Medial axis Research Papers - Academia.edu (original) (raw)

A general algorithm for computing Euclidean skeletons of 2D and 3D data sets in linear time is presented. These skeletons are defined in terms of a new concept, called the integer medial axis (IMA) transform. We prove a number of... more

A general algorithm for computing Euclidean skeletons of 2D and 3D data sets in linear time is presented. These skeletons are defined in terms of a new concept, called the integer medial axis (IMA) transform. We prove a number of fundamental properties of the IMA skeleton, and compare these with properties of the CMD (centers of maximal disks) skeleton. Several pruning methods for IMA skeletons are introduced (constant, linear and square-root pruning) and their properties studied. The algorithm for computing the IMA skeleton is based upon the feature transform, using a modification of a linear-time algorithm for Euclidean distance transforms. The skeletonization algorithm has a time complexity which is linear in the number of input points, and can be easily parallelized. We present experimental results for several data sets, looking at skeleton quality, memory usage and computation time, both for 2D images and 3D volumes.

The investigation of 3D euclidean symmetry sets (SS) and medial axis is an important area, due in particular to their various important applications. The pre-symmetry set of a surface M in 3-space (resp. smooth closed curve in 2D) is the... more

The investigation of 3D euclidean symmetry sets (SS) and medial axis is an important area, due in particular to their various important applications. The pre-symmetry set of a surface M in 3-space (resp. smooth closed curve in 2D) is the set of pairs of points which contribute to the symmetry set, that is, the closure of the set of pairs of distinct points p and q in M, for which there exists a sphere (resp. a circle) tangent to M at p and at q. The aim of this paper is to address problems related to the smoothness and the singularities of the pre-symmetry sets of 3D shapes. We show that the pre-symmetry set of a smooth surface in 3-space has locally the structure of the graph of a function from R^2 to R^2, in many cases of interest.

For compact regions Omega in R^3 with generic smooth boundary B, we consider geometric properties of Omega which lie midway between their topology and geometry and can be summarized by the term "geometric complexity". The... more

For compact regions Omega in R^3 with generic smooth boundary B, we consider geometric properties of Omega which lie midway between their topology and geometry and can be summarized by the term "geometric complexity". The "geometric complexity" of Omega is captured by its Blum medial axis M, which is a Whitney stratified set whose local structure at each point is given by specific standard local types. We classify the geometric complexity by giving a structure theorem for the Blum medial axis M. We do so by first giving an algorithm for decomposing M using the local types into "irreducible components" and then representing each medial component as obtained by attaching surfaces with boundaries to 4--valent graphs. The two stages are described by a two level extended graph structure. The top level describes a simplified form of the attaching of the irreducible medial components to each other, and the second level extended graph structure for each irreduc...

Recent technology advancements on X-ray computed tomography (X-ray CT) offer a nondestructive approach to extract complex three-dimensional geometries with details as small as a few microns in size. This new technology opens the door to... more

Recent technology advancements on X-ray computed tomography (X-ray CT) offer a nondestructive approach to extract complex three-dimensional geometries with details as small as a few microns in size. This new technology opens the door to study the interplay between microscopic properties (e.g., porosity) and macroscopic fluid transport properties (e.g., permeability). To take full advantage of X-ray CT, we introduce a multiscale framework that relates macroscopic fluid transport behavior not only to porosity but also to other important microstructural attributes, such as occluded/connected porosity and geometrical tortuosity, which are extracted using new computational techniques from digital images of porous materials. In particular, we introduce level set methods, and concepts from graph theory, to determine the geometrical tortuosity and connected porosity, while using a Lattice Boltzmann/Finite Element scheme to obtain homogenized effective permeability at specimen-scale. We showcase the applicability and efficiency of this multiscale framework by two examples, one using a synthetic array and another using a sample of natural sandstone with complex pore structure.

A general algorithm for computing Euclidean skeletons of 3D data sets in linear time is presented. These skeletons are defined in terms of a new concept, called the integer medial axis (IMA) transform. The algorithm is based upon the... more

A general algorithm for computing Euclidean skeletons of 3D data sets in linear time is presented. These skeletons are defined in terms of a new concept, called the integer medial axis (IMA) transform. The algorithm is based upon the computation of 3D feature transforms, using a modification of an algorithm for Euclidean distance transforms. The skeletonization algorithm has a time complexity which is linear in the amount of voxels, and can be easily parallelized. The relation of the IMA skeleton to the usual definition in terms of centers of maximal disks is discussed.

This chapter presents an overview on contributions of the Welfenlab to GRK 615. Those contributions partial to computational differential geometry include computations of geodesic medial axis, cut locus, geodesic Voronoi diagrams,... more

This chapter presents an overview on contributions of the Welfenlab to GRK 615. Those contributions partial to computational differential geometry include computations of geodesic medial axis, cut locus, geodesic Voronoi diagrams, (“shortest”) geodesics joining two given points, “focal sets and conjugate loci” in Riemannian manifolds and the application of the medial axis on metal forming simulation. The chapter includes also the computation of Laplace spectra of surfaces, solids and images and the application of those Laplace spectra to recognize the respective objects in large collections of surfaces, solids and images. Beyond that this article touches also on the origin of the afore-mentioned works including research done at the Welfenlab as well as works that can be traced back to the graduate studies of the first author.

In contrast to the extensively researched modeling of plant architecture, the modeling of plant organs largely remains an open problem. In this paper, we propose a method for modeling lobed leaves. This method extends the concept of... more

In contrast to the extensively researched modeling of plant architecture, the modeling of plant organs largely remains an open problem. In this paper, we propose a method for modeling lobed leaves. This method extends the concept of sweeps to branched skeletons. The input of the model is a 2D leaf silhouette, which can be defined interactively or derived from a scanned leaf image. The algorithm computes the skeleton (medial axis) of the leaf and approximates it using spline curves interconnected into a branching structure (sticky splines). The leaf surface is then constructed by sweeping a generating curve along these splines. The orientation of the generating curve is adjusted to properly capture the shape of the leaf blade near the extremities and branching points of the skeleton, and to avoid selfintersections of the surface. The leaf model can be interactively modified by editing the shape of the silhouette and the skeleton. It can be further manipulated in 3D using functions th...

Modelling and reconstruction of tubular objects is a known problem in computer graphics. For computer aided surgical planning the constructed geometrical models need to be consistent and compact at the same time, which known approaches... more

Modelling and reconstruction of tubular objects is a known problem in computer graphics. For computer aided surgical planning the constructed geometrical models need to be consistent and compact at the same time, which known approaches cannot guarantee. In this paper we present a new method for generating compact, topologically consistent, 2-manifold surfaces of branching tubular objects using a two-stage approach. The proposed method is based on connection of polygonal cross-sections along the medial axis and subsequent renemen t. Higher order furcations can be handled correctly.