Curve Skeleton Research Articles

Skeletonization of volumetric vascular images - Distance information utilized for visualization

This paper deals with two techniques to represent relevant information from volumetric vascular im- ages in a more compact format. The images are obtained with contrast-enhanced magnetic resonance angiography (MRA). After segmentation of the vessels, the curve skeleton is extracted by an algorithm based on the distance transformation. The algorithm first reduces the original object to a surface skeleton and then to a curve skeleton, after which "pruning" can be performed to remove irrelevant small branches. Applying this procedure to MRA data from the pelvic arteries resulted in a good description of the tree structure of the vessels with a much smaller number of voxels. To detect stenoses, 2D projections such as maximum intensity projection (MIP) are usually employed, but these often fail to demonstrate a stenosis if the projection angle is not suitably chosen. A new presentation method surrounds each voxel in the distance-labeled curve skeleton of the segmented vascular tree with a ball whose radius represents the minimum vessel radius at that level. Experiments with synthetic data indicate that stenoses invisible in an ordinary projection may be seen with this technique. It is concluded that the distance-labelled curve skeleton seems to be useful for visualizing variations in vessel calibre and in the future possibly also for quantification of arterial stenoses.

Convolution Surfaces for Line Skeletons with Polynomial Weight Distributions

Convolution surfaces generalize point-based implicit surfaces to incorporate higher-dimensional skeletal elements; line segments can be considered the most fundamental skeletal elements since they can approximate curve skeletons. Existing analytical models for line-segment skeletons assume uniform weight distributions, and thus they can produce only constant-radius convolution surfaces. This paper presents an analytical solution for convolving line-segment skeletons with a variable kernel modulated by a polynomial function, allowing generalized cylindrical convolution surfaces to be modeled conveniently. Its computational requirement is competitive with that of uniform weight distribution. The source code of the field computation is available online.

A Parallel 3D 12-Subiteration Thinning Algorithm

Thinning on binary images is an iterative layer by layer erosion until only the “skeletons” of the objects are left. This paper presents an efficient parallel thinning algorithm which produces either curve skeletons or surface skeletons from 3D binary objects. It is important that a curve skeleton is extracted directly (i.e., without creating a surface skeleton). The strategy which is used is called directional: each iteration step is composed of a number of subiterations each of which can be executed in parallel. One iteration step of the proposed algorithm contains 12 subiterations instead of the usual six. The algorithm makes easy implementation possible, since deletable points are given by 3×3×3 matching templates. The topological correctness for (26, 6) binary pictures is proved.

Computing skeletons in three dimensions

Skeletonization will probably become as valuable a tool for shape analysis in 3D, as it is in 2D. We present a topology preserving 3D skeletonization method which computes both surface and curve skeletons whose voxels are labelled with the D 6 distance to the original background. The surface skeleton preserves all shape information, so (close to) complete recovery of the object is possible. The curve skeleton preserves the general geometry of the object. No complex computations, large sets of masks, or extra memory are used, which make implementations efficient. Resulting skeletons for geometric objects in a number of 2 Mbyte images are shown as examples.

Factoring Rational Polynomials over the Complex Numbers

NC algorithms are given for determining the number and degrees of the factors, irreducible over the complex numbers ${\bf C}$, of a multivariate polynomial with rational coefficients and for approximating each irreducible factor. NC is the class of functions computable by logspace-uniform boolean circuits of polynomial size and polylogarithmic depth. The measures of size of the input polynomial are its degree, coefficient length, number of variables (d, c, and n, respectively). If n is fixed, we give a deterministic NC algorithm. If the number of variables is not fixed, we give a random (Monte-Carlo) NC algorithm in these input measures to find the number and degree of each irreducible factor. After reducing to the two-variable, square-free case, we apply the classical algebraic geometry fact that the absolute irreducible factors of $(P(z_1 ,z_2 ) = 0)$ correspond to the connected components of the real surface (or complex curve) $P(z_1 ,z_2 ) = 0$ minus its singular points. In finding the number of connected components of the surface $P = 0$, the surface is projected to the the $z_2 $-plane. The singular points of $P(z_1 ,z_2 )$ lie over the projection’s critical values. The inverse image of a grid isolating the critical values in the $z_2 $-plane lifts to a one-dimensional real curve skeleton on the surface $(P = 0)$ whose number of connected components is precisely the number of connected components of $P = 0$ minus its singular points. The connectivity of this curve skeleton is constructed symbolically using Sturm sequences associated with the various polynomials defining these maps. Given the number of irreducible factors and their degrees, the actual factors can be reconstructed using the recent result of Neff [Proceedings of the 31st Annual Symposium on Foundations of Computer Science, pp. 152–162] on finding zeros of one-variable polynomials in NC.