approximate tree kernels convolution kernels for trees provide simple means for learning with tree structured data the computation time of tree kernels is quadratic in the size of the trees since all pairs of nodes need to be compared thus large parse trees obtained from html documents or structured network data render convolution kernels inapplicable in this article we propose an effective approximation technique for parse tree kernels the approximate tree kernels atks limit kernel computation to a sparse subset of relevant subtrees and discard redundant structures such that training and testing of kernel based learning methods are significantly accelerated we devise linear programming approaches for identifying such subsets for supervised and unsupervised learning tasks respectively empirically the approximate tree kernels attain run time improvements up to three orders of magnitude while preserving the predictive accuracy of regular tree kernels for unsupervised tasks the approximate tree kernels even lead to more accurate predictions by identifying relevant dimensions in feature space

recursive tracking of vascular networks in angiograms based on the detection deletion scheme a computer algorithm was developed for automated identification of 2 d vascular networks in x ray angiograms this was accomplished by using an adaptive tracking algorithm in a three stage recursive procedure first given a starting position and direction a segment in the vascular network was identified second by filling it with the surrounding background pixel values the detected segment was deleted from the angiogram the detection deletion scheme was employed to prevent the problem of tracking path reentry in those areas where vessels overlap third all branch points were detected by use of matched filtering along both edges of the vessel the detected branch points were used as the starting points in the next recursion the recursive procedure terminated when no new branch point was found the algorithm showed a good performance when it was applied to angiograms of coronary and radial arteries to provide a quantitative evaluation vascular networks identified by the algorithm were compared to those identified by a human the algorithm made some false negative errors but very few false positive errors

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

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

cone metric spaces and fixed point theorems of contractive mappings in this paper we introduce cone metric spaces prove some fixed point theorems of contractive mappings on cone metric spaces

reconstructing the 3 d medial axes of coronary arteries in single view cineangiograms describes a technique for reconstructing the skeletal structure of coronary arteries from a succession of frames of a single view cineangiogram the authors use local features in each frame to determine correspondences of arterial segments in successive frames they define a similarity measure in 2d image space as the change in angular coordinates of corresponding pairs they use a form of gradient descent to find those depth coordinates that minimize the average deviation of the 3d angular coordinates of all points on the skeleton from the coordinates produced by a 3d scaling transformation in experiments with software models the reconstruction error was approximately two pixels when the initial guessed reconstruction was as large as 30 pixels

elastic geodesic paths in shape space of parameterized surfaces this paper presents a novel riemannian framework for shape analysis of parameterized surfaces in particular it provides efficient algorithms for computing geodesic paths which in turn are important for comparing matching and deforming surfaces the novelty of this framework is that geodesics are invariant to the parameterizations of surfaces and other shape preserving transformations of surfaces the basic idea is to formulate a space of embedded surfaces surfaces seen as embeddings of a unit sphere in ir3 and impose a riemannian metric on it in such a way that the reparameterization group acts on this space by isometries under this framework we solve two optimization problems one given any two surfaces at arbitrary rotations and parameterizations we use a path straightening approach to find a geodesic path between them under the chosen metric second by modifying a technique presented in 25 we solve for the optimal rotation and parameterization registration between surfaces their combined solution provides an efficient mechanism for computing geodesic paths in shape spaces of parameterized surfaces we illustrate these ideas using examples from shape analysis of anatomical structures and other general surfaces

model based morphological segmentation and labeling of coronary angiograms a method for extraction and labeling of the coronary arterial tree cat using minimal user supervision in single view angiograms is proposed the cat structural description skeleton and borders is produced along with quantitative information for the artery dimensions and assignment of coded labels based on a given coronary artery model represented by a graph the stages of the method are 1 cat tracking and detection 2 artery skeleton and border estimation 3 feature graph creation and iv artery labeling by graph matching the approximate cat centerline and borders are extracted by recursive tracking based on circular template analysis the accurate skeleton and borders of each cat segment are computed based on morphological homotopy modification and watershed transform the approximate centerline and borders are used for constructing the artery segment enclosing area asea where the defined skeleton and border curves are considered as markers using the marked asea an artery gradient image is constructed where all the asea pixels except the skeleton ones are assigned the gradient magnitude of the original image the artery gradient image markers are imposed as its unique regional minima by the homotopy modification method the watershed transform is used for extracting the artery segment borders and the feature graph is updated finally given the created feature graph and the known model graph a graph matching algorithm assigns the appropriate labels to the extracted cat using weighted maximal cliques on the association graph corresponding to the two given graphs experimental results using clinical digitized coronary angiograms are presented

reconstruction of 3 d binary tree like structures from three mutually orthogonal projections a method is developed to generate the 3d binary representation for a tree like object from three mutually orthogonal projections this is done by first backprojecting the binarized images from three directions and then iteratively removing artifacts in the backprojection three different algorithms have been developed the lagrange multiplier algorithm the conjugate gradient algorithm and the minimum voxel representation algorithm mra the performance of these algorithms under noise free conditions is evaluated using mathematically projected images of a 3d tree structure while all three algorithms are capable of producing a relatively accurate reconstruction the mra is superior not only because it requires the least amount of computation but also because it uses binary instead of gray scale information in the input images reconstruction of 3d coronary arterial structures using mra is further verified with x ray images of a human chest phantom and shows a satisfactory performance the result of this study should be valuable for 3d imaging of blood vessels

toward a theory of statistical tree shape analysis to develop statistical methods for shapes with a tree structure we construct a shape space framework for tree shapes and study metrics on the shape space this shape space has singularities which correspond to topological transitions in the represented trees we study two closely related metrics on the shape space ted and qed qed is a quotient euclidean distance arising naturally from the shape space formulation while ted is the classical tree edit distance using gromov s metric geometry we gain new insight into the geometries defined by ted and qed we show that the new metric qed has nice geometric properties that are needed for statistical analysis geodesics always exist and are generically locally unique following this we can also show the existence and generic local uniqueness of average trees for qed ted while having some algorithmic advantages does not share these advantages along with the theoretical framework we provide experimental proof of concept results on synthetic data trees as well as small airway trees from pulmonary ct scans this way we illustrate that our framework has promising theoretical and qualitative properties necessary to build a theory of statistical tree shape analysis

a nonparametric regression model with tree structured response developments in science and technology over the last two decades has motivated the study of complex data objects in this article we consider the topological properties of a population of tree structured objects our interest centers on modeling the relationship between a tree structured response and other covariates for tree structured objects this poses serious challenges since most regression methods rely on linear operations in euclidean space we generalize the notion of nonparametric regression to the case of a tree structured response variable in addition we develop a fast algorithm and give its theoretical justification we implement the proposed method to analyze a dataset of human brain artery trees an important lesson is that smoothing in the full tree space can reveal much deeper scientific insights than the simple smoothing of summary statistics this article has supplementary materials online

object oriented data analysis sets of trees object oriented data analysis is the statistical analysis of populations of complex objects in the special case of functional data analysis these data objects are curves where standard euclidean approaches such as principal component analysis have been very successful recent developments in medical image analysis motivate the statistical analysis of populations of more complex data objects which are elements of mildly non euclidean spaces such as lie groups and symmetric spaces or of strongly non euclidean spaces such as spaces of tree structured data objects these new contexts for object oriented data analysis create several potentially large new interfaces between mathematics and statistics this point is illustrated through the careful development of a novel mathematical framework for statistical analysis of populations of tree structured objects

clustering tree structured data on manifold tree structured data usually contain both topological and geometrical information and are necessarily considered on manifold instead of euclidean space for appropriate data parameterization and analysis in this study we propose a novel tree structured data parameterization called topology attribute matrix t a matrix so the data clustering task can be conducted on matrix manifold we incorporate the structure constraints embedded in data into the non negative matrix factorization method to determine meta trees from the t a matrix and the signature vector of each single tree can then be extracted by meta tree decomposition the meta tree space turns out to be a cone space in which we explore the distance metric and implement the clustering algorithm based on the concepts like frechet mean finally the t a matrix based clustering tambac framework is evaluated and compared using both simulated data and real retinal images to illustrate its efficiency and accuracy

principal components analysis in the space of phylogenetic trees phylogenetic analysis of dna or other data commonly gives rise to a collection or sample of inferred evolutionary trees principal components analysis pca cannot be applied directly to collections of trees since the space of evolutionary trees on a fixed set of taxa is not a vector space this paper describes a novel geometrical approach to pca in tree space that constructs the first principal path in an analogous way to standard linear euclidean pca given a data set of phylogenetic trees a geodesic principal path is sought that maximizes the variance of the data under a form of projection onto the path due to the high dimensionality of tree space and the nonlinear nature of this problem the computational complexity is potentially very high so approximate optimization algorithms are used to search for the optimal path principal paths identified in this way reveal and quantify the main sources of variation in the original collection of trees in terms of both topology and branch lengths the approach is illustrated by application to simulated sets of trees and to a set of gene trees from metazoan animal species

the logarithm map its limits and frechet means in orthant spaces the first part of the paper studies the expression for and the properties of the logarithm map on an orthant space which is a simple stratified space with the aim of analysing frechet means of probability measures on such a space in the second part we use these results to characterise frechet means and to derive various of their properties including the limiting distribution of sample frechet means

relative optimality conditions and algorithms for treespace frechet means recent interest in treespaces as well founded mathematical domains for phylogenetic inference and statistical analysis for populations of anatomical trees has motivated research into efficient and

optimization methods on riemannian manifolds and their application to shape space we extend the scope of analysis for linesearch optimization algorithms on possibly infinite dimensional riemannian manifolds to the convergence analysis of the bfgs quasi newton scheme and the fletcher reeves conjugate gradient iteration numerical implementations for exemplary problems in shape spaces show the practical applicability of these methods

statistical graph space analysis the sample mean is one of the most fundamental concepts in statistics properties of the sample mean that are well defined in euclidean spaces become unclear in graph spaces this paper proposes conditions under which the following properties are valid existence uniqueness and consistency of means the midpoint property necessary conditions of optimality and convergence results of mean algorithms the theoretical results address common misconceptions about the graph mean in graph edit distance spaces serve as a first step towards a statistical analysis of graph spaces and result in a theoretically well founded mean algorithm that outperformed six other mean algorithms with respect to solution quality on different graph datasets representing images and molecules highlightsa first step towards a theory of statistical graph space analysis is proposed mmm algorithm is proposed that outperformed six other mean algorithms necessary conditions of optimality are proved convergence of mmm algorithm is shown basic statistical and geometrical properties are shown

generalized principal component analysis gpca this paper presents an algebro geometric solution to the problem of segmenting an unknown number of subspaces of unknown and varying dimensions from sample data points we represent the subspaces with a set of homogeneous polynomials whose degree is the number of subspaces and whose derivatives at a data point give normal vectors to the subspace passing through the point when the number of subspaces is known we show that these polynomials can be estimated linearly from data hence subspace segmentation is reduced to classifying one point per subspace we select these points optimally from the data set by minimizing certain distance function thus dealing automatically with moderate noise in the data a basis for the complement of each subspace is then recovered by applying standard pca to the collection of derivatives normal vectors extensions of gpca that deal with data in a high dimensional space and with an unknown number of subspaces are also presented our experiments on low dimensional data show that gpca outperforms existing algebraic algorithms based on polynomial factorization and provides a good initialization to iterative techniques such as k subspaces and expectation maximization we also present applications of gpca to computer vision problems such as face clustering temporal video segmentation and 3d motion segmentation from point correspondences in multiple affine views

polynomial time metrics for attributed trees we address the problem of comparing attributed trees and propose four novel distance measures centered around the notion of a maximal similarity common subtree the proposed measures are general and defined on trees endowed with either symbolic or continuous valued attributes and can be applied to rooted as well as unrooted trees we prove that our measures satisfy the metric constraints and provide a polynomial time algorithm to compute them this is a remarkable and attractive property since the computation of traditional edit distance based metrics is in general np complete at least in the unordered case we experimentally validate the usefulness of our metrics on shape matching tasks and compare them with an approximation of edit distance

the gauss bonnet theorem for cone manifolds and volumes of moduli spaces this paper generalizes the gauss bonnet formula to a class of stratied spaces called riemannian cone manifolds as an application we compute the volumes of the moduli spacesm0 n with respect to the complex hyperbolic metrics introduced by picard deligne mostow and thurston

principal component analysis and the locus of the frechet mean in the space of phylogenetic trees evolutionary relationships are represented by phylogenetic trees and a phylogenetic analysis of gene sequences typically produces a collection of these trees one for each gene in the analysis analysis of samples of trees is difficult due to the multi dimensionality of the space of possible trees in euclidean spaces principal component analysis is a popular method of reducing high dimensional data to a low dimensional representation that preserves much of the sample s structure however the space of all phylogenetic trees on a fixed set of species does not form a euclidean vector space and methods adapted to tree space are needed previous work introduced the notion of a principal geodesic in this space analogous to the first principal component here we propose a geometric object for tree space similar to the formula see text th principal component in euclidean space the locus of the weighted frechet mean of formula see text vertex trees when the weights vary over the formula see text simplex we establish some basic properties of these objects in particular showing that they have dimension formula see text and propose algorithms for projection onto these surfaces and for finding the principal locus associated with a sample of trees simulation studies demonstrate that these algorithms perform well and analyses of two datasets containing apicomplexa and african coelacanth genomes respectively reveal important structure from the second principal components

on approximability of optimization problems related to red blue split graphs abstract an edge bicolored graph g v r b is called red blue split graph if there exists a partition i b and i r of v such that i b and i r are independent sets in v b and v r respectively red blue split graphs generalize several well studied graph classes including split graphs bipartite graphs and konig egervary graphs in this paper we consider the algorithmic complexity of various optimization problems like minimum edge or vertex deletion and maximum edge or vertex induced subgraph related to red blue split graphs all these problems are np hard and thus we look at them from algorithmic paradigms that are meant for coping with np hardness we obtain various hardness as well as algorithmic results for these problems in the realm of approximation algorithms and parameterized complexity the main tool we use to obtain all our results is polynomial time transformations between appropriate problems on the way we also resolve some problems related to inapproximability about certain optimization problems mentioned by korach et al 2006 17

generalized median graph computation by means of graph embedding in vector spaces the median graph has been presented as a useful tool to represent a set of graphs nevertheless its computation is very complex and the existing algorithms are restricted to use limited amount of data in this paper we propose a new approach for the computation of the median graph based on graph embedding graphs are embedded into a vector space and the median is computed in the vector domain we have designed a procedure based on the weighted mean of a pair of graphs to go from the vector domain back to the graph domain in order to obtain a final approximation of the median graph experiments on three different databases containing large graphs show that we succeed to compute good approximations of the median graph we have also applied the median graph to perform some basic classification tasks achieving reasonable good results these experiments on real data open the door to the application of the median graph to a number of more complex machine learning algorithms where a representative of a set of graphs is needed

a 3d reconstruction of vascular structures from two x ray angiograms using an adapted simulated annealing algorithm a three dimensional 3d reconstruction of the vessel lumen from two angiographic views based on the reconstruction of a series of cross sections is proposed assuming uniform mixing of contrast medium and background subtraction the cross section of each vessel is reconstructed through a binary representation a priori information about both the slice to be reconstructed and the relationships between adjacent slices are incorporated to lessen ambiguities on the reconstruction taking into account the knowledge of normal vessel geometry an initial solution of each slice is created using an elliptic model based method this initial solution is then deformed to be made consistent with projection data while being constrained into a connected realistic shape for that purpose properties on the expected optimal solution are described through a markov random field to find an optimal solution a specific optimization algorithm based on simulated annealing is used the method performs well both on single vessels and on branching vessels possessing an additional inherent ambiguity when viewed at oblique angles results on 2d slice independent reconstruction and 3d reconstruction of a stack of spatially continuous 2d slices are presented for single vessels and bifurcations

on the hardness of approximating minimum vertex cover we prove the minimum vertex cover problem to be np hard to approximate to within a factor of 1 3606 extending on previous pcp and hardness of approximation technique to that end one needs to develop a new proof framework and to borrow and extend ideas from several fields

reconstructing the cross sections of coronary arteries from biplane angiograms an algorithm that reconstructs the cross sections of the lumens of coronary arteries from two mutually orthogonal x ray projections is described the algorithm accommodates the possibility of elliptical crescent or star shapes it represents each biplane projection of a transverse slice of the arterial lumen as a binary valued image the single coordinate moments of these two projection images are equal to those of the slice since the cross coordinate moments of the slice are not available from the projections an algorithm to estimate these moments based on assumptions of smoothness and connectivity is developed once all the missing moments are estimated the image of the slice can be estimated by inverting these moments using the uniqueness theorem governing the relation between an image and its moments preliminary tests of the algorithm on synthetic data on hardware phantoms and on a segment of a barium enhanced in vitro coronary artery are reported

model guided labeling of coronary structure assigning anatomic labels to coronary arteries in x ray angiograms is an important task in medical imaging motivated by the desire to standardize the assessment of coronary artery disease and to facilitate the three dimensional 3 d reconstruction and visualization of the coronary vasculature however automatic labeling poses a number of significant challenges including the presence of noise artifacts competing structures misleading visual cues and other difficulties associated with a dynamic and inherently complex structure the authors have developed a model guided approach that addresses these challenges and automatically labels the vascular structure in coronary angiographic images the approach consists of two models 1 a symbolic model represented through a directed acyclic graph that captures vascular tree hierarchies and branch interrelationships and 2 a generalized 3 d model that captures spatial and geometric relationships importantly the approach detects ambiguities such as vessel overlaps that may be found in a frame of a cine sequence and resolves these ambiguities by considering the information derived from other unambiguous frames in the temporal sequence employing dynamic programming methods to match the image features found in the different ambiguous and unambiguous frames this paper presents this model guided labeling algorithm and discusses the experimental results obtained from implementing and applying the resulting labeling system to a variety of clinical images the results indicate the feasibility of achieving robust and consistently accurate image labeling through this model guided temporal disambiguation method

computing medians and means in hadamard spaces the geometric median as well as the frechet mean of points in a hadamard space are important in both theory and applications surprisingly no algorithms for their computation are hitherto known to address this issue we use a splitting version of the proximal point algorithm for minimizing a sum of convex functions and prove that this algorithm produces a sequence converging to a minimizer of the objective function which extends a recent result of bertsekas math program 129 2011 pp 163 195 into hadamard spaces the method is quite robust and not only does it yield algorithms for the median and the mean but also it applies to various other optimization problems we moreover show that another algorithm for computing the frechet mean can be derived from the law of large numbers due to sturm ann probab 30 2002 pp 1195 1222 in applications computing medians and means is probably most needed in tree space which is an instance of a hadamard space invented by billera holmes and vo