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

morphological shape decomposition a technique for decomposing a binary shape into a union of simple binary shapes is presented the decomposition is shown to be unique and invariant to translation rotation and scaling the techniques used in the decomposition are based on mathematical morphology the shape description produced can be used in object recognition and in binary image coding

on the cellular convexity of complexes in this paper we discuss cellular convexity of complexes a new definition of cellular convexity is given in terms of a geometric property then it is proven that a regular complex is celiularly convex if and only if there is a convex plane figure of which it is the cellular image hence the definition of cellular convexity by sklansky 7 is equivalent to the new definition for the case of regular complexes the definition of minsky and papert 4 is shown to be equivalent to our definition therefore au definitions are virtually equivalent it is shown that a regular complex is cellularly convex if and only if its minimum perimeter polygon does not meet the boundary of the complex a 0 n time algorithm is presented to determine the cellular convexity of a complex when it resides in n m cells and is represented by the run length code

dade s invariant conjecture for the chevalley groups of type g2 in the defining characteristic abstract this paper is part of a program to study the conjecture of e c dade on counting characters in blocks for several finite groups in this paper we verify dade s invariant conjecture for the chevalley groups g 2 p a in the defining characteristic when p 2 or 3 this implies dade s final conjecture when p 2 or 3

three dimensional euclidean distance transformation and its application to shortest path planning in this paper we present a novel method to obtain the three dimensional euclidean distance transformation edt in two scans of the image the shortest path can be extracted based on the distance maps using the minimum value tracing the edt is obtained correctly and efficiently in a constant time for arbitrary types of images including the existence of obstacles by adopting the new dynamically rotational mathematical morphology we not only guarantee the collision free in the shortest path but also reduce the time complexity dramatically

binary image comparison with local dissimilarity quantification in this paper we present a method for binary image comparison for binary images intensity information is poor and shape extraction is often difficult therefore binary images have to be compared without using feature extraction due to the fact that different scene patterns can be present in the images we propose a modified hausdorff distance hd locally measured in an adaptive way the resulting set of measures is richer than a single global measure the local hd measures result in a local dissimilarity map ldmap including the dissimilarity spatial layout a classification of the images in function of their similarity is carried out on the ldmaps using a support vector machine the proposed method is tested on a medieval illustration database and compared with other methods to show its efficiency

new algorithms for euclidean distance transformation of an n dimensional digitized picture with applications abstract in this paper we propose a new method to obtain the euclidean distance transformation and the voronoi diagram based on the exact euclidean metric for an n dimensional picture we present four algorithms to perform the transformation which are constructed by the serial composition of n dimensional filters when performed by a general purpose computer they are faster than the method by h yamada for a two dimensional picture those algorithms require only one n dimensional array for storing input output pictures and a single one dimensional array for a work area if an input picture needs not be preserved

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

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

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

sketching a grey tone pattern from its distance transform we describe a procedure to create an abstraction of a grey tone pattern by sketching its regions which have locally higher intensities the sketch is a set of simple digital lines qualitatively analogous to the skeleton representation computed in the case of a single valued pattern the grey tone pattern is regarded as constituted by a number of regions with constant grey value and the skeleton is found by detecting suitable pixels on the distance transform of the pattern computation of the distance transform is accomplished according to the city block distance by ordered propagation over regions with increasing grey values neighbourhood conditions are used to detect the set of the skeletal pixels which is subsequently reduced to unit thickness finally the skeleton undergoes a pruning process which removes a part or all of some of its branches

the higher order hamiltonian structures for the modified classical yang baxter equation we consider constructing the higher order hamiltonian structures on the dual of the lie algebra from the first hamiltonian structure of the coadjoint orbit method for this purpose we show that the structure of the lie algebrag is inherited to the algebra of vector fields ong through the solution of the modified classical yang baxter equation classicalr matrix we study the algebra that generates the compatible poisson brackets

linear time euclidean distance transform algorithms two linear time and hence asymptotically optimal algorithms for computing the euclidean distance transform of a two dimensional binary image are presented the algorithms are based on the construction and regular sampling of the voronoi diagram whose sites consist of the unit feature pixels in the image the first algorithm which is of primarily theoretical interest constructs the complete voronoi diagram the second more practical algorithm constructs the voronoi diagram where it intersects the horizontal lines passing through the image pixel centers extensions to higher dimensional images and to other distance functions are also discussed

a list processing approach to compute voronoi diagrams and the euclidean distance transform we propose an efficient voronoi transform algorithm for constructing voronoi diagrams using segment lists of rows a significant feature of the algorithm is that it takes segments rather than pixels as the basic units to represent and propagate the nearest neighbor information the segment lists are dynamically updated as they are scanned a distance map can then be easily computed from the segment list representation of the voronoi diagram experimental results have demonstrated its high efficiency extension of the algorithm to higher dimensions is also discussed

on generalized distance transformation of digitized pictures in this paper we present a generalization of the distance transformation of a digitized picture in two different aspects first we define the generalized distance transformation of a binary picture gdtb a subclass of gdtb called a local minimum filter family of gdtb lmf gdtb characterized by a series of local minimum filters with varying neighborhoods is discussed in detail a skeleton is defined for lmf gdtb and it is proved that any binary picture can be reconstructed exactly from its skeleton with the distance value on it second the gray weighted distance transformation gwdt is extended to a generalized gwdt ggwdt by introducing an arbitrary initial picture after the fundamental equation of ggwdt and its solution are derived it is proved that an arbitrary gray picture is generated by iterative application of ggwdt from a uniquely determined elementary picture and a sequence of initial value pictures

weight evaluation for features via constrained data pairscan t linkq facing the massive amount of data appearing on the web automatic analysis tools have become essential for web users to discover valuable information online precise similarity measurement plays a decisive role in enabling analysis tools to acquire high quality performances because different features contribute diversely to similarity calculation it is necessary to utilize weight to measure feature s contribution and import it into similarity measurement to accurately assign feature s weight constrained data pairs provided by users are usually imported into the weight evaluation procedure whereas conventional plans all fail to consider two challenges a asymmetrical distribution of constrained data pairs and b inconsistency contained by constrained data pairs if these two issues occur conventional plans are incompetent at addressing them or are even unable to work thus this paper proposes a novel constraint based weight evaluation to address these two issues for the former constrained data pairs are partitioned into several equivalent classes and distributing parameters are assigned to constrained data pairs to balance their distributions for the latter constrained data pairs are connected one after another and belief values are thereby formed to indicate their probability of being inconsistent experimental results demonstrate that this type of evaluation is independent of any algorithm with this evaluation similarities can be calculated more accurately

distances defined by neighborhood sequences abstract this paper investigates general properties of distance functions defined over digitized space we assume that a distance between two points is defined as the length of a shortest path connecting them in the underlying graph which is defined by a given neighborhood sequence many typical distance functions can be described in this form but there are cases in which given neighborhood sequence do not define distance functions we first derive a necessary and sufficient condition for a neighborhood sequence to define a distance function we then discuss another important problem of estimating how tight such distances can approximate the euclid distance from the view point of relative error and absolute error

compatible poisson structures and the virasoro algebra this paper is dedicated to henry mckean whose master craftsmanship enriched such diverse areas of mathematics and provided his disciples with a lofty measure by which to gauge their efforts and so redouble them

fast distance transformation on irregular two dimensional grids in this article we propose a new fast algorithm to compute the squared euclidean distance transform e 2dt on every two dimensional 2 d irregular isothetic grids regular square grids quadtree based grids etc our new fast algorithm is an extension of the e 2dt method proposed by breu et al 3 it is based on the implicit order of the cells in the grid and builds a partial voronoi diagram of the centers of background cells thanks to a data structure of lists we compare the execution time of our method with the ones of others approaches we developed in previous works in those experiments we consider various kinds of classical 2 d grids in imagery to show the interest of our methodology and to point out its robustness we also show that our method may be interesting regarding an application where we extract an adaptive medial axis construction based on a quadtree decomposition scheme

estimating shortest paths and minimal distances on digitized three dimensional surfaces abstract the estimation of minimal distances and shortest paths between points on a continuous three dimensional 3d surface given in digitized form is considered the suggested approach combines recently developed length estimators for digitized 3d curves with well known algorithms for computing shortest paths in sparse graphs to estimate the minimal distances and the corresponding shortest paths in a systematic computationally efficient way in comparison to differential geometry techniques for solving the continuous geodesic problem and to computational geometry algorithms for solving the discrete geodesic probelm the suggested hybrid continuous discrete approach is well suited to applications in which continuous surfaces are given in digitized form computation time must be relatively short and an approximate solution is acceptable in particular the suggested method can be used to efficiently obtain a good initial approximation as an input to other algorithms thus improving their speed of convergence and reducing their tendency to converge to insignificant local minima some confusion has recently arisen with respect to the relation between the design of length estimators and the optimization of local distances for distance transformations since 3d length estimation is a central concept in the suggested approach an effort is made to resolve discrepancies and to highlight the similarities and differences between these problems it is shown that while in two dimensions the design of length estimators and the design of local weights for distance transformations are rather similar in three dimensions digital geometry dictates that these problems are different the contexts in which each of the two approaches should be preferred are identified

line based recognition using a multidimensional hausdorff distance a line feature based approach for model based recognition using a four dimensional hausdorff distance is proposed this approach reduces the problem of finding the rotation scaling and translation transformations between a model and an image to the problem of finding a single translation minimizing the hausdorff distance between two sets of points in a four dimensional space the implementation of the proposed algorithm can be naturally extended to higher dimensional spaces to efficiently find correspondences between n dimensional patterns the method performance and sensitivity to segmentation problems are quantitatively characterized using an experimental protocol with simulated data it is shown that the algorithm performs well is robust to occlusion and outliers and that it degrades nicely as the segmentation problems increase experiments with real images are also presented

information geometry for landmark shape analysis unifying shape representation and deformation shape matching plays a prominent role in the comparison of similar structures we present a unifying framework for shape matching that uses mixture models to couple both the shape representation and deformation the theoretical foundation is drawn from information geometry wherein information matrices are used to establish intrinsic distances between parametric densities when a parameterized probability density function is used to represent a landmark based shape the modes of deformation are automatically established through the information matrix of the density we first show that given two shapes parameterized by gaussian mixture models gmms the well known fisher information matrix of the mixture model is also a riemannian metric actually the fisher rao riemannian metric and can therefore be used for computing shape geodesics the fisher rao metric has the advantage of being an intrinsic metric and invariant to reparameterization the geodesic computed using this metric establishes an intrinsic deformation between the shapes thus unifying both shape representation and deformation a fundamental drawback of the fisher rao metric is that it is not available in closed form for the gmm consequently shape comparisons are computationally very expensive to address this we develop a new riemannian metric based on generalized phi entropy measures in sharp contrast to the fisher rao metric the new metric is available in closed form geodesic computations using the new metric are considerably more efficient we validate the performance and discriminative capabilities of these new information geometry based metrics by pairwise matching of corpus callosum shapes we also study the deformations of fish shapes that have various topological properties a comprehensive comparative analysis is also provided using other landmark based distances including the hausdorff distance the procrustes metric landmark based diffeomorphisms and the bending energies of the thin plate tps and wendland splines

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

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

vehicle path planning by using adaptive constrained distance transformation abstract the adaptive constrained distance transformation acdt is proposed to solve the vehicle path planning problem the acdt is a generalization of the constrained distance transformation the incremental distance of the constrained distance transformation is modified into the incremental cost by defining the incremental cost according to the incremental distance the vehicle characteristics and the local spatial properties of the terrain the adaptive constrained distance transformation can be applied to solve the vehicle path planning problem that accounts for more than the euclidean distance and hard constraints

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

shape based averaging a new method for averaging multidimensional images is presented which is based on signed euclidean distance maps computed for each of the pixel values we refer to the algorithm as shape based averaging sba because of its similarity to raya and udupa s shape based interpolation method the new method does not introduce pixel intensities that were not present in the input data which makes it suitable for averaging nonnumerical data such as label maps segmentations using segmented human brain magnetic resonance images sba is compared to label voting for the purpose of averaging image segmentations in a multiclassifier fashion sba on average performed as well as label voting in terms of recognition rates of the averaged segmentations sba produced more regular and contiguous structures with less fragmentation than did label voting sba also was more robust for small numbers of atlases and for low atlas resolutions in particular when combined with shape based interpolation we conclude that sba improves the contiguity and accuracy of averaged image segmentations

uno s invariant conjecture for steinberg s triality groups in defining characteristic abstract we verify uno s invariant conjecture for steinberg s triality groups d 4 3 q q a power of an odd prime p in the defining characteristic p uno s invariant conjecture is a refinement of dade s invariant conjecture

weighted distance transforms generalized to modules and their computation on point lattices this paper presents the generalization of weighted distances to modules and their computation through the chamfer algorithm on general point lattices the first part is dedicated to formalization of definitions and properties distance metric norm of weighted distances on modules it resumes tools found in literature to express the weighted distance of any point of a module and to compute optimal weights in the general case to get rotation invariant distances the second part of this paper proves that for any point lattice the sequential two scan chamfer algorithm produces correct distance maps finally the definitions and computation of weighted distances are applied to the face centered cubic fcc and body centered cubic bcc grids

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 efficient algorithms for achieving euclidean distance transformation euclidean distance transformation edt is used to convert a digital binary image consisting of object foreground and nonobject background pixels into another image where each pixel has a value of the minimum euclidean distance from nonobject pixels in this paper the improved iterative erosion algorithm is proposed to avoid the redundant calculations in the iterative erosion algorithm furthermore to avoid the iterative operations the two scan based algorithm by a deriving approach is developed for achieving edt correctly and efficiently in a constant time besides we discover when obstacles appear in the image many algorithms cannot achieve the correct edt except our two scan based algorithm moreover the two scan based algorithm does not require the additional cost of preprocessing or relative coordinates recording

spatial planning a configuration space approach this paper presents algorithms for computing constraints on the position of an object due to the presence of ther objects this problem arises in applications that require choosing how to arrange or how to move objects without collisions the approach presented here is based on characterizing the position and orientation of an object as a single point in a configuration space in which each coordinate represents a degree of freedom in the position or orientation of the object the configurations forbidden to this object due to the presence of other objects can then be characterized as regions in the configuration space called configuration space obstacles the paper presents algorithms for computing these configuration space obstacles when the objects are polygons or polyhedra

a euclidean distance transform using grayscale morphology decomposition a fast and exact euclidean distance transformation using decomposed grayscale morphological operators is presented applied on a binary image a distance transformation assigns each object pixel a value that corresponds to the shortest distance between the object pixel and the background pixels it is shown that the large structuring element required for the euclidean distance transformation can be easily decomposed into 3 spl times 3 windows this is possible because the square of the euclidean distance matrix changes uniformly both in the vertical and horizontal directions a simple extension for a 3d euclidean distance transformation is discussed a fast distance transform for serial computers is also presented acting like thinning algorithms the version for serial computers focuses operations only on the potential changing pixels and propagates from the boundary of objects significantly reducing execution time nonsquare pixels can also be used in this algorithm an example application shape filtering using arbitrary sized circular dilation and erosion is discussed rotation invariant basic morphological operations can be done using this example application

a matrix integral solution to two dimensional w p gravity thep th gel fand dickey equation and the string equation l p 1 have a common solution expressible in terms of an integral overn n hermitean matrices for largen the integrand being a perturbation of a gaussian generalizing kontsevich s integral beyond the kdv case it is equivalent to showing that is a vacuum vector for aw generated from the coefficients of the vertex operator this connection is established via a quadratic identity involving the wave function and the vertex operator which is a disguised differential version of the fay identity the latter is also the key to the spectral theory for the two compatible symplectic structures of kdv in terms of the stress energy tensor associated with the virasoro algebra given a differential operator l d p q 2 t d p 2 cdots q p t with d frac partial dx t t 1 t 2 t 3 x equiv t 1 consider the deformation equations1 0 1 begin gathered frac partial l partial t n l n p l n 1 2 n 0 mod p hfill p reduced kp equation hfill end gathered ofl for which there exists a differential operatorp possibly of infinite order such that 0 2 l p 1 string equation in this note we give a complete solution to this problem in section 1 we give a brief survey of useful facts about thei function t the wave function t z solution of t n l n p x andl 1 p z and the corresponding infinitedimensional planev 0 of formal power series inz for largez v 0 span psi t z for all t in mathbb c infty in sato s grassmannian the three theorems below form the core of the paper their proof will be given in subseuqent sections each of which lives on its own right

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

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

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

fast 2 d distance transformations the performance of a number of image processing methods depends on the output quality of a distance transformation dt process most of the fast dt methodologies are not accurate whereas other error free dt algorithms are not very fast in this paper a novel fast simple and error free dt algorithm is presented by recording the relative x and y coordinates of the examined image pixels an optimal algorithm can be developed to achieve the dt of an image correctly and efficiently in constant time without any iteration furthermore the proposed method is general since it can be used by any kind of distance function leading to accurate image dts

a morphological approach to shortest path planning for rotating objects this paper presents a mathematical morphology based approach to find the shortest path of an arbitrarily shaped moving object with rotations amid obstacles of arbitrary shapes by utilizing the proposed rotational morphology along with the continuous turning angle constraint the representation framework of the motion planning problem becomes simple and practical two vehicle models with different driving and turning systems are used in solving the path planning problem and their strategies are presented simulation results show that our algorithm works successfully in various conditions