian classifiers with distinctions on error types and reject types a formal analysis is conducted to reveal the parameter redundancy of cost terms when abstaining classifications are enforced the redundancy implies an intrinsic problem of nonconsistency for interpreting cost terms if no data are given to the cost terms we demonstrate the weakness of bayesian classifiers in class imbalanced classifications on the contrary mutual information classifiers are able to provide an objective solution from the given data which shows a reasonable balance among error types and reject types numerical examples of using two types of classifiers are given for confirming the differences including the extremely class imbalanced cases finally we briefly summarize the bayesian and mutual information classifiers in terms of their application advantages and disadvantages respectively

combination of augmented lagrangian technique and st preconditioner for saddle point problems abstract for symmetric indefinite linear systems we introduce a new triangular preconditioner based on symmetric and triangular st decomposition a new 1 1 block is obtained by augmented lagrangian technique the new st preconditioner is introduced by the combination of the new 1 1 block and symmetric and triangular st decomposition then a preconditioned system can be obtained by preconditioning technique which is superior to the original system in terms of condition number we study the spectral properties of preconditioned system such as eigenvalues the estimation of condition number and then give the quasi optimal parameter numerical examples are given to indicate that the new preconditioner has obvious efficiency advantages finally we conclude that the new st preconditioner is a better option to deal with large and sparse problems

block triangular preconditioners for saddle point problems with a penalty term block triangular preconditioners for a class of saddle point problems with a penalty term are considered an important example is the mixed formulation of the pure displacement problem in linear elasticity it is shown that the spectrum of the preconditioned system is contained in a real positive interval and that the interval bounds can be made independent of the discretization and penalty parameters this fact is used to construct bounds of the convergence rate of the gmres method with respect to an energy norm numerical results are given for gmres and bi cgstab

probabilities from fuzzy observations abstract we introduce a new definition for getting the probability of a fuzzy event from a collection of fuzzy observations this definition is based upon the use of the compatibility of the target event with the observation we show that this new definition satisfies the komolgoroff conditions we show how it naturally extends all the properties of the classical probability theory

analysis of data generated from multidimensional type 1 and type 2 fuzzy membership functions due to the numerous applications that utilize different types of fuzzy membership functions mfs it may sometimes be difficult to choose an appropriate mf for a particular application in this paper we establish preliminary guidelines to direct this selection by proposing a three stage method in the forward stage different mfs such as crisp mfs type 1 t1 fuzzy mfs and type 2 t2 fuzzy mfs are generated from multidimensional data sets next in the reverse stage data is generated back from these mfs by considering different bin sizes in doing so various data sets may be generated for different applications which require fuzzy data finally for the similarity analysis stage we propose an iterative algorithm that makes use of the results of wilcoxon signed rank wsr and wilcoxon rank sum wrs tests to compare the original data and the generated data from the results of these tests recommendations concerning the suitability of mfs for a specific application may be suggested by observing the accuracy of representation and the requirements of the application with this analysis the objective is to gain insight on when t2 fuzzy sets may be considered to outperform t1 fuzzy sets and vice versa several examples are provided using synthetic and real data to validate the iterative algorithm for data sets in various dimensions

fuzzy semi markov migration process in credit risk we explore the rating system used by credit agencies with a focus on problems that justify the use of fuzzy set theory we prove that a fuzzy market is viable if and only if an equivalent martingale measure exists from which we construct the forward probability measure and under which the discounted price of a default free bond is a martingale we model the evolution of credit migration of a defaultable bond as an inhomogeneous semi markov process with fuzzy states we study the effects of changing the real probability measure to a forward probability measure in addition we investigate the asymptotic behaviour of the survival probability in each fuzzy state given in the absence of default finally we discuss parameter estimation and calibration of the inhomogeneous markov chain with fuzzy states

on parameterized inexact uzawa methods for generalized saddle point problems abstract for the large sparse saddle point problems bai et al recently studied a class of parameterized inexact uzawa piu methods z z bai b n parlett z q wang on generalized successive overrelaxation methods for augmented linear systems numer math 102 2005 1 38 for a special case that the 1 1 block is solved exactly they determined the convergence domain and computed the optimal iteration parameters and the corresponding optimal convergence factor for the induced method in this paper we develop these methods to the large sparse generalized saddle point problems for the obtained parameterized inexact uzawa method we prove its convergence under suitable restrictions on the iteration parameters in particular we determine its quasi optimal iteration parameters and the corresponding quasi optimal convergence factor for the saddle point problems furthermore this piu method is generalized to obtain a framework of accelerated variants of the parameterized inexact uzawa methods for solving both symmetric and nonsymmetric generalized saddle point problems by using the techniques of vector extrapolation matrix relaxation and inexact iteration

numerical methods for generalized least squares problems usually generalized least squares problems are solved by transforming them into regular least squares problems which can then be solved by well known numerical methods however this approach is not very effective in some cases and besides is very expensive for large scale problems in 1979 paige suggested another approach which consists of solving an equivalent equality constrained least squares problem by the orthogonal decomposition the bnp algorithm or the james implicit nullspace iterative methods in this paper we present some new developments of the numerical methods for example 2 cycle sor method and preconditioned conjugate gradient method for generalized least squares problems some numerical comparisons are included as well

on solving block structured indefinite linear systems we consider 2 2 block indefinite linear systems whose 2 2 block is zero such systems arise in many applications we discuss two techniques that are based on modifying the 1 1 block in a way that makes the system easier to solve the main part of the paper focuses on an augmented lagrangian approach a technique that modifies the 1 1 block without changing the system size the choice of the parameter involved the spectrum of the linear system and its condition number are discussed and some analytical observations are provided a technique of deflating the 1 1 block is then introduced finally numerical experiments that validate the analysis are presented

fast nonsymmetric iterations and preconditioning for navier stokes equations discretization and linearization of the steady state navier stokes equations gives rise to a nonsymmetric indefinite linear system of equations in this paper we introduce preconditioning techniques for such systems with the property that the eigenvalues of the preconditioned matrices are bounded independently of the mesh size used in the discretization we confirm and supplement these analytic results with a series of numerical experiments indicating that krylov subspace iterative methods for nonsymmetric systems display rates of convergence that are independent of the mesh parameter in addition we show that preconditioning costs can be kept small by using iterative methods for some intermediate steps performed by the preconditioner

discussion probability theory and fuzzy logic are complementary rather than competitive the relationship between probability theory and fuzzy logic has long been an object of discussion and some controversy the position articulated in this article is that probability theory by itself is not sufficient for dealing with uncertainty and imprecision in real world settings to enhance its effectiveness probability theory needs an infusion of concepts and techniques drawn from fuzzy logic especially the concept of a linguistic variable and the calculus of fuzzy if then rules in the final analysis probability theory and fuzzy logic are complementary rather than competitive

digital topology and geometry in medical imaging a survey digital topology and geometry refers to the use of topologic and geometric properties and features for images defined in digital grids such methods have been widely used in many medical imaging applications including image segmentation visualization manipulation interpolation registration surface tracking object representation correction quantitative morphometry etc digital topology and geometry play important roles in medical imaging research by enriching the scope of target outcomes and by adding strong theoretical foundations with enhanced stability fidelity and efficiency this paper presents a comprehensive yet compact survey on results principles and insights of methods related to digital topology and geometry with strong emphasis on understanding their roles in various medical imaging applications specifically this paper reviews methods related to distance analysis and path propagation connectivity surface tracking image segmentation boundary and centerline detection topology preservation and local topological properties skeletonization and object representation correction and quantitative morphometry a common thread among the topics reviewed in this paper is that their theory and algorithms use the principle of digital path connectivity path propagation and neighborhood analysis

a ridge following algorithm for finding the skeleton of a fuzzy image abstract a typical method for finding the skeleton of a grayscale image is to threshold the image then thin the result with one of several binary thinning algorithms this process loses information in the intensity dimension by reducing all intensities to either one of two values the loss of intensity information causes the skeleton of non uniform grayscale images to be centrally located within the thresholded image but may not lie along the maximum intensity ridge line in this paper we propose a method that avoids thresholding instead we transform the grayscale image into a fuzzy set and use the degree of membership in the underlying object to find the skeleton along the maximum intensity ridge line topographical labeling methods find a ridge line by topographically labeling pixels the ride line is the union of peak ridge and saddle point pixels these methods require preprocessing or postprocessing to find connected single pixel skeletons we show the results of our algorithm on images varying in uniformity from a gaussian smoothed binary rectangle to a scanned image of a grayscale object we compare results with the threshold and thin and topographical labeling methods

computing skeletons in three dimensions skeletonization will probably become as valuable a tool for shape analysis in 3d as it is in 2d we present a topology preserving 3d skeletonization method which computes both surface and curve sk

the medial scaffold of 3d unorganized point clouds we introduce the notion of the medial scaffold a hierarchical organization of the medial axis of a 3d shape in the form of a graph constructed from special medial curves connecting special medial points a key advantage of the scaffold is that it captures the qualitative aspects of shape in a hierarchical and tightly condensed representation we propose an efficient and exact method for computing the medial scaffold based on a notion of propagation along the scaffold itself starting from initial sources of the flow and constructing the scaffold during the propagation we examine this method specifically in the context of an unorganized cloud of points in 3d e g as obtained from laser range finders which typically involve hundreds of thousands of points but the ideas are generalizable to data arising from geometrically described surface patches the computational bottleneck in the propagation based scheme is in finding the initial sources of the flow we thus present several ideas to avoid the unnecessary consideration of pairs of points which cannot possibly form a medial point source such as the visibility of a point from another given a third point and the interaction of clusters of points an application of using the medial scaffold for the representation of point samplings of real life objects is also illustrated

the multiresolution gradient vector field skeleton many algorithms suppress skeleton associated with boundary perturbation by preventing their formation or by costly branch pruning this work proposes a novel concept of structural and textural skeletons the former is associated with the general shape structure and the latter with boundary perturbations these skeletons remain disconnected to facilitate gross shape matching without the need for branch pruning they are extracted from a multiresolution gradient vector field that is efficiently generated within a pyramidal framework experimental results show that these skeletons are scale and rotation invariant they are less affected by boundary noise compared to skeletons extracted by popular iterative and non iterative techniques

a 3d fully parallel surface thinning algorithm the thinning is an iterative layer by layer erosion until only the skeletons of the objects are left this paper presents a thinning algorithm for extracting medial surfaces from 3d binary pictures the strategy which is used is called fully parallel which means that the same parallel operator is applied at each iteration an efficient implementation of the proposed algorithm on conventional sequential computers is given and the topological correctness for 26 6 binary pictures is proved

characterization of continuous t norms compatible with zadeh s probability of fuzzy events in this work we study the concept of p z compatibility which holds for those t norms for which zadeh s probability of fuzzy events fulfills kolmogorov s axioms whether a probability satisfies kolmogorov s axioms or not depends almost exclusively on the t norm and dual t conorm used to define intersection and union respectively of fuzzy sets in order to better understand fuzzy probability theory we provide a complete characterization of the p z compatible continuous t norms

evaluation of st preconditioners for saddle point problems the general block st decomposition of the saddle point problem is used as a preconditioner to transform the saddle point problem into an equivalent symmetric and positive definite system such a decomposition is called a block st preconditioner two general block st preconditioners are proposed for saddle point problems with symmetric and positive definite 1 1 block some estimations of the condition number of the preconditioned system are given the same study is done for singular 1 1 block

structured preconditioners for nonsingular matrices of block two by two structures for the large sparse block two by two real nonsingular matrices we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations for the specific versions such as modified block jacobi type modified block gauss seidel type and modified block unsymmetric symmetric gauss seidel type preconditioners we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices also we show that when these structured preconditioners are employed to precondition the krylov subspace methods such as gmres and restarted gmres fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two by two coefficient matrices in particular these structured preconditioners can lead to efficient and high quality preconditioning matrices for some typical matrices from the real world applications

surface and curve skeletonization of large 3d models on the gpu we present a gpu based framework for extracting surface and curve skeletons of 3d shapes represented as large polygonal meshes we use an efficient parallel search strategy to compute point cloud skeletons and their distance and feature transforms fts with user defined precision we regularize skeletons by a new gpu based geodesic tracing technique which is orders of magnitude faster and more accurate than comparable techniques we reconstruct the input surface from skeleton clouds using a fast and accurate image based method we also show how to reconstruct the skeletal manifold structure as a polygon mesh and the curve skeleton as a polyline compared to recent skeletonization methods our approach offers two orders of magnitude speed up high precision and low memory footprints we demonstrate our framework on several complex 3d models

new preconditioners for saddle point problems we present new preconditioners based on matrix splittings for the saddle point problems the spectral property of one of the preconditioned matrix is studied in detail numerical examples are used to illustrate the efficiency of the new preconditioners

a preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems this paper provides a preconditioned iterative technique for the solution of saddle point problems these problems typically arise in the numerical approximation of partial differential equations by lagrange multiplier techniques and or mixed methods the saddle point problem is reformulated as a symmetric positive definite system which is then solved by conjugate gradient iteration applications to the equations of elasticity and stokes are discussed and the results of numerical experiments are given

two augmentation preconditioners for nonsymmetric and indefinite saddle point linear systems with singular 1 1 blocks abstract in this paper two preconditioners based on augmentation are introduced for the solution of large saddle point type systems with singular 1 1 blocks we study the spectral characteristics of the preconditioners show that all eigenvalues of preconditioned matrices are strongly clustered finally numerical experiments are also reported for illustrating the efficiency of the presented preconditioners

performance and analysis of saddle point preconditioners for the discrete steady state navier stokes equations we examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the steady state navier stokes equations with a combination of analytic and empirical results we study the effects of fundamental parameters on convergence we demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers the structure of these distributions is independent of the discretization mesh size but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller these characteristics are directly correlated with the convergence properties of iterative solvers

a formal classification of 3d medial axis points and their local geometry this paper proposes a novel hypergraph skeletal representation for 3d shape based on a formal derivation of the generic structure of its medial axis by classifying each skeletal point by its order of contact we show that genetically the medial axis consists of five types of points which are then organized into sheets curves and points 1 sheets manifolds with boundary which are the locus of bitangent spheres with regular tangency a sub 1 sup 2 a sub k sup n notation means n distinct k fold tangencies of the sphere of contact as explained in the text two types of curves 2 the intersection curve of three sheets and the locus of centers of tritangent spheres a sub 1 sup 3 and 3 the boundary of sheets which are the locus of centers of spheres whose radius equals the larger principal curvature i e higher order contact a sub 3 points and two types of points 4 centers of quad tangent spheres a sub 1 sup 4 and 5 centers of spheres with one regular tangency and one higher order tangency a sub 1 a sub 3 the geometry of the 3d medial axis thus consists of sheets a sub 1 sup 2 bounded by one type of curve a sub 3 on their free end which corresponds to ridges on the surface and attached to two other sheets at another type of curve a sub 1 sup 3 which support a generalized cylinder description the a sub 3 curves can only end in a sub 1 a sub 3 points where they must meet an a sub 1 sup 3 curve the a sub 1 sup 3 curves meet together in fours at an a sub 1 sup 4 point this formal result leads to a compact representation for 3d shape referred to as the medial axis hypergraph representation consisting of nodes a sub 1 sup 4 and a sub 1 a sub 3 points links between pairs of nodes a sub 1 sup 3 and a sub 3 curves and hyperlinks between groups of links a sub 1 sup 2 sheets the description of the local geometry at nodes by itself is sufficient to capture qualitative aspects of shapes in analogy to 2d we derive a pointwise reconstruction formula to reconstruct a surface from this medial axis hypergraph together with the radius function thus this information completely characterizes 3d shape and lays the theoretical foundation for its use in recognition morphing design and manipulation of shapes

a mathematical morphology approach to euclidean distance transformation a distance transformation technique for a binary digital image using a gray scale mathematical morphology approach is presented applying well developed decomposition properties of mathematical morphology one can significantly reduce the tremendous cost of global operations to that of small neighborhood operations suitable for parallel pipelined computers first the distance transformation using mathematical morphology is developed then several approximations of the euclidean distance are discussed the decomposition of the euclidean distance structuring element is presented the decomposition technique employs a set of 3 by 3 gray scale morphological erosions with suitable weighted structuring elements and combines the outputs using the minimum operator real valued distance transformations are considered during the processes and the result is approximated to the closest integer in the final output image

automatic and robust computation of 3d medial models incorporating object variability this paper presents a novel processing scheme for the automatic and robust computation of a medial shape model which represents an object population with shape variability the sensitivity of medial descriptions to object variations and small boundary perturbations are fundamental problems of any skeletonization technique these problems are approached with the computation of a model with common medial branching topology and grid sampling this model is then used for a medial shape description of individual objects via a constrained model fit the process starts from parametric 3d boundary representations with existing point to point homology between objects the voronoi skeleton of each sampled object boundary is partitioned into non branching medial sheets and simplified by a novel pruning algorithm using a volumetric contribution criterion using the surface homology medial sheets are combined to form a common medial branching topology finally the medial sheets are sampled and represented as meshes of medial primitives results on populations of up to 184 biological objects clearly demonstrate that the common medial branching topology can be described by a small number of medial sheets and that even a coarse sampling leads to a close approximation of individual objects

a skeletonization algorithm by maxima tracking on euclidean distance transform abstract a simple and efficient algorithm using the maxima tracking approach on euclidean distance transform to detect skeleton points is presented the advantages of the skeleton obtained are 1 connectivity preservation 2 single pixel in width and 3 its locations as close as to the most symmetrical axes besides the condition of the least slope change of skeleton is used to ensure the fairness of the digital medial axes with the least effort the algorithm can be modified to eliminate non significant short skeletal branches originating from the object contour while the critical shape informative medial axes are preserved

euclidean skeletons of digital image and volume data in linear time by the integer medial axis transform a general algorithm for computing euclidean skeletons of 2d and 3d data sets in linear time is presented these skeletons are defined in terms of a new concept called the integer medial axis ima transform we prove a number of fundamental properties of the ima skeleton and compare these with properties of the cmd centers of maximal disks skeleton several pruning methods for ima skeletons are introduced constant linear and square root pruning and their properties studied the algorithm for computing the ima skeleton is based upon the feature transform using a modification of a linear time algorithm for euclidean distance transforms the skeletonization algorithm has a time complexity which is linear in the number of input points and can be easily parallelized we present experimental results for several data sets looking at skeleton quality memory usage and computation time both for 2d images and 3d volumes