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

limit laws for sums of functions of subtrees of random binary search trees we consider sums of functions of subtrees of a random binary search tree and obtain general laws of large numbers and central limit theorems these sums correspond to random recurrences of the quicksort type x n stackrel cal l x i n x n 1 i n y n n ge 1 where in is uniformly distributed on 0 1 n 1 yn is a given random variable x k stackrel cal l x k for all k and given in xin and x n 1 in are independent conditions are derived such that x n mu n sigma sqrt n stackrel cal l rightarrow cal n 0 1 the normal distribution for some finite constants mu and sigma

probabilistic analysis of bucket recursive trees we introduce the bucket recursive tree a generalization of recursive trees the tree grows from a succession of integer labels that join the multitype nodes of the tree according to a stochastic rule we study the multivariate structure of the tree and obtain a multivariate central limit theorem for the joint distribution of the number of nodes of different types for trees with bucket size b 26 for trees with b 26 a phase change in the distribution is detected and the central limit theorem does not hold recent results on the extended polya urn models smythe 1995 provide the basic tool to reach these results two kinds of distances are also studied the tree height and the depth of the nth label strong laws for the height are obtained via a technique based on introducing ghost nodes then removing them the ghost nodes create a companion tree that relates our tree to biggins 1976 first and last birth problem in a multitype process a weak law for the depth is obtained by formulating and asymptotically manipulating the depth probability generating function

fixed points of the smoothing transform two sided solutions given a sequence c t c t1 t2 of real valued random variables with tj 0 for all j 1 and almost surely finite n sup j 1 tj g 0 the smoothing transform associated wi

solutions to complex smoothing equations we consider smoothing equations of the form begin aligned x mathop limits text law sum j ge 1 t j x j c end aligned where c t 1 t 2 ldots is a given sequence of random variables and x 1 x 2 ldots are independent copies of x and independent of the sequence c t 1 t 2 ldots the focus is on complex smoothing equations i e the case where the random variables x c t 1 t 2 ldots are complex valued but also more general multivariate smoothing equations are considered in which the t j are similarity matrices under mild assumptions on c t 1 t 2 ldots we describe the laws of all random variables x solving the above smoothing equation these are the distributions of randomly shifted and stopped levy processes satisfying a certain invariance property called u alpha stability which is related to operator semi stability the results are applied to various examples from applied probability and statistical physics

lowner inequality of indefinite type abstract a selfadjoint involutive matrix j provides c n with an indefinite inner product x y jx y for a pair of j selfadjoint matrices a b the j order relation a j b is defined as ax x bx x for all x we will show that if a b are j selfadjoint matrices such that all eigenvalues of a b are real and contained in an interval then for any operator monotone function f t on the matrices f a f b are well defined by the riesz dunford integral and a j b f a j f b when j i and f t t 1 2 on 0 this is the classical lowner inequality

on ram priority queues priority queues are some of the most fundamental data structures for example they are used directly for task scheduling in operating systems moreover they are essential to greedy algorithms we study the complexity of integer priority queue operations on a ram with arbitrary word size modeling the possibilities in standard imperative programming languages such as c we present exponential improvements over previous bounds and we show tight relations to sorting our first result is a ram priority queue supporting find min in constant time and insert and delete min in time o log log n where n is the current number of keys in the queue this is an exponential improvement over the o sqrt log n bound of fredman and willard proceedings of the 22nd acm symposium on the theory of computing baltimore md pp 1 7 plugging this priority queue into dijkstra s algorithm gives an o mlog log m algorithm for the single source shortest path problem on a graph with m edges as compared with the previous o m sqrt log m bound based on fredman and willard s priority queue the above bounds assume o n 2 varepsilon w space where w is the word length and varepsilon 0 they can however be achieved in linear space using randomized hashing our second result is a general equivalence between sorting and priority queues a priority queue is monotone if the minimum is nondecreasing over time as in many greedy algorithms we show that on a ram the amortized operation cost of a monotone priority queue is equivalent to the per key cost of sorting for example the equivalence implies that the single source shortest paths problem on a graph with m edges is no harder than that of sorting m keys with the current ram sorting this gives an o m log log m time bound as above but the relation holds regardless of the future developments in ram sorting from the equivalence result for any fixed varepsilon 0 we derive a randomized monotone o sqrt log n 1 varepsilon priority queue with expected constant time decrease key plugging this into dijkstra s algorithm gives an o n sqrt log n 1 varepsilon m algorithm for the single source shortest path problem on a graph with n nodes and m edges complementing the above o mlog log m algorithm if m gg n this improves the o nlog n log log n m bound by fredman and willard proceedings of the 31st ieee symposium on the foundations of computer science st louis mo 1990 pp 719 725 based on their o log n log log n priority queue with constant decrease key

second phase changes in random m ary search trees and generalized quicksort convergence rates we study the convergence rate to normal limit law for the space requirement of random m ary search trees while it is known that the random variable is asymptotically normally distributed for 3 m 26 and that the limit law does not exist for m 26 we show that the convergence rate is o n 1 2 for 3 m 19 and is o n 3 3 2 where 4 3 3 2 is a parameter depending on m for 20 m 26 our approach is based on a renemen t to the method of moments and applicable to other recursive random variables we briey mention the applications to quicksort proper and the generalized quicksort of hennequin where more phase changes are given these results provide natural concrete examples for which the berry esseen bounds are not necessarily proportional to the reciprocal of the standard deviation local limit theorems are also derived

a minimum spanning tree algorithm with inverse ackermann type complexity a deterministic algorithm for computing a minimum spanning tree of a connected graph is presented its running time is 0 m m n where is the classical functional inverse of ackermann s function and n respectively m is the number of vertices respectively edges the algorithm is comparison based it uses pointers not arrays and it makes no numeric assumptions on the edge costs

using canny s criteria to derive a recursively implemented optimal edge detector a highly efficient recursive algorithm for edge detection is presented using canny s design 1 we show that a solution to his precise formulation of detection and localization for an infinite extent filter leads to an optimal operator in one dimension which can be efficiently implemented by two recursive filters moving in opposite directions in addition to the noise truncature immunity which results the recursive nature of the filtering operations leads with sequential machines to a substantial saving in computational effort five multiplications and five additions for one pixel independent of the size of the neighborhood the extension to the two dimensional case is considered and the resulting filtering structures are implemented as two dimensional recursive filters hence the filter size can be varied by simply changing the value of one parameter without affecting the time execution of the algorithm performance measures of this new edge detector are given and compared to canny s filters various experimental results are shown

knot segmentation in 3d ct images of wet wood this paper proposes a fully automatic method to segment wood knots from images obtained by an x ray computed tomography scanner wood knot segmentation is known to be a difficult problem in the presence of sapwood because of the quite similar density of knots and wet sapwood classical segmentation techniques produce unsatisfactory results due to the very weak distinction between these two intensities to overcome this limitation caused by physical characteristics we propose to exploit the geometric properties of both the knot shapes and knot sapwood interface based on previous work related to automatic knot detection a new segmentation algorithm is proposed that uses a robust curvature estimation of 2d digital contours the segmentation process is fast easily parallelizable and produces better segmentation results than other state of the art algorithms it may be reproduced from the precise description given here or from source code available online

generalized potlatch and smoothing processes we consider simple generalizations of the potlatch and smoothing processes which were introduced in 8 and studied in 5 these generalizations provide relatively simple examples of infinite interacting systems which exhibit phase transition the original potlatch and smoothing processes do not exhibit phase transition our results show that for the generalized processes phase transition does not usually occur in one or two dimensions but usually does occur in higher dimensions upper and lower bounds for the relevant critical values are obtained as one application of our results we obtain the limiting behavior of the critical values for the linear contact process in d dimensions as d thus answering a question we raised in 3 this is done by comparing the contact process with an appropriate generalized smoothing process

on the internal structure of random recursive circuits we study the joint probability distribution of the number of nodes of fan out k in random recursive circuits for suitable norming we obtain a limiting multivariate normal distribution for the numbers of node of fan out at most k where we compute explicitly the limiting covariance matrix by solving a recurrence satisfied among its entries

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

power watershed a unifying graph based optimization framework in this work we extend a common framework for graph based image segmentation that includes the graph cuts random walker and shortest path optimization algorithms viewing an image as a weighted graph these algorithms can be expressed by means of a common energy function with differing choices of a parameter q acting as an exponent on the differences between neighboring nodes introducing a new parameter p that fixes a power for the edge weights allows us to also include the optimal spanning forest algorithm for watershed in this same framework we then propose a new family of segmentation algorithms that fixes p to produce an optimal spanning forest but varies the power q beyond the usual watershed algorithm which we term the power watershed in particular when q 2 the power watershed leads to a multilabel scale and contrast invariant unique global optimum obtained in practice in quasi linear time placing the watershed algorithm in this energy minimization framework also opens new possibilities for using unary terms in traditional watershed segmentation and using watershed to optimize more general models of use in applications beyond image segmentation

fast approximate energy minimization via graph cuts many tasks in computer vision involve assigning a label such as disparity to every pixel a common constraint is that the labels should vary smoothly almost everywhere while preserving sharp discontinuities that may exist e g at object boundaries these tasks are naturally stated in terms of energy minimization the authors consider a wide class of energies with various smoothness constraints global minimization of these energy functions is np hard even in the simplest discontinuity preserving case therefore our focus is on efficient approximation algorithms we present two algorithms based on graph cuts that efficiently find a local minimum with respect to two types of large moves namely expansion moves and swap moves these moves can simultaneously change the labels of arbitrarily large sets of pixels in contrast many standard algorithms including simulated annealing use small moves where only one pixel changes its label at a time our expansion algorithm finds a labeling within a known factor of the global minimum while our swap algorithm handles more general energy functions both of these algorithms allow important cases of discontinuity preserving energies we experimentally demonstrate the effectiveness of our approach for image restoration stereo and motion on real data with ground truth we achieve 98 percent accuracy

evaluation of hierarchical watersheds this paper aims to understand the practical features of hierarchies of morphological segmentations namely the quasi flat zones hierarchy and watershed hierarchies and to evaluate their potential in the context of natural image analysis we propose a novel evaluation framework for the hierarchies of partitions designed to capture various aspects of those representations precision of their regions and contours possibility to extract high quality horizontal cuts and optimal non horizontal cuts for image segmentation and the ease of finding a set of regions representing a semantic object this framework is used to assess and to optimize hierarchies with respect to the possible pre and post processing steps we show that used in conjunction with a state of the art contour detector watershed hierarchies are competitive with the complex state of the art methods for hierarchy construction in particular the proposed framework allows us to identify a watershed hierarchy based on a novel extinction value the number of parent nodes that outperforms the other hierarchies of morphological segmentations this coupled with the fact that watershed hierarchies satisfy clear global optimality properties and can be efficiently computed on large data make them valuable candidates for various computer vision tasks

spectral spatial classification of hyperspectral data based on a stochastic minimum spanning forest approach in this paper a new method for supervised hyperspectral data classification is proposed in particular the notion of stochastic minimum spanning forest msf is introduced for a given hyperspectral image a pixelwise classification is first performed from this classification map m marker maps are generated by randomly selecting pixels and labeling them as markers for the construction of msfs the next step consists in building an msf from each of the m marker maps finally all the m realizations are aggregated with a maximum vote decision rule in order to build the final classification map the proposed method is tested on three different data sets of hyperspectral airborne images with different resolutions and contexts the influences of the number of markers and of the number of realizations m on the results are investigated in experiments the performance of the proposed method is compared to several classification techniques both pixelwise and spectral spatial using standard quantitative criteria and visual qualitative evaluation

universal limit laws for depths in random trees random binary search trees b ary search trees median of 2k 1 trees quadtrees simplex trees tries and digital search trees are special cases of random split trees for these trees we offer a universal law of large numbers and a limit law for the depth of the last inserted point as well as a law of large numbers for the height

directed connected operators asymmetric hierarchies for image filtering and segmentation connected operators provide well established solutions for digital image processing typically in conjunction with hierarchical schemes in graph based frameworks such operators basically rely on symmetric adjacency relations between pixels in this article we introduce a notion of directed connected operators for hierarchical image processing by also considering non symmetric adjacency relations the induced image representation models are no longer partition hierarchies i e trees but directed acyclic graphs that generalize standard morphological tree structures such as component trees binary partition trees or hierarchical watersheds we describe how to efficiently build and handle these richer data structures and we illustrate the versatility of the proposed framework in image filtering and image segmentation

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

the functional equation of the smoothing transform given a sequence t ti i 1 of non negative random variables a function f on the positive halfline can be transformed to e q i 1 f tti we study the fixed points of this transform within the class of decreasing functions by exploiting the intimate relationship with general branching processes a full description of the set of solutions is established without the moment conditions that figure in earlier studies since the class of functions under consideration contains all laplace transforms of probability distributions on 0 1 the results provide the full description of the set of solutions to the fixed point equation of the smoothing transform x d

martingale convergence and the stopped branching random walk we discuss the construction of stopping lines in the branching random walk and thus the existence of a class of supermartingales indexed by sequences of stopping lines applying a method of lyons 1997 and lyons pemantle and peres 1995 concerning size biased branching trees we establish a relationship between stopping lines and certain stopping times consequently we develop conditions under which these supermartingales are also martingales further we prove a generalization of biggins martingale convergence theorem biggins 1977a within this context

on steady distributions of kinetic models of conservative economies we analyze the large time behavior of various kinetic models for the redistribution of wealth in simple market economies introduced in the pertinent literature in recent years as specific examples we study models with fixed saving propensity introduced by chakraborti and chakrabarti eur phys j b 17 167 170 2000 as well as models involving both exchange between agents and speculative trading as considered by cordier et al j stat phys 120 253 277 2005 we derive a sufficient criterion under which a unique non trivial stationary state exists and provide criteria under which these steady states do or do not possess a pareto tail in particular we prove the absence of pareto tails in pointwise conservative models like the one in eur phys j b 17 167 170 2000 while models with speculative trades introduced in j stat phys 120 253 277 2005 develop fat tails if the market is risky enough the results are derived by a fourier based technique first developed for the maxwell boltzmann equation gabetta et al in j stat phys 81 901 934 1995 bisi et al in j stat phys 118 1 2 301 331 2005 pareschi and toscani in j stat phys 124 2 4 747 779 2006 and from a recursive relation which allows to calculate arbitrary moments of the stationary state

central limit theorem for a class of one dimensional kinetic equations we introduce a class of kinetic type equations on the real line which constitute extensions of the classical kac caricature the collisional gain operators are defined by smoothing transformations with rather general properties by establishing a connection to the central limit problem we are able to prove long time convergence of the equation s solutions toward a limit distribution for example we prove that if the initial condition belongs to the domain of normal attraction of a certain stable law then the limit is a scale mixture of under some additional assumptions explicit exponential rates for the convergence to equilibrium in wasserstein metrics are calculated and strong convergence of the probability densities is shown

convergence to stable laws for a class of multidimensional stochastic recursions we consider a markov chain x n n 0 infty on mathbb r d defined by the stochastic recursion xn mn xn 1 qn where qn mn are i i d random variables taking values in the affine group a mathbb r d mathbb r d rtimes rm gl mathbb r d assume that mn takes values in the group of similarities of mathbb r d and the markov chain has a unique stationary measure which has unbounded support we denote by mn the expansion coefficient of mn and we assume mathbb e m alpha 1 for some positive we show that the partial sums s n sum k 0 n x k properly normalized converge to a normal law 2 or to an infinitely divisible law which is stable in a natural sense 2 or of the tails of and of stationary measure for an associated markov operator 2 we extend the results to the situation where mn is a random generalized similarity

on connectivity properties of grayscale pictures abstract a grayscale digital picture is called connected if it has only one connected component of constant gray level that is maximal i e not adjacent to any component of higher gray level this note establishes some equivalent conditions for connectedness and also defines a grayscale generalization of the genus in terms of sums of local property values

the joint distribution of elastic buckets in multiway search trees random search trees are studied when they grow under a general computer memory management scheme in a general scheme the space is released in buckets of certain predesignated sizes for a search tree with branch factor m the nodes may hold up to m 1 keys suppose the buckets of the memory management scheme that can hold less than m keys have key capacities c 1 ldots c p the search tree must then be implemented with multitype nodes of these capacities after n insertions let x n i be the number of buckets of type i i e of capacity c i 1 leq i leq p the multivariate structure of the tree is investigated for the vector bf x n x n 1 ldots x n p t the asymptotic mean and covariance matrix are determined under practical memory management schemes all variances and covariances experience a phase transition for 3 leq m leq 26 all variances and covariances are asymptotically linear in n for higher branch factors the variances and covariances become a superlinear but subquadratic function of n the joint distribution of bf x n is shown to be multivariate normal in a range of m while the tree is growing conversions between types are necessary a multivariate problem concerning these conversions with an asymptotic multivariate normal distribution is also studied the fixed bucket exact fit and buddy system allocation schemes will serve as illustrating examples

tail homogeneity of stationary measures for some multidimensional stochastic recursions we consider a stochastic recursion x n 1 m n 1 x n q n 1 n in mathbb n where q n m n are i i d random variables such that q n are translations m n are similarities of the euclidean space mathbb r d and x n in mathbb r d in the present paper we show that if the recursion has a unique stationary measure with unbounded support then the weak limit of properly dilated exists and defines a homogeneous tail measure the structure of is studied and the supports of and are compared in particular we obtain a product formula for

diffusion limited aggregation on a tree we study the following growth model on a regular d ary tree points at distance n adjacent to the existing subtree are added with probabilities proportional to n where 1 is a positive real parameter the heights of these clusters are shown to increase linearly with their total size this complements known results that show the height increases only logarithmically when 1 results are obtained using stochastic monotonicity and regeneration results which may be of independent interest our motivation comes from two other ways in which the model may be viewed as a problem in first passage percolation and as a version of diffusion limited aggregation dla adjusted so that fingering occurs

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

on a reverse heinz kato furuta inequality abstract in the set up of minkowski spaces the schwarz inequality holds with the reverse inequality sign as a consequence the same occurs with the triangle inequality in this note extensions of this indefinite version of the schwarz inequality are presented namely a reverse heinz kato furuta inequality valid for timelike vectors is included and related inequalities that also hold with the reverse sign are investigated

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

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