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

the k fundamental group of a closed k surface in this paper we deal with the problem of computing the digital fundamental group of a closed k surface by using various properties of both a simple closed k surface and a digital covering map to be specific let sc k i n i l i be a simple closed k i curve with l i elements in z n i email protected 1 2 then the cartesian product sc k 1 n 1 l 1xsc k 2 n 2 l email protected z n 1 n 2 is not always a closed k surface with some k adjacency of z n 1 n 2 thus we provide a condition for sc k 1 n 1 l 1xsc k 2 n 2 l 2 to be a simple closed k surface with some k adjacency depending on the k i adjacency email protected 1 2 besides even if sc k 1 n 1 l 1xsc k 2 n 2 l 2 is not a closed k surface we show that the k fundamental group of sc k 1 n 1 l 1xsc k 2 n 2 l 2 can be calculated by both a k homotopic thinning and a strong k deformation retract

minimal simple closed 18 surfaces and a topological preservation of 3d surfaces the aim of this paper is to investigate a map which preserves digital topological properties of a digital 3d surface such as the topological number digital k contractibility and so on furthermore the two types of minimal simple closed 18 surfaces mss 1 8 and mss 1 8 are established in z 3 in relation with 18 contractibility which are shown to be different from each other up to digital 18 homotopy moreover it is proved that mss 1 8 is the minimal simple closed 18 surface without 18 contractibility and mss 1 8 is the minimal malgouyres simple closed 18 surface with 18 contractibility finally we show that a digital k 0 k 1 homeomorphism preserves a simple k 0 surface to a simple k 1 surface and vice versa

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

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

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

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

joint physical layer coding and network coding for bidirectional relaying we consider a communication system where two transmitters wish to exchange information through a central relay the transmitter and relay nodes exchange data over synchronized average power constrained additive white gaussian noise channels with a real input with signal to noise ratio snr of snr an upper bound on the capacity is 1 2 log 1 snr bits per transmitter per use of the multiple access phase and broadcast phase of the bidirectional relay channel we show that using lattice codes and lattice decoding we can obtain a rate of 1 2 log 1 2 snr bits per transmitter which is essentially optimal at high snr the main idea is to decode the sum of the codewords modulo a lattice at the relay followed by a broadcast phase which performs slepian wolf coding we also show that if the two transmitters use identical lattices with minimum angle decoding we can achieve the same rate of 1 2 log 1 2 snr the proposed scheme can be thought of as a joint physical layer network layer code which outperforms other recently proposed analog network coding schemes

equivalent k0 k1 covering and generalized digital lifting this paper studies various properties of k 0 k 1 continuity in relation to uniform k 0 k 1 continuity and a pasting theorem which can be used in image synthesis image segmentation and image weaving furthermore we establish an equivalent k 0 k 1 covering and provide some condition to construct the generalized digital lifting of an equivalent k 0 k 1 covering map which is used to calculate the digital fundamental group of a digital image and to classify digital images by a discrete deck s transformation group to be specific let p e e 0 b b 0 be a pointed k 0 k 1 covering map and let f x x 0 b b 0 be a pointed k 2 k 1 continuous map then we can provide a condition to have a concerning pointed k 2 k 0 continuous map email protected x x 0 e e 0 such that email protected email protected f

comparison among digital fundamental groups and its applications the notion of digital fundamental group was originated by khalimsky e khalimsky motion deformation and homotopy in finite spaces proc ieee int conf syst man cybernet 1987 227 234 motivated by this notion three kinds of digital k homotopies as well as the relative k homotopy were established r ayala e dominguez a r frances a quintero homotopy in digital spaces discrete appl math 125 1 2003 3 24 l boxer a classical construction for the digital fundamental group j math imaging vis 10 1999 51 62 s e han connected sum of digital closed surfaces inform sci 176 3 2006a 332 348 t y kong a digital fundamental group comput graphics 13 1989 159 166 r malgouyres homotopy in 2 dimensional digital images theor comput sci 230 2000 221 233 these four notions contributed to the development of three kinds of k fundamental groups of a digital image x k one was established by kong kong 1989 and malgouyres malgouyres 2000 and we denote by p k m k x this digital fundamental group another was developed by boxer boxer 1999 and extended by han han 2006a s e han discrete homotopy of a closed k surface lncs 4040 springer verlag berlin 2006b pp 214 225 s e han equivalent k 0 k 1 covering and generalized digital lifting inform sci 178 2 2008 550 561 by using both the k homotopic thinning han 2006b s e han remarks on digital k homotopy equivalence honam math j 29 1 2007 101 118 and han s digital covering theory s e han digital coverings and their applications j appl math comput 18 1 2 2005 487 495 han 2006b which is denoted as p b h k x in this paper the other was established by ayala et al by using the framework of a multilevel architecture ayala 2003 since each of these digital k fundamental groups has an intrinsic feature of its own and its usages depend on the situation this study is focused on the first two notions p k m k x and p b h k x and intended to show the strong merits of p b h k x in relation to the classification of digital images

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

surfaces in three dimensional digital images this is one of a series of reports on the digital geometry of three dimensional images such as those produced by computed tomography in this report we define simple surface points and simple closed surfaces and show that any connected collection of simple surface points form a simple closed surface thus proving a three dimensional analog of the two dimensional jordan curve theorem we also show that the converse is not a theorem in contrast to the two dimensional case and discuss more complex surface types finally we show that the two dimensional analog of our definition of simple closed surface characterizes simple closed curves but that several other characterizations of 2 d curves when extended to 3 d are not adequate to characterize surfaces

the khalimsky topologies are precisely those simply connected topologies on z n whose connected sets include all 2 n connected sets but no 3 n 1 disconnected sets we give a proof of the result stated in the title here the concepts of 2n and 3n 1 dis connected sets are the natural generalizations to zn of the standard concepts of 4 and 8 dis connected sets in 2d digital topology suppose we have an n dimensional scanner that digitizes n dimensional objects to subsets of zn we are interested in topological spaces zn that might allow standard concepts and methods of general topology to be directly and usefully applied to good digitizations produced by the scanner but our result suggests that if a topological space zn is not a khalimsky space then it will not satisfy our requirement our proof involves some purely discrete arguments and a fact about simply connected polyhedra that is a well known consequence of the simplicial approximation theorem but also uses the following fact which was one of the main results in an earlier paper in r m shortt ed general topology and applications proc 1988 northeast conf marcel dekker new york 1990 pp 153 164 by the author and khalimsky for any t0 topological space in which each point lies in a finite open set and a finite closed set there exists a polyhedron whose vertices are in 1 1 correspondence with the points of the space such that the homotopy classes of continuous maps into the topological space from any metric space are in 1 1 correspondence with the homotopy classes of continuous maps from that metric space into the polyhedron

wyner ziv theory for a general function of the correlated sources corresp a source coding problem is considered for the wyner ziv type system where the decoder is required to estimate the value of some function of the encoder input and the side information the rate distortion function is established for this system and for some binary cases parametric expressions are obtained to enable numerical calculations

on equitable resource allocation problems a lexicographic minimax approach in this expository paper we review a variety of resource allocation problems in which it is desirable to allocate limited resources equitably among competing activities applications for such problems are found in diverse areas including distribution planning production planning and scheduling and emergency services location each activity is associated with a performance function representing for example the weighted shortfall of the selected activity level from a specified target a resource allocation solution is called equitable if no performance function value can be improved without either violating a constraint or degrading an already equal or worse off i e larger performance function value that is associated with a different activity a lexicographic minimax solution determines this equitable solution that is it determines the lexicographically smallest vector whose elements the performance function values are sorted in nonincreasing order the problems reviewed include large scale allocation problems with multiple knapsack resource constraints multiperiod allocation problems for storable resources and problems with substitutable resources the solution of large scale problems necessitates the design of efficient algorithms that take advantage of special mathematical structures indeed efficient algorithms for many models will be described we expect that this paper will help practitioners to formulate and solve diverse resource allocation problems and motivate researchers to explore new models and algorithmic approaches

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

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

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

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

proportional equity flow problem for terminal arcs the proportional equity flow problem extends a class of problems referred to as equity flow problems whose objective is to equitably distribute flow among the arcs in a flow circulation network the proportionally bounded flow circulation problem places lower and upper bounds on each arc flow that are nondecreasing continuous functions of the flow through one special arc and the objective is to maximize the flow through the special arc the proportional equity flow problem for terminal arcs problem ta is then defined as a special case where all the proportional arcs enter a sink vertex applications of both the general problem and problem ta are given two optimality conditions for problem ta are developed as well as an algorithm that is polynomially bounded for many types of nondecreasing continuous proportional bounding functions specifically the algorithm is shown to be polynomially bounded if the largest root of an equation can be found in polynomial time

connected sum of digital closed surfaces during the past 20 years the research of digital surfaces has proceeded to find their properties in the digital space z n such as a topological number a simple k point the 3d jordan theorem a k separating set a boundary detecting algorithm and so on actually unlike surfaces in a continuous space the features of digital surfaces have different characteristics the aim of this paper is to introduce the notion of a digital closed k surface in z n n 3 with the general k adjacency relations as a generalization of malgouyres and morgenthaler s k surfaces in z 3 to establish some minimal simple closed k surfaces in z 3 and to find their digital topological properties in relation with the k fundamental group and k contractibility moreover a connected sum of two digital closed surfaces is introduced and its digital topological properties are investigated

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

compute and forward harnessing interference through structured codes interference is usually viewed as an obstacle to communication in wireless networks this paper proposes a new strategy compute and forward that exploits interference to obtain significantly higher rates between users in a network the key idea is that relays should decode linear functions of transmitted messages according to their observed channel coefficients rather than ignoring the interference as noise after decoding these linear equations the relays simply send them towards the destinations which given enough equations can recover their desired messages the underlying codes are based on nested lattices whose algebraic structure ensures that integer combinations of codewords can be decoded reliably encoders map messages from a finite field to a lattice and decoders recover equations of lattice points which are then mapped back to equations over the finite field this scheme is applicable even if the transmitters lack channel state information

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

digital fundamental group and euler characteristic of a connected sum of digital closed surfaces unlike the connected sum in classical topology its digital version is shown to have some intrinsic feature in this paper we study both the digital fundamental group and the euler characteristic of a connected sum of digital closed k i surfaces i 0 1

lattices for distributed source coding jointly gaussian sources and reconstruction of a linear function consider a pair of correlated gaussian sources x 1 x 2 two separate encoders observe the two components and communicate compressed versions of their observations to a common decoder the decoder is interested in reconstructing a linear combination of x 1 and x 2 to within a mean square distortion of d we obtain an inner bound to the optimal rate distortion region for this problem a portion of this inner bound is achieved by a scheme that reconstructs the linear function directly rather than reconstructing the individual components x 1 and x 2 first this results in a better rate region for certain parameter values our coding scheme relies on lattice coding techniques in contrast to more prevalent random coding arguments used to demonstrate achievable rate regions in information theory we then consider the case of linear reconstruction of k sources and provide an inner bound to the optimal rate distortion region some parts of the inner bound are achieved using the following coding structure lattice vector quantization followed by ldquocorrelatedrdquo lattice structured binning

erratum to non product property of the digital fundamental group informat sci 171 1 3 2005 73 91 proof since the 8 adjacency of msc4 msc4 is compatible with the 4 adjacency of msc4 by definition 2 we see that each projection map pi msc4 msc4 msc4 i 2 1 2 is an 8 4 continuous map thus the proof is completed by the similar proof of that of algebraic topological method of the fundamental group meanwhile there is no compatible k3 adjacency on msc8 msc4 z k3 2 8 32 64 80 with the 8and 4 adjacency of msc8 and msc4 respectively hence we cannot establish the k3 fundamental group of the

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

covering rough set structures for a locally finite covering approximation space abstract the purpose of this paper is to introduce the notion of locally finite covering approximation lfc for brevity space u c which is a generalization of a finite covering approximation space furthermore for a subset x of the universe u we develop two kinds of neighborhood systems derived from the covering c specifically k1 x and k2 x which are not comparable with each other next the paper establishes two topologies on u alexandroff and quasi discrete topologies generated by systems k1 x and k2 x as bases respectively after suggesting many examples explaining the above notions and systems we develop two new types of rough set structures an h rough set operator h h and a khalimsky k for short topological rough set operator d k d k the present paper investigates various properties of these operators and refers to certain advantages of them which can be used in applied science including digital geometry information geometry computer vision pattern recognition image processing and so on in order to accomplish this we use some tools such as granulations neighborhood systems typical pawlak s tools related topologies rough set structures associated with the k topology and so on furthermore through this work we can obtain strong connections between covering rough set theory and applied geometry in the present paper all sets u c and x u need not be finite

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

the equivalence between two definitions of digital surfaces digital surfaces deal with properties of surfaces in digital spaces in the early 80 s researchers began to establish definitions for digital surfaces unlike surfaces in continuous spaces digital surfaces have different characteristics a general and intuitive definition for digital surfaces is still an open problem this paper presents a proof of the equivalence of two digital surface definitions one of the definitions was developed based on simple surface points given by morgenthaler and rosenfeld 1 and the second uses a parallel move concept as given by direct adjacency 2 3 a by product of this proof of equivalence is that any simple surface point in a simple surface belongs to one of six types of simple surface points

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

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

towards equitable distributions via proportional equity constraints we propose a new approach to the formulation of models for solving problems of equitable distribution equity is achieved by the use of proportional equity constraints which require that each recipient must receive ashare of the total distribution which falls within a prespecified range the algorithmic implications of this new approach are illustrated using two mathematical models we consider the problem of maximizing total flow in a multiterminal network with proportional equity constraints an efficient algorithm for solving this problem is provided as a further application we show that the introduction of such constraints still permits easy solutions for the linear knapsack problem

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

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

the bipartite rationing problem in the bipartite rationing problem a set of agents share a single resource available in different types each agent has a claim over only a subset of the resource types and these claims overlap in arbitrary fashion the goal is to divide fairly the various types of resources between the claimants when resources are in short supply with a single type of resource this is the standard rationing problem o neill b 1982 a problem of rights arbitration from the talmud math soc sci 2 4 345 371 of which the three benchmark solutions are the proportional uniform gains and uniform losses methods we extend these methods to the bipartite context imposing the familiar consistency requirement the division is unchanged if we remove an agent respectively a resource and take away at the same time his share of the various resources respectively reduce the claims of the relevant agents the uniform gains and uniform losses methods have infinitely many consistent extensions but the proportional me

lattice strategies for the dirty multiple access channel in costa s dirty paper channel gaussian random binning is able to eliminate the effect of interference which is known at the transmitter and thus achieve capacity we examine a generalization of the dirty paper problem to a multiple access channel mac setup where structured lattice based binning seems to be necessary to achieve capacity in the dirty mac two additive interference signals are present one known to each transmitter but none to the receiver the achievable rates using costa s gaussian binning vanish if both interference signals are strong in contrast it is shown that lattice strategies lattice precoding can achieve positive rates independent of the interference power furthermore in some cases which depend on the noise variance and power constraints high dimensional lattice strategies are in fact optimal in particular they are optimal in the limit of high snr where the capacity region of the dirty mac with strong interference approaches that of a clean mac whose power is governed by the minimum of the users powers rather than their sum the rate gap at high snr between lattice strategies and optimum rather than gaussian random binning is conjectured to be 1 2 log2 e 6 0 254 bit thus the doubly dirty mac is another instance of a network setting like the korner marton problem where linear structured coding is potentially better than random binning

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

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

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

on modulo sum computation over an erasure multiple access channel we study modulo sum computation of two binary source sequences over a two user erasure multiple access channel the channel is modeled as a binary input erasure multiple access channel which can be in one of three states either the channel output is a modulo sum of the two input symbols or the channel output equals the input symbol on the first link and an erasure on the second link or vice versa the associated state sequence is independent and identically distributed unlike previously studied multiple access channels the proposed channel is not matched to the modulo sum function and therefore we expect simple cut set upper bounds to be far from capacity in this paper we establish a new upper bound on the modulo sum capacity that is tighter than the cut set bound the key step in establishing this new bound is to provide suitable side information to the encoders to reduce the setup to a compound multiple access channel and then capture the tension across multiple receivers required to compute the modulo sum function in our lower bound it suffices to use identical linear codebooks at the two encoders when a strictly causal feedback of the channel state is available to the encoders we present a simple coding scheme that can achieve a rate larger than our upper bound for the case without feedback this shows that the modulo sum capacity is increased with feedback an extension to the case of lossy reconstruction is also treated briefly

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 approach to the maximum flow problem all previously known efficient maximum flow algorithms work by finding augmenting paths either one path at a time as in the original ford and fulkerson algorithm or all shortest length augmenting paths at once using the layered network approach of dinic an alternative method based on the preflow concept of karzanov is introduced a preflow is like a flow except that the total amount flowing into a vertex is allowed to exceed the total amount flowing out the method maintains a preflow in the original network and pushes local flow excess toward the sink along what are estimated to be shortest paths the algorithm and its analysis are simple and intuitive yet the algorithm runs as fast as any other known method on dense graphs achieving an o n 3 time bound on an n vertex graph by incorporating the dynamic tree data structure of sleator and tarjan we obtain a version of the algorithm running in o nm log n 2 m time on an n vertex m edge graph this is as fast as any known method for any graph density and faster on graphs of moderate density the algorithm also admits efficient distributed and parallel implementations a parallel implementation running in o n 2 log n time using n processors and o m space is obtained this time bound matches that of the shiloach vishkin algorithm which also uses n processors but requires o n 2 space

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

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

the flow circulation sharing problem the flow circulation sharing problem is defined as a network flow circulation problem with a maximin objective function the arcs in the network are partitioned into regular arcs and tradeoff arcs where each tradeoff arc has a non decreasing tradeoff function associated with it all arcs have lower and upper bounds on their flow while the value of the smallest tradeoff function is maximized the model is useful in equitable resource allocation problems over time which is illustrated in a coal strike example and a submarine assignment example some properties including optimality conditions are developed each cut in the network defines a knapsack sharing problem which leads to an optimality condition similar to the max flow min cut theorem an efficient algorithm for both the continuous and integer versions of the flow circulation sharing problem is developed and computational experience given in addition efficient algorithms are developed for problems where some of the arcs have infinite flow upper bounds

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

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

a fast parametric maximum flow algorithm and applications the classical maximum flow problem sometimes occurs in settings in which the arc capacities are not fixed but are functions of a single parameter and the goal is to find the value of the parameter such that the corresponding maximum flow or minimum cut satisfies some side condition finding the desired parameter value requires solving a sequence of related maximum flow problems in this paper it is shown that the recent maximum flow algorithm of goldberg and tarjan can be extended to solve an important class of such parametric maximum flow problems at the cost of only a constant factor in its worst case time bound faster algorithms for a variety of combinatorial optimization problems follow from the result

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