Computing a compact spline representation of the medial axis transform of a 2D shape

We present a full pipeline for computing the medial axis transform of an arbitrary 2D shape. The instability of the medial axis transform is overcome by a pruning algorithm guided by a user-defined Hausdorff distance threshold. The stable medial axis transform is then approximated by spline curves in 3D to produce a smooth and compact representation. These spline curves are computed by minimizing the approximation error between the input shape and the shape represented by the medial axis transform. Our results on various 2D shapes suggest that our method is practical and effective, and yields faithful and compact representations of medial axis transforms of 2D shapes.     

Shape representation using a generalized potential field model

This paper is concerned with efficient derivation of the medial axis transform of a 2D polygonal region. Instead of using the shortest distance to the region border, a potential field model is used for computational efficiency. The region border is assumed to be charged and the valleys of the resulting potential field are used to estimate the axes for the medial axis transform. The potential valleys are found by following the force field, thus, avoiding 2D search. The potential field is computed in closed form using equations of the border segments. The simple Newtonian potential is shown to be inadequate for this purpose. A higher order potential is defined which decays faster with distance than the inverse of distance. It is shown that as the potential order becomes arbitrarily large, the axes approach those computed using the shortest distance to the border. Algorithms are given for the computation of axes, which can run in linear parallel time for part of the axes having initial guesses. Experimental results are presented for a number of examples     

Robust skeletonization using the discrete λ-medial axis

Medial axes and skeletons are notoriously sensitive to contour irregularities. This lack of stability is a serious problem for applications in e.g. shape analysis and recognition. In 2005, Chazal and Lieutier introduced the λ-medial axis as a new concept for computing the medial axis of a shape subject to single parameter filtering. The λ-medial axis is stable under small shape perturbations, as proved by these authors. In this article, a discrete λ-medial axis (DLMA) is introduced and compared with the recently introduced integer medial axis (GIMA). We show that DLMA provides measurably better results than GIMA, with regard to stability and sensibility to rotations. We give efficient algorithms to compute the DLMA, and we also introduce a variant of the DLMA which may be computed in linear-time.     

平面形体的中线提取及其应用

本文介绍了平面中线的思想来源及中线提取的基本原理,提出了一种解决平面中线提取的新方法,探讨了中线提取在有限元网格剖分中的应用。本文最后给出了平面多边形中线提取及其应用的实例。     

An efficient, accurate approach to medial axis transforms of pockets with closed free-form boundaries

Medial axis transform of a pocket with free-form closed boundaries is a completed, compact representation of the pocket geometric shape and topology. It is very useful to multiple cutters selection and their tool paths generation for CNC machining of complex pockets. In the past decades, much research has been successfully conducted on the topic of finding the medial axis of a shape domain bounded with a polygon or simple geometries, e.g., lines and circles. Currently, more pockets with free-form boundaries are adopted in mechanical parts; however, the prior medial axis generation methods cannot handle this type of pockets well, resulting in long computation time and low medial axis accuracy. To address this problem, an efficient, accurate approach to calculating the medial axis transforms of these pockets is proposed in this work. An original optimization model of bisectors is established, and a new optimization method—the hybrid global optimization method—is developed to efficiently and accurately solve the optimization model of bisectors. The new optimization model and solver have been applied to many examples, and the testing results have demonstrated the advantages of this innovative approach over the prior medial axis methods. It can be an effective solution to the medial axis transforms of complex pockets.     

Stochastic jump-diffusion process for computing medial axes inMarkov random fields

Proposes a statistical framework for computing medial axes of 2D shapes. In the paper, the computation of medial axes is posed as a statistical inference problem not as a mathematical transform. The paper contributes to three aspects in computing medial axes. 1) Prior knowledge is adopted for axes and junctions so that axes around junctions are regularized. 2) Multiple interpretations of axes are possible, each being assigned a probability. 3) A stochastic jump-diffusion process is proposed for estimating both axes and junctions in Markov random fields. We argue that the stochastic algorithm for computing medial axes is compatible with existing algorithms for image segmentation, such as region growing, snake, and region competition. Thus, our method provides a new direction for computing medial axes from texture images. Experiments are demonstrated on both synthetic and real 2D shapes     

Medial Axes and Mean Curvature Motion II: Singularities

What happens to the medial axis of a curve that evolves through MCM (Mean Curvature Motion)? We explore some theoretical results regarding properties of both medial axes and curvature motions. Specifically, using singularity theory, we present all possible topological transitions of a symmetry set (of which the medial axis is a subset) whose originating curve undergoes MCM. All calculations are presented in a clear and organized fashion and are easily generalized for other front motions. A companion article deals with non-singular points of the medial axis through direct calculations.     

Computing the medial axis transform in parallel with eight scanoperations

The main result of this paper shows that the block-based digital medial axis transform can be computed in parallel by a constant number of calls to scan (parallel prefix) operations. This gives time- and/or work-optimal parallel implementations for the distance-based and the block-based medial axis transform in a wide variety of parallel architectures. Since only eight scan operations plus a dozen local operations are performed, the algorithm is very easy to program and use. The originality of our approach is the use of the notion of a derived grid and the oversampling of the image in order to reduce the computation of the block-based medial axis transform in the original grid to the much easier task of computing the distance based medial axis transform of the oversampling of the image on the derived grid     

基于双法线跟踪的形状中轴并行提取算法

为准确而高效地提取出形状的中轴,提出一种利用双法线跟踪算法来并行计算形状中轴的方法。通过离散化将形状的边界离散为由若干样本点连接成的多边形,分别对样本点以及样本点连接成的边界边进行两次的法线跟踪,通过多次的迭代与并行计算后,得到所有样本点对应的中轴点,根据样本点的拓扑联通性连接相应中轴点,生成形状的中轴。通过多次实验,该方法可以快速准确得到形状的中轴,验证了其精确性和高效性。     

线性四元树中轴变换

骨架和中轴变换概念运用于线性四元树,定义线性四元树中轴变换为具有一组棋盘距离值的线性四元树骨架.线性四元树中轴变换提供一种非常紧凑的区域表示法,它导致区域分割成边长为2的幂之和的最大正方形集合.提出两种算法计算一给定线性四元树的线性四元树中轴变换.最坏情况下它们的时间复杂性是O(n~2),其中n为线性四元树中四分形的数目.     

模糊及强噪声条件下中心线的提取

提出一种“重心预取多尺度求精”的中心线提取算法,即利用重心估算大致的中心线,并在线上的关键点附近,特别是分叉处进行多尺度求精,从三维图象中提取树状分支物体的中心线,该算法综合了重心法估算的简单性和尺度空间分析法的准确性等特点,特别适合在边界模糊、强噪声条件下的应用。该算法已成功地应用在医学ＣＴ心脏三维图象的可视化中,文中同时说明了如何从中心线生成曲截面展开图,首次实现了分支血管的曲截面显示。     

Computation of medial axis and offset curves of curved boundaries in planar domain

In this paper, we begin our research from the generating theory of the medial axis. The normal equidistant mapping relationships between two boundaries and its medial axis have been proposed based on the moving Frenet frames and Cesaro’s approach of the differential geometry. Two pairs of adjoint curves have been formed and the geometrical model of the medial axis transform of the planar domains with curved boundaries has been established. The relations of position mapping, scale transform and differential invariants between the curved boundaries and the medial axis have been investigated. Based on this model, a tracing algorithm for the computation of the medial axis has been generated. In order to get the accurate medial axis and branch points, a Two_Tangent_Points_Circle algorithm and a Three_Tangent_Points_Circle algorithm have been generated, which use the results of the tracing algorithm as the initial values to make the iterative process effective. These algorithms can be used for the computation of the medial axis effectively and accurately. Based on the medial axis transform and the envelope theory, the trimmed offset curves of curved boundaries have been investigated. Several numerical examples are given at the end of the paper.     

LSMAT Least Squares Medial Axis Transform

The medial axis transform has applications in numerous fields including visualization, computer graphics, and computer vision. Unfortunately, traditional medial axis transformations are usually brittle in the presence of outliers, perturbations and/or noise along the boundary of objects. To overcome this limitation, we introduce a new formulation of the medial axis transform which is naturally robust in the presence of these artefacts. Unlike previous work which has approached the medial axis from a computational geometry angle, we consider it from a numerical optimization perspective. In this work, we follow the definition of the medial axis transform as ‘the set of maximally inscribed spheres’. We show how this definition can be formulated as a least squares relaxation where the transform is obtained by minimizing a continuous optimization problem. The proposed approach is inherently parallelizable by performing independent optimization of each sphere using Gauss–Newton, and its least‐squares form allows it to be significantly more robust compared to traditional computational geometry approaches. Extensive experiments on 2D and 3D objects demonstrate that our method provides superior results to the state of the art on both synthetic and real‐data.     

On the intrinsic reconstruction of shape from its symmetries

The main question we address is: What is the minimal information required to generate closed, nonintersecting planar boundaries? For this paper, we restrict "shape" to this meaning. More precisely, we examine whether the medial axis, together with dynamics, can serve as a language to design shapes and to effect shape changes. We represent the medial axis together with a direction of flow along the axis as the shock graph and examine the reconstruction of shape along each of the three types of medial axis points, A/sub 1//sup 2/, A/sub 1//sup 3/, A/sub 3/, and the associated six types of shock points. First, we show that the tangent and curvature of the medial axis and the speed and acceleration of the shock with respect to time of propagation are sufficient to determine the boundary tangent and curvature at corresponding points of the boundary. This implies that a rather coarse sampling of the symmetry axis, its tangent, curvature, speed, and acceleration is sufficient to regenerate accurately a local neighborhood of shape at regular axis points (A/sub 1//sup 2/). Second, we examine the reconstruction of shape at branch points (A/sub 1//sup 3/) where three regular branches are joined. We show that the three pairs of geometry (that is, curvature) and dynamics (that is, acceleration) must satisfy certain constraints. Finally, we derive similar results for the end points of shock branches (A/sub 3/ points). These formulas completely specify the local reconstruction of a shape from its shock-graph or medial axis and the conditions required to form a coherent shape from the medial axis.     

Hyperbolic Hausdorff Distance for Medial Axis Transform

Although the Hausdorff distance is a popular device to measure the differences between sets, it is not natural for some specific classes of sets, especially for the medial axis transform which is defined as the set of all pairs of the centers and the radii of the maximal balls contained in another set. In spite of its many advantages and possible applications, the medial axis transform has one great weakness, namely its instability under the Hausdorff distance when the boundary of the original set is perturbed. Though many attempts have been made for the resolution of this phenomenon, most of them are heuristic in nature and lack precise error analysis. In this paper, we show that this instability can be remedied by introducing a new metric called the hyperbolic Hausdorff distance, which is most natural for measuring the differences between medial axis transforms. Using the hyperbolic Hausdorff distance, we obtain error bounds, which make the operation of medial axis transform almost an isometry. By various examples, we also show that the bounds obtained are sharp. In doing so, we show that bounding both the Hausdorff distance between domains and the Hausdorff distance between their boundaries is necessary and sufficient for bounding the hyperbolic Hausdorff distance between their medial axis transforms. These results drastically improve the previous results and open a new way to practically control the Hausdorff distance error of the domains under its medial axis transform error, and vice versa.     

Topology Preserving Simplification of Medial Axes in 3D Models

We propose an efficient method for topology‐preserving simplification of medial axes of 3D models. Existing methods either cannot preserve the topology during medial axes simplification or have the problem of being geometrically inaccurate or computationally expensive. To tackle these issues, we restrict our topology‐checking to the areas around the topological holes to avoid unnecessary checks in other areas. Our algorithm can keep high precision even when the medial axis is simplified to be in very few vertices. Furthermore, we parallelize the medial axes simplification procedure to enhance the performance significantly. Experimental results show that our method can preserve the topology with highly efficient performance, much superior to the existing methods in terms of topology preservation, accuracy and performance.