Medial axis

In this undergraduate research project report, a fast sampling based algorithm and surface reconstruction strategy is presented to compute the medial transform of 2D and 3D geometrical shapes. In addition, a method to isolate the junction points of the 2D medial axis is provided and iterative approaches to surface sampling in 3D are discussed.

We present an algorithm for the generation of coarse and fine finite element (FE) meshes on multiply connected surfaces, based on the medial axis transform (MAT). The MAT is employed to automatically decompose a complex shape into topologically simple subdomains, and to extract important shape characteristics and their length scales. Using this technique, we can create a coarse subdivision of a complex surface and select local element size to generate fine triangular meshes within those subregions in an automated manner. Therefore, this approach can lead to integration of fully automatic FE mesh generation functionality into FE preprocessing systems. We can also classify existing FE mesh generators depending on their underlying algorithmic approaches: 9 mapping mesh generation [6-12]; 9 node insertion followed by area/volume triangulation [13-201;

We present an algorithm for the generation of coarse and fine finite element (FE) meshes on multiply connected surfaces, based on the medial axis transform (MAT). The MAT is employed to automatically decompose a complex shape into topologically simple subdomains, and to extract important shape characteristics and their length scales. Using this technique, we can create a coarse subdivision of a complex surface and select local element size to generate fine triangular meshes within those subregions in an automated manner. Therefore, this approach can lead to integration of fully automatic FE mesh generation functionality into FE preprocessing systems. We can also classify existing FE mesh generators depending on their underlying algorithmic approaches: 9 mapping mesh generation [6-12]; 9 node insertion followed by area/volume triangulation [13-201;

In this paper we study the problem of reconstructing orthogonal polyhedra from a putative vertex set, i.e., we are given a set of points and want to find an orthogonal polyhedron for which this is the set of vertices. We are especially interested in testing whether such a polyhedron is unique. Since this is not the case in general, we focus on orthogonally convex polyhedra, and give an O(n log n) algorithm to find the answer. We then consider the case where the given set of points may be rotated beforehand. For 2D, we prove uniqueness and provide an O(n log n) algorithm to obtain the answer, which can then be used for an O(n 2 log n) algorithm in 3D.

Many foods contain structural features (such as air spaces, cells, inclusions) that are manifested over a large range of dimensions, including the nanometer range. To make significant advances in delivering foods with excellent quality, the role of microstructure, and its interactions with composition and external conditions must be understood and controlled. This can only be achieved by first developing accurate techniques that detect changes in the internal structure of foods. X-ray CT is a powerful tool for obtaining highly detailed 3D information on internal food structures. Moreover, because of the penetrating capabilities of X-rays, samples can be investigated in a non-destructive manner, thus not majorly altering internal structures by invasive sample preparation. Another advantage of this technology is that it can be used in a range of relevant resolutions to obtain images in the range of millimeter to micron (X-ray micro-CT) and even nanometer (Xray nano-CT) pixel resolution. A selection of moist (sugar foam, apple) and dry foods (cereal products) were scanned using X-ray micro-and nano-CT. Images, morphometric characteristics and geometric 3D models provided a detailed insight in the various foods in terms of porosity, connectivity and anisotropy of microstructural features. X-ray nano-CT proved to be a valuable yet challenging technique considering sample representability. Nevertheless, X-ray nano-CT enabled the investigation of the 3D microstructure of samples in a near-native state at unprecedented resolutions.

In this paper, we present the first nontrivial theoretical bound on the quality of the 3D solids generated by any contour interpolation method. Given two arbitrary parallel contour slices with n vertices in 3D, let α be the smallest angle in the constrained Delaunay triangulation of the corresponding 2D contour overlay, we present a contour interpolation method which reconstructs a 3D solid with the minimum dihedral angle of at least α/8. Our algorithm runs in O(nlogn) time where n is the size of the contour overlay.We also present a heuristic algorithm that optimizes the dihedral angles of a mesh representing a surface in 3D.

There are so many applications of ellipse equation, such that graphical representation and finding problem of any elliptical shapes object, computer graphics, space science, engineering design and so on. In this paper state about general equation of ellipse. Generally, there are various equations for drawing an ellipse in 2d plane but not exactly the equations are general, must have some condition and rule. So, here observed those things and found a problem to create a general equation of ellipse. And tried to invent a polynomial equation of curve that can represent an ellipse and focus of ellipse act as variable of equation (and a starting point of major axis (vertex) also act as a variable).

In this paper we describe the architecture of a simple evacuation model which is based on a graph representation of the scene. Such graphs are typically constructed using Medial Axis Transform (MAT) or Straight Medial Axis Transform (S-MAT) transformations, the former being a part of the Voronoi diagram (Dirichlet tessellation) of the floor plan. In our work we construct such graphs for floor plans using Voronoi diagram along with the dual Delaunay triangulation of a set of points approximating the scene. Information supplied by Delaunay triangulation complements the graph in two ways: it determines capacities of some paths associated with edges, and provides a bijection between graph vertices and a set of regions forming a partition of the floor plan. We call the representation granular for this reason. We provide an exposition of the representation of fire scene that aids egress time calculations, discuss the algorithm of construction of this representation and briefly discuss the applicability of network flow models (e.g. the Ford-Fulkerson method or the push-relabel method) in our setting.

In this paper we describe an evacuation modeling framework based on a graph representation of the scene which is derived from its geometric description. Typically such graphs (geometric networks) are constructed using Medial Axis Transform (MAT) or Straight Medial Axis Transform (S-MAT). In our work we use Voronoi tessellation of a set of points approximating the scene (a single floor plan) along with the dual graph -Delaunay triangulation. Using these two graphs we extract not only the information about paths in the building, but also information about path widths and areas assigned to vertices. Typically only path lengths from MAT or S-MAT based geometric networks are used in evacuation modeling. Our approach enables us to include flow analysis and e.g. locate bottlenecks. We discuss a typical density-based evacuation model coupled with a partial behavioral evacuation model within proposed framework.

Classical differential geometry is often considered as an “art of manipulating with indices”. In these lectures we develop a more geometric approach by explaining the true mathematical meaning of all introduced notions. Where possible, we try to avoid coordinates totally. But the correspondence to the traditional coordinate presentation is also explained. The usage of invariant language not only simplifies many arguments but also reduces the amount of computations in particular problems. The best example is a simple formula for the Gaussian curvature of a surface based on the concept of connection 1-form.

We present a novel algorithm to perform continuous collision detection for articulated models. Given two discrete configurations of the links of an articulated model, we use an "arbitrary in-between motion" to interpolate its motion between two successive time steps and check the resulting trajectory for collisions. Our approach uses a three-stage pipeline: (1) dynamic bounding-volume hierarchy (D-BVH) culling based on interval arithmetic; (2) culling refinement using the swept volume of line swept sphere (LSS) and graphics hardware accelerated queries; (3) exact contact computation using OBB-trees and continuous collision detection between triangular primitives. The overall algorithm computes the time of collision, contact locations and prevents any interpenetration between the articulated model with the environment. We have implemented the algorithm and tested its performance on a 2.4 GHz Pentium PC with 1 Gbyte of RAM and a NVIDIA GeForce FX 5800 graphics card. In practice, our algorithm is able to perform accurate and continuous collision detection between articulated models and complex environments at nearly interactive rates.

Applications of of the medial axis have been limited because of its instability and algebraic complexity. In this paper, we use a simplification of the medial axis, the θ-SMA, that is parameterized by a separation angle (θ) formed by the vectors connecting a point on the medial axis to the closest points on the boundary. We present a formal characterization of the degree of simplification of the θ-SMA as a function of θ, and we quantify the degree to which the simplified medial axis retains the features of the original polyhedron. We present a fast algorithm to compute an approximation of the θ-SMA. It is based on a spatial subdivision scheme, and uses fast computation of a distance field and its gradient using graphics hardware. The complexity of the algorithm varies based on the error threshold that is used, and is a linear function of the input size. We have applied this algorithm to approximate the SMA of models with tens or hundreds of thousands of triangles. Its running time varies from a few seconds, for a model consisting of hundreds of triangles, to minutes for highly complex models.

Geometric and neural models of illusory-contour (IC) synthesis currently use only local contour geometry to derive the shape of ICs. Work on the visual representation of shape, by contrast, points to the importance of both contour and surface geometry. We investigated the influence of surfacebased geometric factors on IC shape. The local geometry of inducing-contour pairs was equated in stereoscopic IC displays, and the shape of the enclosed surface was varied by manipulating sign of curvature, cross-axial shape width, and medial-axis geometry. IC shapes were measured using a parametric shape-adjustment task (Experiment 1) and a dot-adjustment task (Experiment 2). Both methods revealed large influences of surface geometry. ICs enclosing locally concave regions were perceived to be systematically more angular than those enclosing locally convex regions. Importantly, the influence of sign of curvature was modulated significantly by shape width and medial-axis geometry: IC shape difference between convex and concave inducers was greater for narrow shapes than wider ones, and greater for shapes with straight axis and symmetric contours (diamond versus bowtie), than those with curved axis and parallel contours (bent tubes). Even at the level of illusory ''contours,'' there is a contribution of region-based geometry which is sensitive to nonlocal shape properties involving medial geometry and part decomposition. Models of IC synthesis must incorporate the role of nonlocal region-based geometric factors in a way that parallels their role in organizing visual shape representation more generally.

In this papel; the Morphological Skeleton Interpolation (MSI) algorithm is presented. It is an eficient shape-based interpolation method used for interpolating slices between successive slices of a 3 -0 object. It is based on the well known mathematical morphology procedure of morphological skeletonization, that is used as a representation of an object. The proposed morphological skeleton matching process provides translation, rotation and scaling information at the same time. The interpolated slices preserve the shape of the original object slices, when the slices have similar shapes. It can also modify the shape of an object when the successive slices have not similar shapes. Applications on artijicial and real data are also presented.

In this paper, the morphological skeleton interpolation (MSI) algorithm is presented. It is an efficient, shape-based interpolation method used for interpolating slices in a three-dimensional (3-D) binary object. It is based on morphological skeletonization, which is used for two-dimensional (2-D) slice representation. The proposed morphological skeleton matching process provides translation, rotation, and scaling information at the same time. The interpolated slices preserve the shape of the original object slices, when the slices have similar shapes. It can also modify the shape of an object when the successive slices do not have similar shapes. Applications on artificial and real data are also presented.

Boundary representation models reverse engineered from 3D range data suffer from various inaccuracies caused by noise in the measured data and the model building software. Beautification aims to improve such models in a post-processing step solely working with the boundary representation model. The improved model should exhibit topological and geometric regularities representing the original, ideal design intent. This paper gives an overview of algorithms for a complete beautification system suitable for improving the topology and the geometry of low to medium complexity reverse engineered models.

Automatic building extraction and delineation from airborne LiDAR point cloud data of urban environments is still a challenging task due to the variety and complexity at which buildings appear. The Medial Axis Transform (MAT) is able to describe the geometric shape and topology of an object, but has never been applied for building roof outline extraction. It represents the shape of an object by its centerline, or skeleton structure instead of its boundary. Notably, end points of the MAT in principle coincide with corner points of building outlines. However, the MAT is sensitive to small boundary irregularities, which makes shape detection in airborne point clouds challenging. We propose a robust MAT-based method for detecting building corner points, which are then connected to form a building boundary polygon. First, we approximate the 2D MAT of a set of building edge points acquired by the alpha-shape algorithm to derive a so-called building roof skeleton. We then propose a hierarchical corner-aware segmentation to cluster skeleton points based on their properties which are the so-called separation angle, radius of the maximally inscribe circle, and defining edge point indices. From each segment, a corner point is then estimated by extrapolating the position of the zero radius inscribed circle based on the skeleton point positions within the segment. Our experiment uses point cloud datasets of Makassar, Indonesia and EYE-Amsterdam, The Netherlands. The average positional accuracy of the building outline results for Makassar and EYE-Amsterdam is 65 cm and 70 cm, respectively, which meet one-meter base map accuracy criteria. The results imply that skeletonization is a promising tool to extract relevant geometric information on e.g. building outlines even from far from perfect geographical point cloud data.

The notion of skeleton plays a major role in shape analysis since the introduction of the medial axis. The continuous medial axis is a skeleton with the following characteristics: centered, thin, homotopic, and reversible (sufficient for the reconstruction of the original object). The discrete Euclidean medial axis (MA) is also reversible and centered, but no longer homotopic nor thin. To preserve topology and reversibility, the MA is usually combined with homotopic thinning algorithms. Since there is a robust and well defined framework for fast homotopic thinning defined in the domain of abstract complexes, some authors have extended the MA to a doubled resolution grid and defined the discrete Euclidean Medial Axis in Higher Resolution (HMA), which can be combined to the framework defined on abstract complexes. Other authors gave an alternative definition of medial axis, which is a reversible subset of the MA, and is called Reduced Discrete Medial Axis (RDMA). The RDMA is thinner than the MA and can be computed in optimal time. In this paper we extend the RDMA to the doubled resolution grid and we define the High-resolution RDMA (HRDMA). The HRDMA is reversible and it can be computed in optimal time. The HRDMA can be combined with the algorithms in abstract complexes, so a reversible and homotopic Euclidean skeleton can be computed in optimal time.

Critical kernels constitute a general framework settled in the category of abstract complexes for the study of parallel thinning in any dimension. It allows to easily design parallel thinning algorithms which produce new types of skeletons, with specific geometrical properties, while guaranteeing their topological soundness. In this paper, we demonstrate that it is possible to define a skeleton based on the Euclidean distance, rather than on the common discrete distances, in the context of critical kernels. We provide the necessary definitions as well as an efficient algorithm to compute this skeleton.

The notion of skeleton plays a major role in shape analysis. Some usually desirable characteristics of a skeleton are: sufficient for the reconstruction of the original object, centered, thin and homotopic. The Euclidean Medial Axis presents all these characteristics in a continuous framework. In the discrete case, the Exact Euclidean Medial Axis (MA) is also sufficient for reconstruction and centered. It no longer preserves homotopy but it can be combined with a homotopic thinning to generate homotopic skeletons. The thinness of the MA, however, may be discussed. In this paper we present the definition of the Exact Euclidean Medial Axis on Higher Resolution which has the same properties as the MA but with a better thinness characteristic, against the price of rising resolution. We provide and prove an efficient algorithm to compute it. Definition 4 (Medial Axis). Let X ⊂ Z n , the medial axis of X, denoted by MA(X), is the set of the centers of all the maximal balls for X.

Skeletons have been playing an important role in shape analysis since the introduction of the medial axis in the sixties. The original medial axis definition is in the continuous Euclidean space. It is a skeleton with the following characteristics: centered, thin, homotopic (it has the same topology as the object), and reversible (sufficient for the reconstruction of the object). The discrete version of the Euclidean medial axis (MA) is also reversible and centered, but no longer homotopic nor thin. The combination of the MA with homotopic thinning algorithms can result in a topology preserving, reversible skeleton. However, such combination may generate thicker skeletons, and the choice of the thinning algorithm is not obvious. A robust and well defined framework for fast homotopic thinning available in the literature is defined in the domain of abstract complexes. Since the abstract complexes are represented in a doubled resolution grid, some authors have extended the MA to a doubled resolution grid and defined the discrete Euclidean medial axis in higher resolution (HMA). The HMA can be combined with the thinning framework defined on abstract complexes. Other authors have given an alternative definition of medial axis, which is a subset of the MA, called reduced discrete medial axis (RDMA). The RDMA is reversible, thinner than the MA, and it can be computed in optimal time. In this paper, we extend the RDMA to the doubled resolution grid and we define the high-resolution RDMA (HRDMA). We provide an optimal algorithm to compute the HRDMA. The HRDMA can be combined with the thinning framework defined on abstract complexes. The skeleton obtained by such combination is provided with strong characteristics, not found in the literature.

The notion of skeleton plays a major role in shape analysis. Some usually desirable characteristics of a skeleton are: centered, thin, homotopic, and sufficient for the reconstruction of the original object. The Euclidean medial axis presents all these characteristics in a continuous framework. In the discrete case, the exact Euclidean medial axis (MA) is also sufficient for reconstruction and centered. It no longer preserves homotopy but it can be combined with a homotopic thinning to generate homotopic skeletons. The thinness of the MA, however, may be discussed. In this paper, we present the definition of the exact Euclidean medial axis in higher resolution, which has the same properties as the MA but with a better thinness characteristic, against the price of rising resolution. We provide and prove an efficient algorithm to compute it.

Figure 1: Our processing pipeline in a nutshell (left to right): we first compute the rotation axis on the input mesh; we then partition the mesh in millable height-field portions (including the top and bottom ones) taking also into account the information of the saliency map; we compute a milling sequence and fabricate the object using the 4-axis milling machine; we clean up the result to obtain the final real object.

In this papel; the Morphological Skeleton Interpolation (MSI) algorithm is presented. It is an eficient shape-based interpolation method used for interpolating slices between successive slices of a 3-0 object. It is based on the well known mathematical morphology procedure of morphological skeletonization, that is used as a representation of an object. The proposed morphological skeleton matching process provides translation, rotation and scaling information at the same time. The interpolated slices preserve the shape of the original object slices, when the slices have similar shapes. It can also modify the shape of an object when the successive slices have not similar shapes. Applications on artijicial and real data are also presented.

In this paper, the morphological skeleton interpolation (MSI) algorithm is presented. It is an efficient, shape-based interpolation method used for interpolating slices in a three-dimensional (3-D) binary object. It is based on morphological skeletonization, which is used for two-dimensional (2-D) slice representation. The proposed morphological skeleton matching process provides translation, rotation, and scaling information at the same time. The interpolated slices preserve the shape of the original object slices, when the slices have similar shapes. It can also modify the shape of an object when the successive slices do not have similar shapes. Applications on artificial and real data are also presented.

Ghys has recently discovered a beautiful theorem: given a diffeomorphism of the projective line, there exist at least four distinct points in which the diffeomorphism is unusually well approximated by projective transformations - . The points in question are the ones in which the 3-jet of the diffeomorphism is that of a projective transformation; in a generic point the order of approximation is only 2. In other words, the Schwarzian derivative of every diffeomorphism of RP 1 has at least four distinct zeroes. The theorem of Ghys is analogous to the classical four vertex theorem: a closed convex plane curve has at least four curvature extrema . The proof presented by E. Ghys was purely geometrical: it was inspired by the Kneser theorem on osculating circles of a plane curve . We will prove an amazing strengthening of the classical Sturm comparison theorem and deduce Ghys' theorem from it. Sturm theory is related to the geometry of curves (see ). This relation is based on the following general idea. A geometrical problem is reformulated as the problem on the least number of zeroes of

Bicipital root and proximal tendon disorders are an important symptom generator in the shoulder. The accuracy of the diagnosis of many shoulder disorders visually without quantitative shape analysis is limited, motivating a clinical need for some ancillary method to access the proximal biceps. Because of the known interrelationship of the bicipital groove (BG) with several types of disorders, we propose an approach to the 3D shape description of the BG that captures information relevant to disorders of the shoulder (e.g. width, depth, angles of walls, presence of spurs). Our approach is medial-axis based and captures intuitive aspects of shape such as thickness, bending, and elongation. Our proposed method overcomes the well-known problem of boundary sensitivity in the medial representation as it is applied to representation and analysis of BG shape. We give preliminary quantitative results indicating that this representation does capture shape variation within our experimental data, providing motivation to explore more sophisticated statistical analysis based on this representation in future work. We also provide a method for semi-automatic segmentation of the BG from computed tomography (CT) scans of the shoulder; an important precursor step to BG shape analysis.

Discrete m-reps use tolerance-based, sampled medial skeleta as an underlying framework for boundaries defined by displaced subdivision surfaces. They provide local and global shape information, combining the strengths of multiscale skeletal modeling with the multi-resolution, deformation and shading properties afforded by subdivision surfaces. Hierarchically linked medial figures allow figural, object-based deformation, and their stability of object representation with respect to boundary perturbation shows advantages of tolerance-based medial representations over Blum axis and Voronoi-based skeletal models. M-rep models provide new approaches to traditional computer graphics modeling, to physically based modeling and simulation, and to image-analysis, segmentation and display, by combining local and object-level deformability and by explicitly including object-scale, tolerance and hierarchical level-of-detail. Sampled medial representations combine the solid modeling capabilities of constructive solid geometry with the flexibility of traditional b-reps, to which they add multiscale medial and boundary deformations parameterized by an object-based coordinate system. This thesis research encompassed conceptual development on discrete m-reps and their implementation for MIDAG (the Medical Image Display and Analysis Group) at UNC-Chapel Hill. Prototype and application code was created to support the following: medial atoms in 3D that included a quaternion frame to establish a local coordinate system; data structures for medial mesh topologies; a new algorithm for interpolating Catmull-Clark subdivision surfaces for m-rep boundaries; a medially based coordinate system parameterizing the m-rep boundary, interior, and local exterior; displacement texturing and displacement meshing of m-rep boundaries; methods of medially based deformation; figure/subfigure blending by implicit surface methods or (with Qiong Han) using remeshing of subdivision boundaries; and C++ code libraries for m-rep modeling in geometric design and image-segmentation applications. In a project spanning so many years, there are countless people to whom thanks are due. First and foremost, to Steve Pizer-the work would never have been done without his insights, his goading, his advice and friendship; to my other committee members, as well as to Lee Westover, who sat on my committee in the early years; to Turner Whitted, for ideas and advice in the early days of the work; to Lee Nackman, for the inspiration of his early work and the pleasure of a conversation in 2001; To everyone at MIDAG who worked with me on medial matters, especially the inestimably helpful Graham Gash; to Tom Fletcher and Paul Yushkevich and Jessica Crouch, my comrades-in-arms, for lots of ideas and discussion along the way. to Qiong Han, my collaborator on the m-rep figure-subfigure blending work; to Jim Damon, a superb teacher and colleague; to Eitan Grinspun at Caltech for his advice on proximity tests on subdivision surfaces; to Rob Katz, as my spiritual advisor, and to Janet Jones for her excellent friendship and for helping me to survive my too numerous run-ins with University Administration; to Matt Lavoie, old friend, for giving me a place to crash when I was in town to finish off the writing; to all my friends in the department, whether former office-mates, students or faculty. to Dr. Robert Cupper, my department chairman and mentor at Allegheny College, for his faith that my Ph.D. was just a matter of time and for convincing the Allegheny administration of that fact. Special thanks to John, Paul, George and Ringo.

We introduce a multiscale approach to modeling and rendering deformable solid objects, replacing polygonal or spline-based B-reps with a medial representation (M-rep) based on hierarchically linked, discretely sampled medial figures. Each medial figure is a widthproportionally sampled medial surface which implies object boundaries within a width-proportional tolerance; boundary variations within this tolerance are specified separately by surface displacements or texture-maps. M-reps share advantages of other skeletal techniques for modeling deformable and articulated figures, but by decoupling object-shape from fine surface detail, they (a) eliminate skeletal instabilities due to small-scale boundary-perturbation, and (b) induce a natural hierarchy and level-of-detail for object-based deformations. This work discusses the theoretical foundation of M-reps, their data structures and implementation, their current use in medical-image analysis and volume rendering, and how model building and rendering are done via M-reps, using texture maps and subdivision surfaces for fine detail.

A dissertation submitted to the faculty of The University of North Carolina at Chapel Hill in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Computer Science. ... Copyright c 2004 Andrew Lewis Thall All rights reserved

M-reps (formerly called DSLs) are a multiscale medial means for modeling and rendering 3D solid geometry. They are particularly well suited to model anatomic objects and in particular to capture prior geometric information effectively in deformable models segmentation approaches. The representation is based on figural models, which define objects at coarse scale by a hierarchy of figures-each figure generally a slab representing a solid region and its boundary simultaneously. This paper focuses on the use of single figure models to segment objects of relatively simple structure. A single figure is a sheet of medial atoms, which is interpolated from the model formed by a net, i.e., a mesh or chain, of medial atoms (hence the name m-reps), each atom modeling a solid region via not only a position and a width but also a local figural frame giving figural directions and an object angle between opposing, corresponding positions on the boundary implied by the m-rep. The special capability of an m-rep is to provide spatial and orientational correspondence between an object in two different states of deformation. This ability is central to effective measurement of both geometric typicality and geometry to image match, the two terms of the objective function optimized in segmentation by deformable models. The other ability of m-reps central to effective segmentation is their ability to support segmentation at multiple levels of scale, with successively finer precision. Objects modeled by single figures are segmented first by a similarity transform augmented by object elongation, then by adjustment of each medial atom, and finally by displacing a dense sampling of the m-rep implied boundary. While these models and approaches also exist in 2D, we focus on 3D objects. The segmentation of the kidney from CT and the hippocampus from MRI serve as the major examples in this paper. The accuracy of segmentation as compared to manual, slice-by-slice segmentation is reported.

We introduce a multiscale approach to modeling and rendering deformable solid objects, replacing polygonal or spline-based B-reps with a medial representation (M-rep) based on hierarchically linked, discretely sampled medial figures. Each medial figure is a widthproportionally sampled medial surface which implies object boundaries within a width-proportional tolerance; boundary variations within this tolerance are specified separately by surface displacements or texture-maps. M-reps share advantages of other skeletal techniques for modeling deformable and articulated figures, but by decoupling object-shape from fine surface detail, they (a) eliminate skeletal instabilities due to small-scale boundary-perturbation, and (b) induce a natural hierarchy and level-of-detail for object-based deformations. This work discusses the theoretical foundation of M-reps, their data structures and implementation, their current use in medical-image analysis and volume rendering, and how model building and rendering are done via M-reps, using texture maps and subdivision surfaces for fine detail.

We present a simple computational model for planar shape decomposition that naturally captures most of the rules and salience measures suggested by psychophysical studies, including the minima and shortcut rules, convexity, and symmetry. It is based on a medial axis representation in ways that have not been explored before and sheds more light into the connection between existing rules like minima and convexity. In particular, vertices of the exterior medial axis directly provide the position and extent of negative minima of curvature, while a traversal of the interior medial axis directly provides a small set of candidate endpoints for part-cuts. The final selection follows a prioritized processing of candidate part-cuts according to a local convexity rule that can incorporate arbitrary salience measures. Neither global optimization nor differentiation is involved. We provide qualitative and quantitative evaluation and comparisons on ground-truth data from psychophysical experiments. With our single computational model, we outperform even an ensemble method on several other competing models.

This paper presents a geological application in a virtual immersive environment : the Immersive Geological Pilot. This application allows a user to interactively build a structured geological model and its associated geological evolution scheme (a directed acyclic graph) by setting the various geological surfaces of the model one after the other in reverse chronological order, beginning with the youngest one. This immersive system runs on a workbench where users work directly within a virtual world. The main goal has been to prove that this geological application can benefit from working immersed and from D interaction techniques. After introducing the geological application on workstation, this paper describes the immersive version and presents the interaction techniques which have been used for selection, manipulation, navigation and application control. The resulting application in a D virtual environment is much more intuitive and ergonomic than the one on workstation.

This paper proposes a unified approach towards computing geometry structures viz. Voronoi diagram, medial axis, Delaunay graph and α-hull of planar closed curves. It initially presents an algorithm for computing the Voronoi diagram of a set of planar freeform closed curves without approximating the curves using points, lines or biarcs. The algorithm starts by computing the minimum antipodal discs (MADs) for all pairs of curves and these MADs are systematically processed to identify all branch points. The key feature of the algorithm is that it computes a branch point without computing any of the bisectors a priori. Local computations of Voronoi segments are then done using the identified pairs of the segments of curves. The theoretical foundation of the algorithm has been first laid for a set of convex curves and then extended to non-convex curves. It has also been shown that the developed algorithm for the Voronoi diagram can also be used to compute related structures such as medial axis, Delaunay graph and α-hull. They have also been addressed without computing Voronoi edges/segments. Results of the implementation have been provided along with a detailed discussion of the algorithm.

This paper proposes a unified approach towards computing geometry structures viz. Voronoi diagram, medial axis, Delaunay graph and α-hull of planar closed curves. It initially presents an algorithm for computing the Voronoi diagram of a set of planar freeform closed curves without approximating the curves using points, lines or biarcs. The algorithm starts by computing the minimum antipodal discs (MADs) for all pairs of curves and these MADs are systematically processed to identify all branch points. The key feature of the algorithm is that it computes a branch point without computing any of the bisectors a priori. Local computations of Voronoi segments are then done using the identified pairs of the segments of curves. The theoretical foundation of the algorithm has been first laid for a set of convex curves and then extended to non-convex curves. It has also been shown that the developed algorithm for the Voronoi diagram can also be used to compute related structures such as medial axis, Delaunay graph and α-hull. They have also been addressed without computing Voronoi edges/segments. Results of the implementation have been provided along with a detailed discussion of the algorithm.

Voronoi diagram (VD) is an extensively studied geometric entity since it has applications in fields such as computer graphics, computer vision, geometric modeling etc. In this paper, an algorithm for computing the VD of a set of planar freeform closed convex curves has been developed, without approximating the curves using points or lines. Algorithms for VD predominantly lie on computing the bisectors, a geometrically complex and a high degree curve even for inputs of low degree. Hence, a lot of processing is required to compute bisector segments that contribute to VD as well as in the computation of branch points. In this paper, it has been shown that computation of a branch point is possible without first computing either the bisector or segments of it. The algorithm uses the minimum antipodal discs (MADs) for all pairs of curves. Three touch discs (TTD) (circles touching three curves) are computed only for a specific set of three curves. Decision criteria for a TTD to become a branch disc (BD, i.e. empty TTD) have been addressed without using explicit curve containment check. Local computations of Voronoi segments are then done. The algorithm gives all branch points via centers of the branch discs along with the segments of curves that will contribute to VD. Results of implementation are provided along with analysis of the algorithm.

Deformable modeling with medial axis representation is a useful means of segmenting and parametrically describing the shape of anatomical structures in medical images. Continuous medial representation (cm-rep) is a "skeleton-first" approach to deformable medial modeling that explicitly parameterizes an object's medial axis and derives the object's boundary algorithmically. Although cm-rep has effectively been used to segment and model a number of anatomical structures with non-branching medial topologies, the framework is challenging to apply to objects with branching medial geometries since branch curves in the medial axis are difficult to parameterize. In this work, we demonstrate the first clinical application of a new "boundary-first" deformable medial modeling paradigm, wherein an object's boundary is explicitly described and constraints are imposed on boundary geometry to preserve the branching configuration of the medial axis during model deformation. This "boundary-first" framework is leveraged to segment and morphologically analyze the aortic valve apparatus in 3D echocardiographic images. Relative to manual tracing, segmentation with deformable medial modeling achieves a mean boundary error of 0.41 ± 0.10 mm (approximately one voxel) in 22 3DE images of normal aortic valves at systole. Deformable medial modeling is additionally demonstrated on pathological cases, including aortic stenosis, Marfan syndrome, and bicuspid aortic valve disease. This study demonstrates a promising approach for quantitative 3DE analysis of aortic valve morphology.

Geometric and neural models of illusory-contour (IC) synthesis currently use only local contour geometry to derive the shape of ICs. Work on the visual representation of shape, by contrast, points to the importance of both contour and surface geometry. We investigated the influence of surfacebased geometric factors on IC shape. The local geometry of inducing-contour pairs was equated in stereoscopic IC displays, and the shape of the enclosed surface was varied by manipulating sign of curvature, cross-axial shape width, and medial-axis geometry. IC shapes were measured using a parametric shape-adjustment task (Experiment 1) and a dot-adjustment task (Experiment 2). Both methods revealed large influences of surface geometry. ICs enclosing locally concave regions were perceived to be systematically more angular than those enclosing locally convex regions. Importantly, the influence of sign of curvature was modulated significantly by shape width and medial-axis geometry: IC shape difference between convex and concave inducers was greater for narrow shapes than wider ones, and greater for shapes with straight axis and symmetric contours (diamond versus bowtie), than those with curved axis and parallel contours (bent tubes). Even at the level of illusory ''contours,'' there is a contribution of region-based geometry which is sensitive to nonlocal shape properties involving medial geometry and part decomposition. Models of IC synthesis must incorporate the role of nonlocal region-based geometric factors in a way that parallels their role in organizing visual shape representation more generally.

In this paper, we show a graph isomorphism between a dual graph of the Delaunay graph of the sampled points and the medial axis of the sampled features. This dual graph captures the fact that two Delaunay triangles share two vertices or an edge. Then, we apply it to the computation of the medial axis of the features selected in an image. The computation of the medial axis of images is of interest in applications such as mapping, climatology, change detection, medicine, etc. This research work provides a way to automate the computation of the medial axis transform of the features of color 2D images. In color images, various features can be distinguished based on their color. The features are thus extracted as object borders, which are sampled in order to compute the medial axis transform. We present also a prototype application for the completely automated or semiautomated processing of (satellite) imagery and scanned maps. Applications include coastline extraction, extraction of fields, clear cuts, clouds, as well as heating or pollution monitoring and dense forest mapping among others.

Areal features are of great importance in applications like shore line mapping, boundary delineation and change detection. This research work is an attempt to automate the process of extracting feature boundaries from satellite imagery. This process is intended to eventually replace manual digitization by computer assisted boundary detection and conversion to a vector layer in a Geographic Information System. Another potential application is to be able to use the extracted linear features in image matching algorithms. In multi-spectral satellite imagery, various features can be distinguished based on their colour. There has been a good amount of work already done as far as boundary detection and skeletonization is concerned, but this research work is different from the previous ones in the way that it uses the Delaunay graph and the Voronoi tessellation to extract boundary and skeletons that are guaranteed to be topologically equivalent to the segmented objects. The features thus extracted as object border can be stored as vector maps in a Geographic Information System after labelling and editing. Here we present a complete methodology of the skeletonization process from satellite imagery using a colour image segmentation algorithm with examples of road networks and hydrographic networks.

We propose an algorithm to find piecewise linear skeletons of handwritten characters by using principal curves. The development of the method was inspired by the apparent similarity between the definition of principal curves (smooth curves which pass through the "middle" of a cloud of points) and the medial axis (smooth curves that go equidistantly from the contours of a character image). The central fitting-and-smoothing step of the algorithm is an extension of the polygonal line algorithm [1, 2] which approximates principal curves of data sets by piecewise linear curves. The polygonal line algorithm is extended to find principal graphs and complemented with two steps specific to the task of skeletonization: an initialization method to capture the approximate topology of the character, and a collection of restructuring operations to improve the structural quality of the skeleton produced by the initialization method. An advantage of our approach over existing methods is that we optimize the skeleton graph by minimizing an intuitive and explicit objective function that captures the two competing criteria of smoothing the skeleton and fitting it closely to the pixels of the character image. We tested the algorithm on isolated handwritten digits and images of continuous handwriting. The results indicate that the proposed algorithm finds a smooth medial axis of the great majority of a wide variety of character templates, and substantially improves the pixelwise skeleton obtained by traditional thinning methods.

In this paper, we present the Delta Medial Axis (DMA), a quasi-linear algorithmic solution addressing several of the main concerns of discrete medial axes (MA) computation. First, its sensitivity to small shape perturbations is counterbalanced by a single parameter (delta), used in a pruning strategy that implicitly takes into account the local topology of the boundary. Second, the discrete nature of images is addressed by approximating the original MA definition in continuous space. Third, to allow real-time performances, a set of algorithmic optimizations is proposed. We compare our approach both qualitatively and quantitatively to recent state of the art solutions, and show that the DMA presents an excellent choice for a wide range of applications. Finally, to allow fast and efficient use of our algorithm, we propose the complete pseudo-code.

After habituation of looking at an object that repeatedly approached and receded through 50 cm along the medial axis, infants of about 18 weeks of age were presented with both the same moving object and one of a different size falling within the same range of visual angles as those projected during habituation. The starting point for movement varied widely from trial to trial to "desensitize" subjects to a range of changing distances and visual angles. Recovery from habituation of looking was greater for the moving object of a different size than for the same object, indicating that the infants perceived an object's invariant size with changing visual angle (i.e., that visual size constancy was operative). A secondary finding was that following habituation to the smaller object there was a markedly greater recovery with the larger object than with the smaller, but following habituation with the larger there was no difference between recovery for the two. These outcomes were attributed to the additive effects of object salience and recovery from habituation. The assistance of G. Coleman and R. McDonell with computing and of H. Tootell with data collection is gratefully acknowledged. Requests for reprints should be sent to R.

In this paper we introduce a second order differential operator in order to partition discrete objects into meaningful parts. One of these parts contains all points of the object's medial axis which is a widely used shape descriptor. Since aliasing is a fundamental problem for methods based on the boundary of discrete objects we use a bilateral filter to reduce the artefacts significantly. As a result of the partitioning and filtering we obtain a lattice point set (a superset of the medial axis) which is robust against noise and aliasing and which is rotationally invariant. The lattice point set can be used for instance to compute the medial axis or similar shape descriptors.

This paper proposes a new algorithm to generate a disconnected, three-dimensional (3D) skeleton and an application of such a skeleton to generate a finite element (FE) mesh sizing function of a solid. The mesh sizing function controls the element size and the gradient, and it is crucial in generating a desired FE mesh. Here, a geometry-based mesh sizing function is generated using a skeleton. A discrete skeleton is generated by propagating a wave from the boundary towards the interior on an octree lattice of an input solid model. As the wave propagates, the distance from the boundary and direction of the wave front are calculated at the lattice-nodes (vertices) of the new front. An approximate Euclidean distance metric is used to calculate the distance traveled by the wave. Skeleton points are generated at the region where the opposing fronts meet. The distance at these skeleton points is used to measure both proximity between geometric entities and feature size, and is utilized to ...