the image foresting transform theory algorithms and applications the image foresting transform ift is a graph based approach to the design of image processing operators based on connectivity it naturally leads to correct and efficient implementations and to a better understanding of how different operators relate to each other we give here a precise definition of the ift and a procedure to compute it a generalization of dijkstra s algorithm with a proof of correctness we also discuss implementation issues and illustrate the use of the ift in a few applications

voxel based surface area estimation from theory to practice abstract consider a complex highly convoluted three dimensional object that has been digitized and is available as a set of voxels how can one estimate the original continuous area of a region of interest on the surface of the object the problem appears in the analysis of segmented mri brain data and in other three dimensional imaging applications several difficulties arise first due to the complexity of the surface and its foldings the region of interest and its intended boundary can be concealed and are therefore difficult to delineate second the correct surface topology on intricate voxel sets may not be obvious third the surface area of a digital voxel world is generally very different than the area of the underlying continuous surface these problems can be partly circumvented by transforming the voxel data to a polyhedral surface representation our challenge is to accomplish the task while maintaining the original voxel representation estimators for the continuous surface area of digital objects have been available for some time however the known methods are limited to fairly smooth and well behaved surfaces this research bridges the gap between the available surface area estimation theory that applies to idealized settings and the reality of mri brain data surface connectivity ambiguities are alleviated by considering the object background boundary voxel faces rather than the border voxels themselves the region of interest on the surface is delimited by growing geodesic paths between user provided anchor points surface estimation is extended to admit surfaces with higher curvature than previously considered performance evaluation results are provided and operation on mri brain data is demonstrated

a new approach to computing the euler characteristic in this paper we describe a new approach to computing the euler characteristic of a three dimensional digital image our approach is based on computing the change in numbers of black components tunnels and cavities in 3 x 3 x 3 neighborhood of an object black point due to its deletion the existing algorithms to computing the euler characteristic of a 3d digital image are based on counting the numbers of all k dimensional elements 0 k 3 in a polyhedral representation of the image our approach can be modified for 6 26 18 6 6 18 and other connectivity relations of grid points a parallel implementation of the algorithm is described using the concept of sub fields

improved estimates of statistical regularization parameters in fourier differentiation and smoothing we investigate the statistical methods of cross validation cv and maximum likelihood ml for estimating optimal regularization parameters in the numerical differentiation and smoothing of non exact data various criteria for optimality are examined and the asymptotic notions of strong optimality weak optimality and suboptimality are introduced relative to these criteria by restricting attention to certainn dimensional hilbert spaces of smooth and stochastic functions wheren is the number of data we give regularity conditions on the data under which cv the regularization parameter predicted by cv is strongly optimal with respect to the predictive mean square signal error we show that ml is at best weakly optimal with respect to this criterion but is strongly optimal with respect to the innovation variance of the data for numerical differentiation cv and ml are both shown to be suboptimal with respect to the predictive mean square derivative error

smoothing splines regression derivatives and deconvolution abstract the statistical properties of a cubic smoothing spline and its derivative are analyzed it is shown that unless unnatural boundary conditions hold the integrated squared bias is dominated by local effects near the boundary similar effects are shown to occur in the regularized solution of a translation kernel integral equation these results are derived by developing a fourier representation for a smoothing spline author

geometrically accurate topology correction of cortical surfaces using nonseparating loops in this paper we focus on the retrospective topology correction of surfaces we propose a technique to accurately correct the spherical topology of cortical surfaces specifically we construct a mapping from the original surface onto the sphere to detect topological defects as minimal nonhomeomorphic regions the topology of each defect is then corrected by opening and sealing the surface along a set of nonseparating loops that are selected in a bayesian framework the proposed method is a wholly self contained topology correction algorithm which determines geometrically accurate topologically correct solutions based on the magnetic resonance imaging mri intensity profile and the expected local curvature applied to real data our method provides topological corrections similar to those made by a trained operator

a cellular model for multi objects multi dimensional homotopic deformations abstract we deal in this paper with complex three dimensional scenes partitioned into several objects of any shape and with varying local dimensions we propose a consistent topological modeling of such scenes based on a cellular model we introduce a definition of simple elements for any adjacency graph based geometrical structure then we prove a local characterization of simple cells of a cellular model in the case of two objects and of any number of objects this characterization allows to define homotopic deformations of a complex scene

brief paper h filtering for uncertain stochastic time delay systems with sector bounded nonlinearities in this paper we deal with the robust h filtering problem for a class of uncertain nonlinear time delay stochastic systems the system under consideration contains parameter uncertainties ito type stochastic disturbances time varying delays as well as sector bounded nonlinearities we aim at designing a full order filter such that for all admissible uncertainties nonlinearities and time delays the dynamics of the filtering error is guaranteed to be robustly asymptotically stable in the mean square while achieving the prescribed h disturbance rejection attenuation level by using the lyapunov stability theory and ito s differential rule sufficient conditions are first established to ensure the existence of the desired filters which are expressed in the form of a linear matrix inequality lmi then the explicit expression of the desired filter gains is also characterized finally a numerical example is exploited to show the usefulness of the results derived

a proportional derivative control strategy for restarting the gmres m algorithm abstract restarted gmres or gmres m is normally used for solving large linear systems a x b with a general possibly nonsymmetric matrix a although the restarted gmres consumes less computational time than its counterpart full gmres if the restarting parameter is not correctly chosen its convergence cannot be guaranteed and the method may converge slowly unfortunately it is difficult to know how to choose this parameter a priori in this article we regard the gmres m method as a control problem in which the parameter m is the controlled variable and propose a new control inspired strategy for choosing the parameter m adaptively at each iteration the advantage of this control strategy method is that only a few additional vectors need to be stored and the controller has the capacity to modify the dimension of the krylov subspace whenever any convergence problem is detected numerical experiments based on benchmark problems show that the proposed control strategy accelerates the convergence of gmres m

a comparison of algorithms for connected set openings and closings the implementation of morphological connected set operators for image filtering and pattern recognition is discussed two earlier algorithms based on priority queues and hierarchical queues respectively are compared to a more recent union find approach unlike the earlier algorithms which process regional extrema in the image sequentially the union find method allows simultaneous processing of extrema in the context of area openings closings and pattern spectra the union find algorithm outperforms the previous methods on almost all natural and synthetic images tested finally extensions to pattern spectra and the more general class of attribute operators are presented for all three algorithms and memory usages are compared

interpolation of 3 d binary images based on morphological skeletonization the morphological skeleton interpolation msi algorithm is presented it is an efficient shape based interpolation method used for interpolating slices in a three dimensional 3 d binary object it is based on morphological skeletonization which is used for two dimensional 2 d slice representation the proposed morphological skeleton matching process provides translation rotation and scaling information at the same time the interpolated slices preserve the shape of the original object slices when the slices have similar shapes it can also modify the shape of an object when the successive slices do not have similar shapes applications on artificial and real data are also presented

nonlinear image labeling for multivalued segmentation we describe a framework for multivalued segmentation and demonstrate that some of the problems affecting common region based algorithms can be overcome by integrating statistical and topological methods in a nonlinear fashion we address the sensitivity to parameter setting the difficulty with handling global contextual information and the dependence of results on analysis order and on initial conditions we develop our method within a theoretical framework and resort to the definition of image segmentation as an estimation problem we show that thanks to an adaptive image scanning mechanism there is no need of iterations to propagate a global context efficiently the keyword multivalued refers to a result property which spans over a set of solutions the advantage is twofold first there is no necessity for setting a priori input thresholds secondly we are able to cope successfully with the problem of uncertainties in the signal model to this end we adopt a modified version of fuzzy connectedness which proves particularly useful to account for densitometric and topological information simultaneously the algorithm was tested on several synthetic and real images the peculiarities of the method are assessed both qualitatively and quantitatively

a locally constrained watershed transform the watershed transform from mathematical morphology is a powerful and flexible tool for segmentation however it does not allow a priori knowledge relating to characteristics of region boundaries to be included in the way that other approaches do this paper introduces the locally constrained watershed transform which includes border constraints by modifying the underlying path definition upon which the watershed transform depends this approach maintains many of the desirable properties of the watershed transform such as well defined stopping conditions and efficient implementation while offering more stable segmentation in the presence of noisy or incomplete boundaries

multiobjective output feedback control via lmi optimization this paper presents an overview of a linear matrix inequality lmi approach to the multiobjective synthesis of linear output feedback controllers the design objectives can be a mix of h sub spl infin performance h sub 2 performance passivity asymptotic disturbance rejection time domain constraints and constraints on the closed loop pole location in addition these objectives can be specified on different channels of the closed loop system when all objectives are formulated in terms of a common lyapunov function controller design amounts to solving a system of linear matrix inequalities the validity of this approach is illustrated by a realistic design example

flat zones filtering connected operators and filters by reconstruction this correspondence deals with the notion of connected operators starting from the definition for operator acting on sets it is shown how to extend it to operators acting on function typically a connected operator acting on a function is a transformation that enlarges the partition of the space created by the flat zones of the functions it is shown that from any connected operator acting on sets one can construct a connected operator for functions however it is not the unique way of generating connected operators for functions moreover the concept of pyramid is introduced in a formal way it is shown that if a pyramid is based on connected operators the flat zones of the functions increase with the level of the pyramid in other words the flat zones are nested filters by reconstruction are defined and their main properties are presented finally some examples of application of connected operators and use of flat zones are described

snakes active contour models a snake is an energy minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges snakes are active contour models they lock onto nearby edges localizing them accurately scale space continuation can be used to enlarge the capture region surrounding a feature snakes provide a unified account of a number of visual problems including detection of edges lines and subjective contours motion tracking and stereo matching we have used snakes successfully for interactive interpretation in which user imposed constraint forces guide the snake near features of interest

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

determining simplicity and computing topological change in strongly normal partial tilings of r2 or r3 abstract a convex polygon in r2 or a convex polyhedron in r3 will be called a tile a connected set p of tiles is called a partial tiling if the intersection of any two of the tiles is either empty or is a vertex or edge in r3 or face of both p is called strongly normal sn if for any partial tiling p p and any tile p p the neighborhood n p p of p the union of the tiles of p that intersect p is simply connected let p be sn and let n p p be the excluded neighborhood of p in p i e the union of the tiles of p other than p itself that intersect p we call p simple in p if n p p and n p p are topologically equivalent this paper presents methods of determining for an sn partial tiling p whether a tile p p is simple and if not of counting the numbers of components and holes in r3 components tunnels and cavities in n p p

a control theoretic study on iterative solutions to nonlinear equations for applications in embedded systems in this paper the fixed point iteration and newton s methods for iteratively solving nonlinear equations are studied in the control theoretical framework this work is motivated by the ever increasing demands for integrating iterative solutions of nonlinear functions into embedded control systems the use of the well established control theoretical methods for our application purpose is inspired by the recent control theoretical study on numerical analysis our study consists of two parts in the first part the existing fixed point iteration and newton s methods are analysed using the stability theory for the sector bounded lure s systems the second part is devoted to the modified iteration methods and the integration of sensor signals into the iterative computations the major results achieved in our study are besides some academic examples applied to the iterative computation of the air path model embedded in the engine control systems

connected shape size pattern spectra for rotation and scale invariant classification of gray scale images in this paper we describe a multiscale and multishape morphological method for pattern based analysis and classification of gray scale images using connected operators compared with existing methods which use structuring elements our method has three advantages first in our method the time needed for computing pattern spectra does not depend on the number of scales or shapes used i e the computation time is independent of the dimensions of the pattern spectrum second size and strict shape attributes can be computed which we use for the construction of joint 2d shape size pattern spectra third our method is significantly less sensitive to noise and is rotation invariant although rotation invariance can also be approximated by methods using structuring elements at different angles this tends to be computationally intensive the classification performance of these methods is discussed using four image sets brodatz coil 20 coil 100 and diatoms the new method obtains better or equal classification performance to the best competitor with a 5 to 9 fold speed gain

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

system theory for numerical analysis many numerical schemes can be suitably studied from a system theoretic point of view this paper studies the relationship between the two disciplines that is numerical analysis and system theory we first see that various iterative solution schemes for linear and nonlinear equations can be suitably transformed into the form of a closed loop feedback system and show the crucial role of the internal model principle in such a context this leads to new stability criteria for newton s method we then study runge kutta type methods for solving differential equations and also derive new stability criteria based on recent results on lmi a numerical example is given to illustrate the advantage of the present theory

discrete representation of straight lines if a continuous straight line segment is digitized on a regular grid obviously a loss of information occurs as a result the discrete representation obtained e g a chaincode string can be coded more conveniently than the continuous line segment but measurements of properties such as line length performed on the representation have an intrinsic inaccuracy due to the digitization process in this paper two fundamental properties of the quantization of straight line segments are treated 1 it is proved that every straight chaincode string can be represented by a set of four unique integer parameters definitions of these parameters are given 2 a mathematical expression is derived for the set of all continuous line segments which could have generated a given chaincode string the relation with the chord property is briefly discussed

extensive partition operators gray level connected operators and region merging classification segmentation algorithms theoretical links the relation between morphological gray level connected operators and segmentation algorithms based on region merging classification strategies has been pointed out several times in the literature however to the best of our knowledge the formal relation between them has not been established this paper presents the link between the two domains based on the observation that both connected operators and segmentation algorithms share a key mechanism they simultaneously operate on images and on partitions and therefore they can be described as operations on a joint image partition model as a result we analyze both segmentation algorithms and connected operators by defining operators on complete product lattices that explicitly model gray level and partition attributes in the first place starting with a complete lattice of partitions we initially define the concept of the segmentation model as a mapping in a product lattice whose elements are three tuples consisting of a partition an image that models the partition attributes and an image that represents the gray level model associated to the segmentation then assuming a conditional ordering relation we show that any region merging classification segmentation algorithm can be defined as an extensive operator in such a complete product lattice in the second place we proposed a very similar lattice based extended representation of gray level functions in the context of connected operators that highlights the mathematical analogy with segmentation algorithms but in which the ordering relation is different we use this framework to show that every region merging classification segmentation algorithm indeed corresponds to a connected operator while this result provides an explanation to previous work in the area it also opens possibilities for further analysis in the two domains from this perspective we additionally study some theoretical properties of a general region merging segmentation algorithm

topology preservation in 3d digital space abstract in a 3d two tone digital space the conditions under which sequential transformation of an object point to a non object point or its inverse does not affect the image topology are investigated the concept of border connectivity is introduced and an efficient algorithm for checking a simple point is proposed in this connection some other interesting properties of 3d digital geometry are also discussed this investigation may be used in thinning and topology preserving morphological transformations in 3d digital space

a topology preserving parallel 3d thinning algorithm for extracting the curve skeleton we introduce a new topology preserving 3d thinning procedure for deriving the curve voxel skeleton from 3d binary digital images based on a rigorously defined classification procedure the algorithm consists of sequential thinning iterations each characterized by six parallel directional sub iterations followed by a set of sequential sub iterations the algorithm is shown to produce concise and geometrically accurate 3d curve skeletons the thinning algorithm is also insensitive to object rotation and only moderately sensitive to noise although this thinning procedure is valid for curve skeleton extraction of general elongated objects in this paper we specifically discuss its application to the orientation modeling of trabecular biological tissues

watershed cuts thinnings shortest path forests and topological watersheds we recently introduced watershed cuts a notion of watershed in edge weighted graphs in this paper our main contribution is a thinning paradigm from which we derive three algorithmic watershed cut strategies the first one is well suited to parallel implementations the second one leads to a flexible linear time sequential implementation whereas the third one links the watershed cuts and the popular flooding algorithms we state that watershed cuts preserve a notion of contrast called connection value on which several morphological region merging methods are implicitly based we also establish the links and differences between watershed cuts minimum spanning forests shortest path forests and topological watersheds finally we present illustrations of the proposed framework to the segmentation of artwork surfaces and diffusion tensor images

non product property of the digital fundamental group for given two digital images both equipped with each digital connectedness we show that the digital fundamental groups of both images do not necessarily lead to the digital fundamental group of the product image in order to prove the non product property of the digital fundamental group of the digital product image we need the notion of product image and some calculation of the digital 8 fundamental group of the digital 8 pseudotorus in z4 by means of a counterexample

deformable templates using large deformation kinematics a general automatic approach is presented for accommodating local shape variation when mapping a two dimensional 2 d or three dimensional 3 d template image into alignment with a topologically similar target image local shape variability is accommodated by applying a vector field transformation to the underlying material coordinate system of the template while constraining the transformation to be smooth globally positive definite jacobian smoothness is guaranteed without specifically penalizing large magnitude deformations of small subvolumes by constraining the transformation on the basis of a stokesian limit of the fluid dynamical navier stokes equations this differs fundamentally from quadratic penalty methods such as those based on linearized elasticity or thin plate splines in that stress restraining the motion relaxes over time allowing large magnitude deformations kinematic nonlinearities are inherently necessary to maintain continuity of structures during large magnitude deformations and are included in all results after initial global registration final mappings are obtained by numerically solving a set of nonlinear partial differential equations associated with the constrained optimization problem automatic regridding is performed by propagating templates as the nonlinear transformations evaluated on a finite lattice become singular application of the method to intersubject registration of neuroanatomical structures illustrates the ability to account for local anatomical variability

antiextensive connected operators for image and sequence processing this paper deals with a class of morphological operators called connected operators these operators filter the signal by merging its flat zones as a result they do not create any new contours and are very attractive for filtering tasks where the contour information has to be preserved this paper shows that connected operators work implicitly on a structured representation of the image made of flat zones the max tree is proposed as a suitable and efficient structure to deal with the processing steps involved in antiextensive connected operators a formal definition of the various processing steps involved in the operator is proposed and as a result several lines of generalization are developed first the notion of connectivity and its definition are analyzed several modifications of the traditional approach are presented they lead to connected operators that are able to deal with texture they also allow the definition of connected operators with less leakage than the classical ones second a set of simplification criteria are proposed and discussed they lead to simplicity entropy and motion oriented operators the problem of using a nonincreasing criterion is analyzed its solution is formulated as an optimization problem that can be very efficiently solved by a viterbi 1979 algorithm finally several implementation issues are discussed showing that these operators can be very efficiently implemented

binary partition tree as an efficient representation for image processing segmentation and information retrieval this paper discusses the interest of binary partition trees as a region oriented image representation binary partition trees concentrate in a compact and structured representation a set of meaningful regions that can be extracted from an image they offer a multiscale representation of the image and define a translation invariant 2 connectivity rule among regions as shown in this paper this representation can be used for a large number of processing goals such as filtering segmentation information retrieval and visual browsing furthermore the processing of the tree representation leads to very efficient algorithms finally for some applications it may be interesting to compute the binary partition tree once and to store it for subsequent use for various applications in this context the paper shows that the amount of bits necessary to encode a binary partition tree remains moderate

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

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

computing distance transformations in convex and non convex domains abstract it is shown that the well known two pass sequential local transformation algorithm for computing a distance transformation in rectangular domains may fail in some convex integer domains but that a four pass algorithm is sufficient in all two dimensional convex domains for non convex domains the number of passes necessary is shown to be generally greater two propagation algorithms for computing the distance transformation are described and shown theoretically and experimentally to be computationally more efficient than the sequential local transformation algorithm in non convex domains of complex shape the relationship of the distance transformation in non convex domains to some more general transformations is explored

a topological characterization of thinning abstract a large number of skeletonization algorithms for binary images use the method of thinning successive layers of pixels are deleted from the figure until it becomes one pixel thick in this paper we analyze the topological properties of the set d of pixels to be deleted from a figure f in order to get a skeleton we characterize them by the concept of strong k deletability k 4 or 8 for individual pixels strong k deletability is equivalent to a more general property that we call k deletability which is well known connectivity requirement assumed at least implicity in all existing thinning algorithms we show then that a strongly k deletable subset d of a figure f can be deleted by a succession of deletions of individual pixels p 1 p t where each p i is k deletable from f p j j i this justifies our definition of strong deletability and shows that any topologically valid skeleton can be obtained by some thinning process

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

a comparison of gcv and gml for choosing the smoothing parameter in the generalized spline smoothing problem the partially improper prior behind the smoothing spline model is used to obtain a generalization of the maximum likelihood gml estimate for the smoothing parameter then this estimate is compared with the generalized cross validation gcv estimate both analytically and by monte carlo methods the comparison is based on a predictive mean square error criteria it is shown that if the true unknown function being estimated is smooth in a sense to be defined then the gml estimate undersmooths relative to the gcv estimate and the predictive mean square error using the gml estimate goes to zero at a slower rate than the mean square error using the gcv estimate if the true function is rough then the gcv and gml estimates have asymptotically similar behavior a monte carlo experiment was designed to see if the asymptotic results in the smooth case were evident in small sample sizes mixed results were obtained for n 32 gcv was somewhat better than gml for n 64 and gcv was decidedly superior for n 128 in the n 32 case gcv was better for smaller sigma 2 and the comparison close for larger sigma 2 the theoretical results are shown to extend to the generalized spline smoothing model which includes the estimate of functions given noisy values of various integrals of them

a topology preserving level set method for geometric deformable models active contour and surface models also known as deformable models are powerful image segmentation techniques geometric deformable models implemented using level set methods have advantages over parametric models due to their intrinsic behavior parameterization independence and ease of implementation however a long claimed advantage of geometric deformable models the ability to automatically handle topology changes turns out to be a liability in applications where the object to be segmented has a known topology that must be preserved we present a new class of geometric deformable models designed using a novel topology preserving level set method which achieves topology preservation by applying the simple point concept from digital topology these new models maintain the other advantages of standard geometric deformable models including subpixel accuracy and production of nonintersecting curves or surfaces moreover since the topology preserving constraint is enforced efficiently through local computations the resulting algorithm incurs only nominal computational overhead over standard geometric deformable models several experiments on simulated and real data are provided to demonstrate the performance of this new deformable model algorithm

shape based interpolation of multidimensional grey level images shape based interpolation as applied to binary images causes the interpolation process to be influenced by the shape of the object it accomplishes this by first applying a distance transform to the data this results in the creation of a grey level data set in which the value at each point represents the minimum distance from that point to the surface of the object by convention points inside the object are assigned positive values points outside are assigned negative values this distance transformed data set is then interpolated using linear or higher order interpolation and is then thresholded at a distance value of zero to produce the interpolated binary data set here the authors describe a new method that extends shape based interpolation to grey level input data sets this generalization consists of first lifting the n dimensional n d image data to represent it as a surface or equivalently as a binary image in an n 1 dimensional n 1 d space the binary shape based method is then applied to this image to create an n 1 d binary interpolated image finally this image is collapsed inverse of lifting to create the n d interpolated grey level data set the authors have conducted several evaluation studies involving patient computed tomography ct and magnetic resonance mr data as well as mathematical phantoms they all indicate that the new method produces more accurate results than commonly used grey level linear interpolation methods although at the cost of increased computation

minimal non deletable sets and minimal non codeletable sets in binary images the concepts of strongly 8 deletable and strongly 4 deletable sets of 1s in binary images on the 2d cartesian grid were introduced by ronse in the mid 1980s to formalize the connectivity preservation conditions that parallel thinning algorithms are required to satisfy in this paper we call these sets deletable and codeletable respectively to establish that a proposed parallel thinning algorithm for binary images on the 2d cartesian grid preserves 8 4 connected foreground components and 4 8 connected background components it is enough to prove that the set of 1s which are changed to 0s at each pass of the algorithm is always a deletable codeletable set ronse established results that are very useful in this context for proving that a finite set d of 1s is deletable or codeletable in particular he showed that d and its proper subsets are all codeletable in a binary image if each singleton and each pair of 8 adjacent pixels in d is codeletable he further showed that d and its proper subsets are all deletable in a binary image if 1 each singleton and each pair of 4 adjacent pixels in d is deletable and 2 no set of 2 3 or 4 pairwise 8 adjacent pixels that is an 8 connected foreground component of the image is entirely contained in d in the 1990s and early 2000s analogous results were obtained by hall ma gau and the author for binary images on the 2d hexagonal grid the 3d cartesian and face centered cubic grids and the 4d cartesian grid this paper extends the above mentioned work to binary images on almost any polytopal complex whose union is n dimensional euclidean space for n 4 our main results generalize and unify the corresponding results of the earlier work

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

automatic detection of optic disc based on pca and mathematical morphology the algorithm proposed in this paper allows to automatically segment the optic disc from a fundus image the goal is to facilitate the early detection of certain pathologies and to fully automate the process so as to avoid specialist intervention the method proposed for the extraction of the optic disc contour is mainly based on mathematical morphology along with principal component analysis pca it makes use of different operations such as generalized distance function gdf a variant of the watershed transformation the stochastic watershed and geodesic transformations the input of the segmentation method is obtained through pca the purpose of using pca is to achieve the grey scale image that better represents the original rgb image the implemented algorithm has been validated on five public databases obtaining promising results the average values obtained a jaccard s and dice s coefficients of 0 8200 and 0 8932 respectively an accuracy of 0 9947 and a true positive and false positive fractions of 0 9275 and 0 0036 demonstrate that this method is a robust tool for the automatic segmentation of the optic disc moreover it is fairly reliable since it works properly on databases with a large degree of variability and improves the results of other state of the art methods

shape based interpolation of multidimensional objects a shape based interpolation scheme for multidimensional images is presented this scheme consists of first segmenting the given image data into a binary image converting the binary image back into a gray image wherein the gray value of a point represents its shortest distance positive value for points of the object and negative for those outside from the cross sectional boundary and then interpolating the gray image the set of all points with nonnegative values associated with them in the interpolated image constitutes the interpolated object the method not only minimizes user involvement in interactive segmentation but also leads to more accurate representation and depiction of dynamic as well as static objects the general methodology and the implementation details of the method are presented and compared on a qualitative and quantitative basis to the existing methods the generality of the proposed scheme is illustrated with a number of medical imaging examples

three dimensional computer aided diagnosis scheme for detection of colonic polyps we have developed a three dimensional 3 d computer aided diagnosis scheme for automated detection of colonic polyps in computed tomography ct colonographic data sets and assessed its performance based on colonoscopy as the gold standard in this scheme a thick region encompassing the entire colonic wall is extracted from an isotropic volume reconstructed from the ct images in ct colonography polyp candidates are detected by first computing of 3 d geometric features that characterize polyps folds and colonic walls at each voxel in the extracted colon and then segmenting of connected components corresponding to suspicious regions by hysteresis thresholding based on these geometric features we apply fuzzy clustering to these connected components to obtain the polyp candidates false positive fp detections are then reduced by computation of several 3 d volumetric features characterizing the internal structures of the polyp candidates followed by the application of discriminant analysis to the feature space generated by these volumetric features the locations of the polyps detected by our computerized method were compared to the gold standard of conventional colonoscopy the performance was evaluated based on 43 clinical cases including 12 polyps determined by colonoscopy our computerized scheme was shown to have the potential to detect polyps in ct colonography with a clinically acceptable high sensitivity and a low fp rate

new characterizations of simple points in 2d 3d and 4d discrete spaces a point of a discrete object is called simple if it can be deleted from this object without altering topology in this article we present new characterizations of simple points which hold in dimensions 2 3 and 4 and which lead to efficient algorithms for detecting such points in order to prove these characterizations we establish two confluence properties of the collapse operation which hold in the neighborhood of a point in spaces of low dimension this work is settled in the framework of cubical complexes which provides a sound topological basis for image analysis and allows to retrieve the main notions and results of digital topology in particular the notion of simple point

a width independent fast thinning algorithm the skeleton of a digital figure can often be regarded as a convenient alternative to the figure itself it is useful both to diminish drastically the amount of data to be handled and to simplify the computational procedures required for description and classification purposes thinning a digital figure down to its skeleton is a time consuming process when conventional sequential computers are employed the procedure we propose allows one to speed up the thinning transformation and to get a well shaped skeleton after cleaning of the input picture has been performed the pixels of the figure are labeled according to their distance from the background and a set whose pixels are symmetrically placed with respect to distinct contour parts of the figure is found this set is then given a linear structure by applying topology preserving removal operations finally a pruning step regarding branches not relevant in the framework of the problem domain completes the process the resulting skeleton is a labeled set of pixels which is shown to possess all the required properties particularly those concerning connectedness topology and shape moreover the original figure can almost completely be recovered by means of a reverse distance transformation only a fixed and small number of sequential passes through the picture is necessary to achieve the goal the computational effort is rather modest and the use of the proposed algorithm turns out to be more advantageous the greater the width of the figure to be thinned

spiral ct based assessment of tracheal stenoses using 3 d skeletonization demonstration of a technique for three dimensional 3 d assessment of tracheal stenoses regarding site length and degree based on spiral computed tomography s ct s ct scanning and automated segmentation of the laryngo tracheal tract ltt was followed by the extraction of the ltt medial axis using a skeletonization algorithm orthogonal to the medial axis the ltt 3 d cross sectional profile was computed and presented as line charts where degree and length was obtained values for both parameters were compared between 36 patients and 18 normal controls separately accuracy and precision was derived from 17 phantom studies average degree and length of tracheal stenoses was found to be 60 5 and 4 32 cm in patients compared with minor caliber changes of 8 8 and 2 31 cm in normal controls p spl lt 0 0001 for the phantoms an excellent correlation between the true and computed 3 d cross sectional profile was found p spl lt 0 005 and an accuracy for length and degree measurements of 2 14 mm and 2 53 respectively could be determined the corresponding figures for the precision were found to be 0 92 mm and 2 56 ltt 3 d cross sectional profiles permit objective accurate and precise assessment of ltt caliber changes minor ltt caliber changes can be observed even in normals and in case of an otherwise normal s ct study can be regarded as artifacts

computational experience with the spectral smoothing method for differentiating noisy data abstract when applied to non exact noisy data numerical methods for calculating derivatives in particular derivatives of order higher than the first based on model functions fitted to exact data become unsatisfactory the spectral smoothing method of anderssen and bloomfield developed to solve this problem entails calculation of a smoothing parameter and the choice of an optimal order sobolev norm that is used as regularizer this method is used to differentiate smooth and integrate noisy data a likelihood function is minimized to determine the smoothing parameter we present numerical results suggesting that this function can be jointly minimized with respect to the smoothing parameter and the order of the regularizing norm thus yielding a fully automatic numerical differentiation procedure

numerical experimentation with time series methods for convolution integral equation iii posedness of integral equations of the first kind is well known in this paper we investigate the linear fredholm integral equation of convolution type defined by kf x email protected k x y f y dy g x

a control theoretic approach to the design of zero finding numerical methods in this paper it is shown how standard iterative methods for solving linear and nonlinear equations can be designed from the point of view of control appropriate choices of control lyapunov functions clfs lead to both continuous and discrete time versions of the newton raphson and conjugate gradient algorithms as well as new variants

distance driven skeletonization in voxel images a distance driven method to compute the surface and curve skeletons of 3d objects in voxel images is described the method is based on the use of the weighted distance transform on the detection of anchor points and on the application of topology preserving removal operations the obtained surface and curve skeletons are centered within the object have the same topology as the object and have unit thickness the object can be almost completely recovered from the surface skeleton since this includes almost all of the centers of maximal balls of the object hence the surface skeleton is a faithful representation in turn though only partial recovery is possible from the curve skeleton this still provides an appealing representation of the object

survey interpolation methods in medical image processing image interpolation techniques often are required in medical imaging for image generation e g discrete back projection for inverse radon transform and processing such as compression or resampling since the ideal interpolation function spatially is unlimited several interpolation kernels of finite size have been introduced this paper compares 1 truncated and windowed sine 2 nearest neighbor 3 linear 4 quadratic 5 cubic b spline 6 cubic g lagrange and 7 gaussian interpolation and approximation techniques with kernel sizes from 1 spl times 1 up to 8 spl times 8 the comparison is done by 1 spatial and fourier analyses 2 computational complexity as well as runtime evaluations and 3 qualitative and quantitative interpolation error determinations for particular interpolation tasks which were taken from common situations in medical image processing for local and fourier analyses a standardized notation is introduced and fundamental properties of interpolators are derived successful methods should be direct current dc constant and interpolators rather than dc inconstant or approximators each method s parameters are tuned with respect to those properties this results in three novel kernels which are introduced in this paper and proven to be within the best choices for medical image interpolation the 6 spl times 6 blackman harris windowed sinc interpolator and the c2 continuous cubic kernels with n 6 and n 8 supporting points for quantitative error evaluations a set of 50 direct digital x rays was used they have been selected arbitrarily from clinical routine in general large kernel sizes were found to be superior to small interpolation masks except for truncated sine interpolators all kernels with n 6 or larger sizes perform significantly better than n 2 or n 3 point methods p spl lt 0 005 however the differences within the group of large sized kernels were not significant summarizing the results the cubic 6 spl times 6 interpolator with continuous second derivatives as defined in 24 can be recommended for most common interpolation tasks it appears to be the fastest six point kernel to implement computationally it provides eminent local and fourier properties is easy to implement and has only small errors the same characteristics apply to b spline interpolation but the 6 spl times 6 cubic avoids the intrinsic border effects produced by the b spline technique however the goal of this study was not to determine an overall best method but to present a comprehensive catalogue of methods in a uniform terminology to define general properties and requirements of local techniques and to enable the reader to select that method which is optimal for his specific application in medical imaging

homotopy in two dimensional digital images we recall the basic definitions concerning homotopy in 2d digital topology and we set and prove several results concerning homotopy of subsets then we introduce an explicit isomorphism between the fundamental group and a free group as a consequence we provide an algorithm for deciding whether two closed path are homotopic

watershed cuts minimum spanning forests and the drop of water principle we study the watersheds in edge weighted graphs we define the watershed cuts following the intuitive idea of drops of water flowing on a topographic surface we first establish the consistency of these watersheds they can be equivalently defined by their catchment basinsrdquo through a steepest descent property or by the dividing linesrdquo separating these catchment basins through the drop of water principle then we prove through an equivalence theorem their optimality in terms of minimum spanning forests afterward we introduce a linear time algorithm to compute them to the best of our knowledge similar properties are not verified in other frameworks and the proposed algorithm is the most efficient existing algorithm both in theory and in practice finally the defined concepts are illustrated in image segmentation leading to the conclusion that the proposed approach improves on the tested images the quality of watershed based segmentations

shapes shocks and deformations i the components of two dimensional shape and the reaction diffusion space we undertake to develop a general theory of two dimensional shape by elucidating several principles which any such theory should meet the principles are organized around two basic intuitions first if a boundary were changed only slightly then in general its shape would change only slightly this leads us to propose an operational theory of shape based on incremental contour deformations the second intuition is that not all contours are shapes but rather only those that can enclose physical material a theory of contour deformation is derived from these principles based on abstract conservation principles and hamilton jacobi theory these principles are based on the work of sethian 1985a c the osher sethian 1988 level set formulation the classical shock theory of lax 1971 1973 as well as curve evolution theory for a curve evolving as a function of the curvature and the relation to geometric smoothing of gage hamilton grayson 1986 1989 the result is a characterization of the computational elements of shape deformations parts bends and seeds which show where to place the components of a shape the theory unifies many of the diverse aspects of shapes and leads to a space of shapes the reaction diffusion space which places shapes within a neighborhood of similar ones such similarity relationships underlie descriptions suitable for recognition

shape based interpolation of tree like structures in three dimensional images many three dimensional 3 d medical images have lower resolution in the z direction than in the x or y directions before extracting and displaying objects in such images an interpolated 3 d gray scale image is usually generated via a technique such as linear interpolation to fill in the missing slices unfortunately when objects are extracted and displayed from the interpolated image they often exhibit a blocky and generally unsatisfactory appearance a problem that is particularly acute for thin treelike structures such as the coronary arteries two methods for shape based interpolation that offer an improvement to linear interpolation are presented in shape based interpolation the object of interest is first segmented extracted from the initial 3 d image to produce a low z resolution binary valued image and the segmented image is interpolated to produce a high resolution binary valued 3 d image the first method incorporates geometrical constraints and takes as input a segmented version of the original 3 d image the second method builds on the first in that it also uses the original gray scale image as a second input tests with 3 d images of the coronary arterial tree demonstrate the efficacy of the methods

two symmetrical thinning algorithms for 3d binary images based on p simple points in this paper we propose two 3d symmetrical thinning algorithms based on the parallel deletion of p simple points the first one permits surface skeletons to be obtained the second one permits curve skeletons to be extracted and as far as we know this is the only symmetrical curve thinning algorithm which preserves topology these algorithms have been conceived in order to obtain precise results on simple specific objects parallelepipeds consequently we can predict the number of deletion iterations the number of deleted points and the skeleton of these objects obtained by the two algorithms which is hardly ever possible to assess with other algorithms even on these simple objects

global minimum for active contour models a minimal path approach a new boundary detection approach for shape modeling is presented it detects the global minimum of an active contour model s energy between two end points initialization is made easier and the curve is not trapped at a local minimum by spurious edges we modify the snake energy by including the internal regularization term in the external potential term our method is based on finding a path of minimal length in a riemannian metric we then make use of a new efficient numerical method to find this shortest path it is shown that the proposed energy though based only on a potential integrated along the curve imposes a regularization effect like snakes we explore the relation between the maximum curvature along the resulting contour and the potential generated from the image the method is capable to close contours given only one point on the objects boundary by using a topology based saddle search routine we show examples of our method applied to real aerial and medical images

building the component tree in quasi linear time the level sets of a map are the sets of points with level above a given threshold the connected components of the level sets thanks to the inclusion relation can be organized in a tree structure that is called the component tree this tree under several variations has been used in numerous applications various algorithms have been proposed in the literature for computing the component tree the fastest ones considering the worst case complexity have been proven to run in o nln n in this paper we propose a simple to implement quasi linear algorithm for computing the component tree on symmetric graphs based on tarjan s union find procedure we also propose an algorithm that computes the n most significant lobes of a map

a new shape preserving parallel thinning algorithm for 3d digital images this paper is concerned with a new parallel thinning algorithm for three dimensional digital images that preserves the topology and maintains their shape we introduce an approach of selecting shape points and outer layer used for erosion during each iteration the approach produces good skeleton for different types of corners the concept of using two image versions in thinning is introduced and its necessity in parallel thinning is justified the robustness of the algorithm under pseudo random noise as well as rotation with respect to shape properties is studied and the results are found to be satisfactory

tubular structure segmentation based on minimal path method and anisotropic enhancement we present a new interactive method for tubular structure extraction the main application and motivation for this work is vessel tracking in 2d and 3d images the basic tools are minimal paths solved using the fast marching algorithm this allows interactive tools for the physician by clicking on a small number of points in order to obtain a minimal path between two points or a set of paths in the case of a tree structure our method is based on a variant of the minimal path method that models the vessel as a centerline and surface this is done by adding one dimension for the local radius around the centerline the crucial step of our method is the definition of the local metrics to minimize we have chosen to exploit the tubular structure of the vessels one wants to extract to built an anisotropic metric the designed metric is well oriented along the direction of the vessel admits higher velocity on the centerline and provides a good estimate of the vessel radius based on the optimally oriented flux this measure is required to be robust against the disturbance introduced by noise or adjacent structures with intensity similar to the target vessel we obtain promising results on noisy synthetic and real 2d and 3d images and we present a clinical validation

digital straight lines and convexity of digital regions it is shown that a digital region is convex if and only if every pair of points in the region is connected by a digital straight line segment contained in the region the midpoint property is shown to be a necessary but not a sufficient condition for the convexity of digital regions however it is shown that a digital region is convex if and only if it has the median point property

an efficient uniform cost algorithm applied to distance transforms the uniform cost algorithm is a special case of the a algorithm for finding the shortest paths in graphs in the uniform cost algorithm nodes are expanded in order of increasing cost an efficient version of this algorithm is developed for integer cost values nodes are sorted by storing them at predefined places bucket sort keeping the overhead low the algorithm is applied to general distance transformation a constrained distance transform is an operation which calculates at each pixel of an image the distance to the nearest pixel of a reference set distance being defined as minimum path length the uniform cost algorithm in the constrained case proves to be the best solution for distance transformation it is fast the processing time is independent of the complexity of the image and memory requirements are moderate

interpolation revisited medical images application based on the theory of approximation this paper presents a unified analysis of interpolation and resampling techniques an important issue is the choice of adequate basis functions the authors show that contrary to the common belief those that perform best are not interpolating by opposition to traditional interpolation the authors call their use generalized interpolation they involve a prefiltering step when correctly applied the authors explain why the approximation order inherent in any basis function is important to limit interpolation artifacts the decomposition theorem states that any basis function endowed with approximation order ran be expressed as the convolution of a b spline of the same order with another function that has none this motivates the use of splines and spline based functions as a tunable way to keep artifacts in check without any significant cost penalty the authors discuss implementation and performance issues and they provide experimental evidence to support their claims

geodesic saliency of watershed contours and hierarchical segmentation the watershed is one of the latest segmentation tools developed in mathematical morphology in order to prevent its oversegmentation the notion of dynamics of a minimum based on geodesic reconstruction has been proposed in this paper we extend the notion of dynamics to the contour arcs this notion acts as a measure of the saliency of the contour contrary to the dynamics of minima our concept reflects the extension and shape of the corresponding object in the image this representation is also much more natural because it is expressed in terms of partitions of the plane i e segmentations a hierarchical segmentation process is then derived which gives a compact description of the image containing all the segmentations one can obtain by the notion of dynamics by means of a simple thresholding finally efficient algorithms for computing the geodesic reconstruction as well as the dynamics of contours are presented

a new algorithm for three dimensional voronoi tessellation abstract a new efficient algorithm for computing the voronoi tessellation in three dimensions is presented the voronoi tessellation is obtained by constructing delaunay tetrahedra the complete set of these tetrahedra corresponds to that of a single voronoi polyhedron

nonlinear observers a circle criterion design and robustness analysis globally convergent observers are designed for a class of systems with monotonic nonlinearities the approach is to represent the observer error system as the feedback interconnection of a linear system and a time varying multivariable sector nonlinearity using lmi software observer gain matrices are computed to satisfy the circle criterion and hence to drive the observer error to zero in output feedback design the observer is combined with control laws that ensure input to state stability with respect to the observer error robustness to unmodeled dynamics is achieved with a small gain assignment design as illustrated on a jet engine compressor example

geodesic methods in quantitative image analysis abstract let x be a part of an image to be analysed given two arbitrary points x and y of x let us define the number d x x y as follows d x x y is the lower bound of the lengths of the arcs in x ending at points x and y if such arcs exist and if not the function d x is an x intrinsic distance function called geodesic distance note that if x and y belong to two disjoint connected components of x d x x y in other words d x seems to be an appropriate distance function to deal with connectivity problems in the metric space x d x all the classical morphological transformations dilation erosion skeletonization etc can be defined the geodesic distance d x also provides rigorous definitions of topological transformations which can be performed by automatic image analysers with the help of parallel iterative algorithms all these notions are illustrated by several examples definition of the length of a fibre and of an effective length factor automatic detection of cells having at least one nucleus or having one single nucleus definitions of the geodesic center and of the ends of an object without a hole etc the corresponding algorithms are described