Medial Axis Research Articles

A Hessian-based filter for vascular segmentation of noisy hepatic CT scans

Extraction and enhancement of tubular structures are important in image processing applications, especially in the analysis of liver CT scans where delineation of vascular structures is needed for surgical planning. Portal vein cross-sections have circular or elliptical shapes, so an algorithm must accommodate both. A vessel segmentation method based on medial-axis points was developed and tested on portal veins in CT images. A medial-axis enhancement filter was developed. Consider a line passing through a point inside a tube and intersecting the edges of the tube. If the point is located on the medial axis, the distance of the point in the direction of the line to the edges of the tube will be equal. This feature was employed in a multi-scale framework to identify liver vessels. Dynamic thresholding was used to reduce noise sensitivity. The isotropic coefficient introduced by Pock etal. was used to reduce the response of the filter for asymmetric cross-sections. Quantitative and qualitative evaluation of the proposed method were performed using both 2D/3D and synthetic/clinical datasets. Compared to other methods for medial-axis enhancement, our method produces better results in low-resolution CT images. Detection rate of the medial axis by the proposed method in a noisy image of standard deviation equal to 0.3 is 68% higher than prior methods. A new Hessian-based method for medial axis vessel segmentation was developed and tested. This method produced superior results compared to prior methods. This new method has the potential for many applications of medial-axis enhancement.

A misunderstanding of principal and medial axes? Reply to Sturz & Bodily

Sturz & Bodily [[1][1]] propose an alternative interpretation for the results presented by Kelly et al. [[2][2]], who trained pigeons and chicks in a rectangular environment to find food located in one of two geometrically identical corners. Once the birds were locating the food accurately, they

Fast equal and biased distance fields for medial axis transform with meshing in mind

A method for robust and efficient medial axis transform (MAT) of arbitrary domains using distance solutions (or level sets) is presented. The distance field, d, is calculated by solving the hyperbolic-natured eikonal equation. The solution is obtained on Cartesian grids. Both the fast-marching method (FMM) and fast-sweeping method (FSM) are used to calculate d. Medial axis point clouds are then extracted based on the distance solution via a simple criterion: the Laplacian or the Hessian determinant of d( x). These point clouds in the pixel/voxel space are further thinned to single pixel wide so that medial axis curves or surfaces can be connected and splined. As an alternative to other methods, the current d-MAT procedure bypasses difficulties that are usually encountered by pure geometric methods (e.g. the Voronoi approach), especially in three dimensions, and provides better accuracy than pure thinning methods. It is also shown that the d-MAT approach provides the potential to sculpt/control the MAT form for specialized solution purposes. Various examples are given to demonstrate the current approach.

An atlas-navigated optimal medial axis and deformable model algorithm (NOMAD) for the segmentation of the optic nerves and chiasm in MR and CT images

In recent years, radiation therapy has become the preferred treatment for many types of head and neck tumors. To plan the procedure, vital structures, including the optic nerves and chiasm, must be identified using CT/MR imagery. In this work we present a novel method for automatically localizing the optic nerves and chiasm using a tubular structure localization algorithm in which a statistical model and image registration are used to incorporate a priori local intensity and shape information. The method results in mean Dice coefficients of 0.8 when compared to manual segmentations over ten test cases. This suggests that our method is more accurate than existing techniques developed for the segmentation of these structures.

Quantification of coronary arterial stenoses in CTA using fuzzy distance transform

Quantification of coronary arterial stenoses is useful for the diagnosis of several coronary heart diseases. Being noninvasive, economical and informative, computed tomographic angiography (CTA) has become a common modality for monitoring disease status and treatment effects. Here, we present a new method for detecting and quantifying coronary arterial stenosis in CTA using fuzzy distance transform (FDT) approach and a new coherence analysis of observed data for computing expected local diameter. FDT allows computing local depth at each image point in the presence of partial voluming and thus, eliminates the need for binarization, commonly, associated with inclusion of additional errors. In the current method, coronary arterial stenoses are detected and their severities are quantified by analyzing FDT values along the medial axis of an arterial tree obtained by its skeletonization. A new skeletal pruning algorithm has been developed toward improving the quality of medial axes and thereby, enhancing the accuracy of stenosis detection and quantification. Further, we have developed a new method to estimate “expected diameter” along a given arterial branch using a new coherence analysis of observed diameter values along the branch. The overall method is completed in the following steps – (1) fuzzy segmentation of coronary artery in CTA, (2) FDT computation of coronary arteries, (3) medial axis computation, (4) estimation of observed and expected diameters along arteries and (5) detection of stenoses and quantification of arterial blockage. The performance of this method has been quantitatively evaluated on a realistic coronary artery phantom dataset with randomly simulated stenoses and the results have been compared with a binary distance transform based and a conventional binary algorithm. The method has also been applied on a clinical CTA dataset from thirteen heart patients and the results have been compared with an expert's quantitative assessment of stenoses. Results of the phantom experiment indicate that the new method (error: 0.53%) is significantly more accurate as compared to both binary distance transform based (error 2.11%) and conventional binary (error 3.71%) methods. Also, the results of the clinical study indicate that the new FDT-based method (kappa coefficient = 87.9%) is highly in agreement with the expert's assessments and, in this respect, outperforms the other two methods (kappa coefficients = 75.2% and 69.5%).

Extraction of three-dimensional soil pore space from microtomography images using a geometrical approach

This paper presents a geometrical model of soil pore space based on the quantitative analysis of synchrotron X-ray microtomography data. Our model calculated the minimal set of balls that recovered the skeleton of the pore space using Delaunay tessellation, and the simply connected sets of balls that could be considered as potential pore channels. This model (DTM software) was then applied to three-dimensional tomography reconstructions of soil aggregates (~ 5 mm diameter) from two management systems (conventionally tilled soil, namely CTT and grassland soil, namely GL) with a voxel edge length of 3.2 μm and 5.4 μm, respectively. Geometric characteristics such as the frequency distribution of pore radius, length, and tortuosity as well as the retention curve were calculated using our model. The organic matter decomposition was also simulated using DTM approach. The results were compared with pore space statistics obtained during a previously published study on the same data using algorithms based on the medial axis and throat computation ( 3dma software). The same tendency on the geometrical statistic was obtained using both methods, with more pores of smaller length and diameter calculated for the aggregate from the conventionally tilled site compared to the grassland aggregate. However, the 3dma method generated a larger quantity of voxels (385,673 and 189,250 for CTT and GL, respectively) compared to the amount of balls in DTM (170,250 and 64,273 for CTT and GL, respectively) and shorter channels because of the presence of throats.

Developing a line-of-sight based algorithm for urban street network generalization

This paper presents a generalization algorithm focusing specifically on urban street networks. A street map that uses outlines of blocks to implicitly delineate the shape of streets is essentially a complex polygon with holes in computer graphics. Given such a street map, this algorithm first applies medial axis transformation to derive a rudimentary skeleton of the street network. Because the resulting medial axes tend to exaggeratedly reflect even a minor change in the street shapes, the rudimentary skeleton must undergo a generalization process to become a representation that does not simply resemble the street shapes but is actually informative for understanding its deeper structure. Instead of discarding the street shapes and using only cartographic generalization techniques to simplify the skeleton, the algorithm uses the medial axes as the guide to partition the shape of the streets into individual convex spaces. These initial convex spaces are then successively merged into larger convex spaces following the line-of-sight and least-angle-change principles until the least set of convex space has been achieved. Next, the algorithm uses the same line-of-sight and least-angle-change principles to successively group convex spaces into mutually exclusive convex-space sets until no further grouping can occur. Finally, original medial axes inside each final convex-space set are replaced with a straight-line segment. Collectively, these new line segments form a generalized street network representation that is more favorable in network theory. This new algorithm points out an improved and likely automatic solution of generating appropriate street network representations for spatial configuration analyses of the urban environment.

LOCAL GEOMETRICAL ANALYSIS OF 3D POROUS NETWORKS BASED ON THE MEDIAL AXIS: APPLICATION TO BONE MICROARCHITECTURE MICROTOMOGRAPHY IMAGES

This paper proposes a new method for local evaluation the geometry of complex 3D porous networks such as bone micro-architecture. The method described here allows local quantification of the geometry of the entire volume. It is well adapted to high resolution images where the voxel size is smaller than the structure thickness. The classification of each point on the medial axis is based on a local topological analysis of a neighbourhood, the size of which is adapted locally to the studied point. This property makes this classification more robust to the small irregularities that may appear in a real biological data set. After classification, new quantitative parameters of bone micro-architecture characterizing plate and rod structures are independently defined. The method is quantitatively evaluated on numerical phantoms presenting shape irregularities. Application to experimental 3D bone microtomographic images shows that the method provides visually correct results, even on complex natural trabecular structures. The additional geometrical parameters provided by this method allow discrimination of volumes whose porosity and mean thickness are similar.

Numerical Simulation of Flow over a Circular Cylinder at Low Reynolds Number

The hydrodynamic characteristics of a circular cylinder in two-dimensional unsteady uniform cross flow was simulated numerically by the laminar model with the reasonable mesh used the method of fluent. The focus of this numerical simulation was to research the characteristics of pressure distribution, drag coefficient and lift coefficient, and the Strouhal number was calculated at Reynolds-numbers value of 200. The results agree well with experimental data and other numerical results according to the reference. In order to study the control measures of the flow over a circular cylinder, the different baffles inserted at various locations downstream of the cylinder have been compared. The results shows that the vortex shedding of flow over a circular cylinder could be well controlled by place the baffle at a right position of the downstream medial axis of the cylinder, which could reduce drag and resist vibration.

3D medial axis point approximation using nearest neighbors and the normal field

We present a novel method to approximate medial axis points given a set of points sampled from a surface and the normal vectors to the surface at those points. For each sample point, we find its maximal tangent ball containing no other sample points, by iteratively reducing its radius using nearest neighbor queries. We prove that the center of the ball constructed by our algorithm converges to a true medial axis point as the sampling density increases to infinity. We also propose a simple heuristic to handle noisy samples. By simple extensions, our method is applied to medial axis point simplification, local feature size estimation and feature-sensitive point decimation. Our algorithm is simple, easy to implement, and suitable for parallel computation using GPU because the iteration process for each sample point runs independently. Experimental results show that our method is efficient both in time and in space.

Curvilinear tool path generation for pocket machining

In this paper, a curvilinear tool path generation method with implicit moving boundaries is presented for pocket machining. The pocket profile is implicitly embedded into a higher-dimensional function, namely the level set function, and is represented with the zero level set as a moving boundary. In order to generate low-curvature parallel boundaries, curvature flow motion is adopted during the evolution of the moving boundaries. The topological changes of the profile offsets are naturally handled by propagating the moving boundaries. The curvilinear tool path is then constructed between adjacent offsets. For complex pockets with elongated narrow sections, the implicit medial axis transformation is proposed to partition the large rounded region and elongated regions. The large rounded pocket region employs the proposed curvilinear tool path, and the elongated narrow regions use continuous arcs, respectively. The proposed tool path generation method is suitable for high-speed machining of pockets with islands and consists of both rounded and elongated narrow regions.

Subjectively interpreted shape dimensions as privileged and orthogonal axes in mental shape space.

The shape of an object is fundamental in object recognition but it is still an open issue to what extent shape differences are perceived analytically (i.e., by the dimensional structure of the shapes) or holistically (i.e., by the overall similarity of the shapes). The dimensional structure of a stimulus is available in a primary stage of processing for separable dimensions, although it can also be derived cognitively from a perceived stimulus consisting of integral dimensions. Contrary to most experimental paradigms, the present study asked participants explicitly to analyze shapes according to two dimensions. The dimensions of interest were aspect ratio and medial axis curvature, and a new procedure was used to measure the participants' interpretation of both dimensions (Part I, Experiment 1). The subjectively interpreted shape dimensions showed specific characteristics supporting the conclusion that they also constitute perceptual dimensions with objective behavioral characteristics (Part II): (1) the dimensions did not correlate in overall similarity measures (Experiment 2), (2) they were more separable in a speeded categorization task (Experiment 3), and (3) they were invariant across different complex 2-D shapes (Experiment 4). The implications of these findings for shape-based object processing are discussed.

Computation of the medial axis of planar domains based on saddle point programming

This paper presents a saddle point programming approach to compute the medial axis (MA). After exploring the saddle point properties of the medial axis transform (MAT), the mathematical programming method is employed to establish the saddle point programming model of the MAT. By using the optimal conditions, i.e., the number and distribution of the tangent points between the boundary and medial axis disk, the one- and two-dimensional saddle point algorithms are developed. In order to determine the branch point, it is better to consider its generating mechanism. Here, we identify the branch point according to the sudden changes of the solutions to the one-dimensional saddle point algorithm. Hence, all the regular and irregular points of MA can be computed by a general algorithm, and it is proved to be efficient and accurate by the numerical examples.

Bone graphs: Medial shape parsing and abstraction

The recognition of 3-D objects from their silhouettes demands a shape representation which is stable with respect to minor changes in viewpoint and articulation. This can be achieved by parsing a silhouette into parts and relationships that do not change across similar object views. Medial descriptions, such as skeletons and shock graphs, provide part-based decompositions but suffer from instabilities. As a result, similar shapes may be represented by dissimilar part sets. We propose a novel shape parsing approach which is based on identifying and regularizing the ligature structure of a medial axis, leading to a bone graph, a medial abstraction which captures a more stable notion of an object’s parts. Our experiments show that it offers improved recognition and pose estimation performance in the presence of within-class deformation over the shock graph.

Medial Axis Transform Method for Shape Design of Hub and Shroud Contours of Impellers

A novel method of designing hub and shroud contours is presented. The method, based on the medial axis transform theory in differential geometry, gives a uniform description of hub and shroud contours and the formula of cross section area. Through solving the formula of cross section area with an additional constraint, the hub and shroud contours can be determined numerically. The constraint is exposed through a curvature equation, which allows the medial axis or hub (shroud) contour to be a certain form. Using this method, various optimization criteria relating to the cross section area can be conveniently introduced into the design.

An optimal approach to multiple tool selection and their numerical control path generation for aggressive rough machining of pockets with free-form boundaries

To machine pockets, especially ones with closed free-form boundary curves, roughing is crucial to part productivity, for this operation alone could take more than 60% of the total machining time. At present, there is a high demand from industry for a new machining technique that can efficiently cut pockets. Aggressive rough machining, in which the largest possible cutters are always employed and are fully immersed in workpieces, can be a solution. Although aggressive roughing is by far the most efficient machining strategy, compared to prior pocketing methods, no computer numerical control (CNC) programming technique has been developed to support it, resulting in few applications in machine shops. To address this urgent industrial need, based on the medial axis transform of a pocket, this work proposes an optimal approach to multiple tool selection and their numerical control (NC) path generation for aggressive roughing of the pocket. First, the NC paths of a specific tool are quickly generated using the pocket’s medial axis transform. Thanks to the unique characteristic of the medial axis transform, the paths can ensure the tool the largest accessible space for pocketing. At the same time, they can guarantee the tool to be free of gouging and interference. Then, an optimization model of selecting multiple cutters and generating their NC paths is built in order to achieve the highest efficiency of the aggressive rough machining. To demonstrate the advantages of this innovative approach, two examples are rendered, and their results are compared to those obtained by the existing methods. This approach can be directly implemented into current CAD/CAM software to promote aggressive rough machining of pockets in industry.

A unified Pythagorean hodograph approach to the medial axis transform and offset approximation

Algorithms based on Pythagorean hodographs (PH) in the Euclidean plane and in Minkowski space share common goals, the main one being rationality of offsets of planar domains. However, only separate interpolation techniques based on these curves can be found in the literature. It was recently revealed that rational PH curves in the Euclidean plane and in Minkowski space are very closely related. In this paper, we continue the discussion of the interplay between spatial MPH curves and their associated planar PH curves from the point of view of Hermite interpolation. On the basis of this approach we design a new, simple interpolation algorithm. The main advantage of the unifying method presented lies in the fact that it uses, after only some simple additional computations, an arbitrary algorithm for interpolation using planar PH curves also for interpolation using spatial MPH curves. We present the functionality of our method for G 1 Hermite data; however, one could also obtain higher order algorithms.

COMPUTATIONAL AND STRUCTURAL ADVANTAGES OF CIRCULAR BOUNDARY REPRESENTATION

Boundary approximation of planar shapes by circular arcs has quantitative and qualitative advantages compared to using straight-line segments. We demonstrate this by way of three basic and frequent computations on shapes – convex hull, decomposition, and medial axis. In particular, we propose a novel medial axis algorithm that beats existing methods in simplicity and practicality, and at the same time guarantees convergence to the medial axis of the original shape.

Curve Skeletons of Planar Domains

AbstractCurve skeletons are 1D entities that capture the essential topology and geometry of a shape in a simple and very compact form, and have found a multitude of applications in engineering and science. The concept itself is ill-defined, which is why the many existing definitions are ad-hoc and employ heuristic algorithms revolving around the (unique) medial axis. We propose a framework for computing curve skeletons for arbitrary planar domains that relies on a modified medial axis, and is based on R-functions that can be thought of as continuous forms of the Boolean logic functions. Furthermore, we propose one particular definition of such a curve skeleton that preserves the homotopy of the domain, is stable in the presence of noise and is well-suited to downstream applications. The framework can be implemented in any commercial geometric kernel for planar domains, and has attractive computational properties. Furthermore, the mathematical concepts are extendable to medial surfaces and curve skeletons ...

Identification of the Dental Root Canal from Micro-CT Records Using 3D Curve Skeleton Extraction

AbstractThis paper presents a novel image processing procedure dedicated to the automated detection of the medial axis of the root canal from dental micro CT records. The 3D model of root canal is built up from several hundreds of parallel cross sections, using image enhancement and an enhanced fuzzy c-means based partitioning, center point detection in the segmented slice, three dimensional inner surface reconstruction and curve skeleton extraction in three dimensions. The central line of the root canal can finally be approximated as a 3D spline curve. The proposed procedure can help prepare several kinds of endodontic interventions.