fast methods for the eikonal and related hamilton jacobi equations on unstructured meshes the fast marching method is a numerical algorithm for solving the eikonal equation on a rectangular orthogonal mesh in o m log m steps where m is the total number of grid points the scheme relies on an upwind finite difference approximation to the gradient and a resulting causality relationship that lends itself to a dijkstra like programming approach in this paper we discuss several extensions to this technique including higher order versions on unstructured meshes in rn and on manifolds and connections to more general static hamilton jacobi equations

curve like structure extraction using minimal path propagation with backtracking minimal path techniques can efficiently extract geometrically curve like structures by finding the path with minimal accumulated cost between two given endpoints though having found wide practical applications e g line identification crack detection and vascular centerline extraction minimal path techniques suffer from some notable problems the first one is that they require setting two endpoints for each line to be extracted endpoint problem the second one is that the connection might fail when the geodesic distance between the two points is much shorter than the desirable minimal path shortcut problem in addition when connecting two distant points the minimal path connection might become inefficient as the accumulated cost increases over the propagation and results in leakage into some non feature regions near the starting point accumulation problem to address these problems this paper proposes an approach termed minimal path propagation with backtracking we found that the information in the process of backtracking from reached points can be well utilized to overcome the above problems and improve the extraction performance the whole algorithm is robust to parameter setting and allows a coarse setting of the starting point extensive experiments with both simulated and realistic data are performed to validate the performance of the proposed method

a viscosity solutions approach to shape from shading the problem of recovering a lambertian surface from a single two dimensional image may be written as a first order nonlinear equation which presents the disadvantage of having several continuous and even smooth solutions a new approach based on hamilton jacobi bellman equations and viscosity solutions theories enables one to study non uniqueness phenomenon and thus to characterize the surface among the various solutions a consistent and monotone scheme approximating the surface is constructed thanks to the dynamic programming principle and numerical results are presented

active contours without edges we propose a new model for active contours to detect objects in a given image based on techniques of curve evolution mumford shah 1989 functional for segmentation and level sets our model can detect objects whose boundaries are not necessarily defined by the gradient we minimize an energy which can be seen as a particular case of the minimal partition problem in the level set formulation the problem becomes a mean curvature flow like evolving the active contour which will stop on the desired boundary however the stopping term does not depend on the gradient of the image as in the classical active contour models but is instead related to a particular segmentation of the image we give a numerical algorithm using finite differences finally we present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable also the initial curve can be anywhere in the image and interior contours are automatically detected

automatic 3d segmentation and quantification of lenticulostriate arteries from high resolution 7 tesla mra images we propose a novel hybrid approach for automatic 3d segmentation and quantification of high resolution 7 tesla magnetic resonance angiography mra images of the human cerebral vasculature our approach consists of two main steps first a 3d model based approach is used to segment and quantify thick vessels and most parts of thin vessels second remaining vessel gaps of the first step in low contrast and noisy regions are completed using a 3d minimal path approach which exploits directional information we present two novel minimal path approaches the first is an explicit approach based on energy minimization using probabilistic sampling and the second is an implicit approach based on fast marching with anisotropic directional prior we conducted an extensive evaluation with over 2300 3d synthetic images and 40 real 3d 7 tesla mra images quantitative and qualitative evaluation shows that our approach achieves superior results compared with a previous minimal path approach furthermore our approach was successfully used in two clinical studies on stroke and vascular dementia

retinal artery vein classification via topology estimation we propose a novel graph theoretic framework for distinguishing arteries from veins in a fundus image we make use of the underlying vessel topology to better classify small and midsized vessels we extend our previously proposed tree topology estimation framework by incorporating expert domain specific features to construct a simple yet powerful global likelihood model we efficiently maximize this model by iteratively exploring the space of possible solutions consistent with the projected vessels we tested our method on four retinal datasets and achieved classification accuracies of 91 0 93 5 91 7 and 90 9 outperforming existing methods our results show the effectiveness of our approach which is capable of analyzing the entire vasculature including peripheral vessels in wide field of view fundus photographs this topology based method is a potentially important tool for diagnosing diseases with retinal vascular manifestation

progressive minimal path method for segmentation of 2d and 3d line structures we propose a novel minimal path method for the segmentation of 2d and 3d line structures minimal path methods perform propagation of a wavefront emanating from a start point at a speed derived from image features followed by path extraction using backtracing usually the computation of the speed and the propagation of the wave are two separate steps and point features are used to compute a static speed we introduce a new continuous minimal path method which steers the wave propagation progressively using dynamic speed based on path features we present three instances of our method using an appearance feature of the path a geometric feature based on the curvature of the path and a joint appearance and geometric feature based on the tangent of the wavefront these features have not been used in previous continuous minimal path methods we compute the features dynamically during the wave propagation and also efficiently using a fast numerical scheme and a low dimensional parameter space our method does not suffer from discretization or metrication errors we performed qualitative and quantitative evaluations using 2d and 3d images from different application areas

the design and use of steerable filters the authors present an efficient architecture to synthesize filters of arbitrary orientations from linear combinations of basis filters allowing one to adaptively steer a filter to any orientation and to determine analytically the filter output as a function of orientation steerable filters may be designed in quadrature pairs to allow adaptive control over phase as well as orientation the authors show how to design and steer the filters and present examples of their use in the analysis of orientation and phase angularly adaptive filtering edge detection and shape from shading one can also build a self similar steerable pyramid representation the same concepts can be generalized to the design of 3 d steerable filters

coherence enhancing diffusion filtering the completion of interrupted lines or the enhancement of flow like structures is a challenging task in computer vision human vision and image processing we address this problem by presenting a multiscale method in which a nonlinear diffusion filter is steered by the so called interest operator second moment matrix structure tensor an m dimensional formulation of this method is analysed with respect to its well posedness and scale space properties an efficient scheme is presented which uses a stabilization by a semi implicit additive operator splitting aos and the scale space behaviour of this method is illustrated by applying it to both 2 d and 3 d images

global minimum for a finsler elastica minimal path approach in this paper we propose a novel curvature penalized minimal path model via an orientation lifted finsler metric and the euler elastica curve the original minimal path model computes the globally minimal geodesic by solving an eikonal partial differential equation pde essentially this first order model is unable to penalize curvature which is related to the path rigidity property in the classical active contour models to solve this problem we present an eikonal pde based finsler elastica minimal path approach to address the curvature penalized geodesic energy minimization problem we were successful at adding the curvature penalization to the classical geodesic energy caselles et al in int j comput vis 22 1 61 79 1997 cohen and kimmel in int j comput vis 24 1 57 78 1997 the basic idea of this work is to interpret the euler elastica bending energy via a novel finsler elastica metric that embeds a curvature penalty this metric is non riemannian anisotropic and asymmetric and is defined over an orientation lifted space by adding to the image domain the orientation as an extra space dimension based on this orientation lifting the proposed minimal path model can benefit from both the curvature and orientation of the paths thanks to the fast marching method the global minimum of the curvature penalized geodesic energy can be computed efficiently we introduce two anisotropic image data driven speed functions that are computed by steerable filters based on these orientation dependent speed functions we can apply the proposed finsler elastica minimal path model to the applications of closed contour detection perceptual grouping and tubular structure extraction numerical experiments on both synthetic and real images show that these applications of the proposed model indeed obtain promising results

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

anisotropic fast marching on cartesian grids using lattice basis reduction we introduce a modification of the fast marching algorithm which solves the anisotropic eikonal equation associated to an arbitrary continuous riemannian metric cal m on a two or three dimensional domain the algorithm has a complexity cal o n ln n n ln kappa cal m where n is the discrete domain cardinality the logarithmic dependency in the maximum anisotropy ratio kappa cal m of the riemannian metric allows us to handle extreme anisotropies for a limited numerical cost we prove the convergence of the algorithm and illustrate its efficiency by numerical experiments the algorithm relies on the computation at each grid point z of a special system of coordinates a reduced basis of the lattice bbb z m with respect to the symmetric positive definite matrix cal m z encoding the desired anisotropy at this point

a variational method for geometric regularization of vascular segmentation in medical images in this paper a level set based geometric regularization method is proposed which has the ability to estimate the local orientation of the evolving front and utilize it as shape induced information for anisotropic propagation we show that preserving anisotropic fronts can improve elongations of the extracted structures while minimizing the risk of leakage to that end for an evolving front using its shape offset level set representation a novel energy functional is defined it is shown that constrained optimization of this functional results in an anisotropic expansion flow which is useful for vessel segmentation we have validated our method using synthetic data sets 2 d retinal angiogram images and magnetic resonance angiography volumetric data sets a comparison has been made with two state of the art vessel segmentation methods quantitative results as well as qualitative comparisons of segmentations indicate that our regularization method is a promising tool to improve the efficiency of both techniques

vessels as 4 d curves global minimal 4 d paths to extract 3 d tubular surfaces and centerlines in this paper we propose an innovative approach to the segmentation of tubular structures this approach combines all of the benefits of minimal path techniques such as global minimizers fast computation and powerful incorporation of user input while also having the capability to represent and detect vessel surfaces directly which so far has been a feature restricted to active contour and surface techniques the key is to represent the trajectory of a tubular structure not as a 3 d curve but to go up a dimension and represent the entire structure as a 4 d curve then we are able to fully exploit minimal path techniques to obtain global minimizing trajectories between two user supplied endpoints in order to reconstruct tubular structures from noisy or low contrast 3 d data without the sensitivity to local minima inherent in most active surface techniques in contrast to standard purely spatial 3 d minimal path techniques however we are able to represent a full tubular surface rather than just a curve which runs through its interior our representation also yields a natural notion of a tube s central curve we demonstrate and validate the utility of this approach on magnetic resonance mr angiography and computed tomography ct images of coronary arteries

tree topology estimation tree like structures are fundamental in nature and it is often useful to reconstruct the topology of a tree what connects to what from a two dimensional image of it however the projected branches often cross in the image the tree projects to a planar graph and the inverse problem of reconstructing the topology of the tree from that of the graph is ill posed we regularize this problem with a generative parametric tree growth model under this model reconstruction is possible in linear time if one knows the direction of each edge in the graph which edge endpoint is closer to the root of the tree but becomes np hard if the directions are not known for the latter case we present a heuristic search algorithm to estimate the most likely topology of a rooted three dimensional tree from a single two dimensional image experimental results on retinal vessel plant root and synthetic tree data sets show that our methodology is both accurate and efficient

efficient algorithms for globally optimal trajectories we present serial and parallel algorithms for solving a system of equations that arises from the discretization of the hamilton jacobi equation associated to a trajectory optimization problem of the following type a vehicle starts at a prespecified point x sub o and follows a unit speed trajectory x t inside a region in spl rscr sup m until an unspecified time t that the region is exited a trajectory minimizing a cost function of the form spl int sub 0 sup t r x t dt q x t is sought the discretized hamilton jacobi equation corresponding to this problem is usually solved using iterative methods nevertheless assuming that the function r is positive we are able to exploit the problem structure and develop one pass algorithms for the discretized problem the first algorithm resembles dijkstra s shortest path algorithm and runs in time o n log n where n is the number of grid points the second algorithm uses a somewhat different discretization and borrows some ideas from a variation of dial s shortest path algorithm 1969 that we develop here it runs in time o n which is the best possible under some fairly mild assumptions finally we show that the latter algorithm can be efficiently parallelized for two dimensional problems and with p processors its running time becomes o n p provided that p o spl radic n log n

regularized laplacian zero crossings as optimal edge integrators we view the fundamental edge integration problem for object segmentation in a geometric variational framework first we show that the classical zero crossings of the image laplacian edge detector as suggested by marr and hildreth inherently provides optimal edge integration with regard to a very natural geometric functional this functional accumulates the inner product between the normal to the edge and the gray level image gradient along the edge we use this observation to derive new and highly accurate active contours based on this functional and regularized by previously proposed geodesic active contour geometric variational models we also incorporate a 2d geometric variational explanation to the haralick edge detector into the geometric active contour framework

design of steerable filters for feature detection using canny like criteria we propose a general approach for the design of 2d feature detectors from a class of steerable functions based on the optimization of a canny like criterion in contrast with previous computational designs our approach is truly 2d and provides filters that have closed form expressions it also yields operators that have a better orientation selectivity than the classical gradient or hessian based detectors we illustrate the method with the design of operators for edge and ridge detection we present some experimental results that demonstrate the performance improvement of these new feature detectors we propose computationally efficient local optimization algorithms for the estimation of feature orientation we also introduce the notion of shape adaptable feature detection and use it for the detection of image corners

efficient fast marching with finsler metrics we study the discretization of the escape time problem find the length of the shortest path joining an arbitrary point z z of a domain omega to the boundary partial omega path length is measured locally via a finsler metric potentially asymmetric and strongly anisotropic this optimal control problem can be reformulated as a static hamilton jacobi partial differential equation or as a front propagation model it has numerous applications ranging from motion planning to image segmentation we introduce a new algorithm fast marching using anisotropic stencil refinement fm asr which addresses this problem on a two dimensional domain discretized on a cartesian grid the local stencils used in our discretization are produced by arithmetic means like in the fm lbr mirebeau in anisotropic fast marching on cartesian grids using lattice basis reduction preprint 2012 a method previously introduced by the author in the special case of riemannian metrics the complexity of the fm asr in an average sense over all grid orientations only depends poly logarithmically on the anisotropy ratio of the metric while most alternative approaches have a polynomial dependence numerical experiments show in several occasions that the accuracy complexity compromise is improved by an order of magnitude or more

reconstructing curvilinear networks using path classifiers and integer programming we propose a novel approach to automated delineation of curvilinear structures that form complex and potentially loopy networks by representing the image data as a graph of potential paths we first show how to weight these paths using discriminatively trained classifiers that are both robust and generic enough to be applied to very different imaging modalities we then present an integer programming approach to finding the optimal subset of paths subject to structural and topological constraints that eliminate implausible solutions unlike earlier approaches that assume a tree topology for the networks ours explicitly models the fact that the networks may contain loops and can reconstruct both cyclic and acyclic ones we demonstrate the effectiveness of our approach on a variety of challenging datasets including aerial images of road networks and micrographs of neural arbors and show that it outperforms state of the art techniques

new possibilities with sobolev active contours recently the sobolev metric was introduced to define gradient flows of various geometric active contour energies it was shown that the sobolev metric outperforms the traditional metric for the same energy in many cases such as for tracking where the coarse scale changes of the contour are important some interesting properties of sobolev gradient flows include that they stabilize certain unstable traditional flows and the order of the evolution pdes are reduced when compared with traditional gradient flows of the same energies in this paper we explore new possibilities for active contours made possible by sobolev metrics the sobolev method allows one to implement new energy based active contour models that were not otherwise considered because the traditional minimizing method render them ill posed or numerically infeasible in particular we exploit the stabilizing and the order reducing properties of sobolev gradients to implement the gradient descent of these new energies we give examples of this class of energies which include some simple geometric priors and new edge based energies we also show that these energies can be quite useful for segmentation and tracking we also show that the gradient flows using the traditional metric are either ill posed or numerically difficult to implement and then show that the flows can be implemented in a stable and numerically feasible manner using the sobolev gradient

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

weighted local variance based edge detection and its application to vascular segmentation in magnetic resonance angiography accurate detection of vessel boundaries is particularly important for a precise extraction of vasculatures in magnetic resonance angiography mra in this paper we propose the use of weighted local variance wlv based edge detection scheme for vessel boundary detection in mra the proposed method is robust against changes of intensity contrast of edges and capable of giving high detection responses on low contrast edges these robustness and capabilities are essential for detecting the boundaries of vessels in low contrast regions of images which can contain intensity inhomogeneity such as bias field interferences induced from other tissues or fluctuation of the speed related vessel intensity the performance of the wlv based edge detection scheme is studied and shown to be able to return strong and consistent detection responses on low contrast edges in the experiments the proposed edge detection scheme can be embedded naturally in the active contour models for vascular segmentation the wlv based vascular segmentation method is tested using mra image volumes it is experimentally shown that the wlv based edge detection approach can achieve high quality segmentation of vasculatures in mra images

flux based anisotropic diffusion applied to enhancement of 3 d angiogram we present a new approach to anisotropic diffusion based on a multidirectional diffusion flux the diffusion flux is decomposed in an orthogonal basis effectively enabling enhancement of contours as well as diffusion along the contours to this end we have selected a three dimensional basis that depicts the directions of principal curvature and has an interesting interpretation in the context of the vessels the diffusion function associated to each vector of the basis depends on the first order derivative of the intensity in this direction instead of the traditional norm of the smoothed gradient accordingly we present the results of a restoration of computed tomography data of the liver

an alternative estimator for the censored quantile regression model the paper introduces an alternative estimator for the linear censored quantile regression model the objective function is globally convex and the estimator is a solution to a linear programming problem hence a global minimizer is obtained in a finite number of simplex iterations the suggested estimator also applies to the case where the censoring point is an unknown function of a set of regressors it is shown that under fairly weak conditions the estimator has a n convergence rate and is asymptotically normal in the case of a fixed censoring point its asymptotic property is nearly equivalent to that of the estimator suggested by powell 1984 1986a a monte carlo study performed shows that the suggested estimator has very desirable small sample properties it precisely corrects for the bias induced by censoring even when there is a large amount of censoring and for relatively small sample sizes

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

proper actions of lattices on contractible manifolds every lattice in a connected semi simple lie group g acts properly discontinuously by isometries on the contractible manifold g k k a maximal compact subgroup of g we prove that if acts on a contractible manifold w and if either 1 the action is properly discontinuous or 2 w is equipped with a complete riemannian metric the action is by isometries and with unbounded orbits g is simple with finite center and rank 1 then dimw dimg k

finsler active contours in this paper we propose an image segmentation technique based on augmenting the conformal or geodesic active contour framework with directional information in the isotropic case the euclidean metric is locally multiplied by a scalar conformal factor based on image information such that the weighted length of curves lying on points of interest typically edges is small the conformal factor that is chosen depends only upon position and is in this sense isotropic although directional information has been studied previously for other segmentation frameworks here we show that if one desires to add directionality in the conformal active contour framework then one gets a well defined minimization problem in the case that the factor defines a finsler metric optimal curves may be obtained using the calculus of variations or dynamic programming based schemes finally we demonstrate the technique by extracting roads from aerial imagery blood vessels from medical angiograms and neural tracts from diffusion weighted magnetic resonance imagery

segmentation of thin structures in volumetric medical images we present a new segmentation method for extracting thin structures embedded in three dimensional medical images based on modern variational principles we demonstrate the importance of the edge alignment and homogeneity terms in the segmentation of blood vessels and vascular trees for that goal the chan vese minimal variance method is combined with the boundary alignment and the geodesic active surface models an efficient numerical scheme is proposed in order to simultaneously detect a number of different objects in the image a hierarchal approach is applied

vessel tractography using an intensity based tensor model with branch detection in this paper we present a tubular structure segmentation method that utilizes a second order tensor constructed from directional intensity measurements which is inspired from diffusion tensor image dti modeling the constructed anisotropic tensor which is fit inside a vessel drives the segmentation analogously to a tractography approach in dti our model is initialized at a single seed point and is capable of capturing whole vessel trees by an automatic branch detection algorithm developed in the same framework the centerline of the vessel as well as its thickness is extracted performance results within the rotterdam coronary artery algorithm evaluation framework are provided for comparison with existing techniques 96 4 average overlap with ground truth delineated by experts is obtained in addition to other measures reported in the paper moreover we demonstrate further quantitative results over synthetic vascular datasets and we provide quantitative experiments for branch detection on patient computed tomography angiography cta volumes as well as qualitative evaluations on the same cta datasets from visual scores by a cardiologist expert

minimal stencils for discretizations of anisotropic pdes preserving causality or the maximum principle we consider discretizations of anisotropic diffusion and of the anisotropic eikonal equation on two dimensional cartesian grids which preserve their structural properties the maximum principle for diffusion and causality for the eikonal equation these two pdes embed geometric information in the form of a field of diffusion tensors and of a riemannian metric respectively it is common knowledge that when these tensors are strongly anisotropic monotonous or causal discretizations of these pdes cannot be strictly local numerical schemes need to involve interactions between each point and the elements of a stencil which is not limited to its immediate neighbors on the discretization grid using tools from discrete geometry we identify the smallest valid stencils in the sense of convex hull inclusion we also estimate for a fixed condition number but a random tensor orientation the worst case and average case radius of these minimal stencils which is relevant for numerical error analysis