Chapter 19 - The Medial Axis Transform

The medial axis can be viewed as a compact representation for an arbitrary model; it is an essential geometric structure in many applications. A number of practical algorithms for its computation have been aimed at speeding up its computation and at addressing its instabilities. In this paper we propose a new algorithm to compute the medial axis with arbitrary precision. It exhibits several desirable properties not previously combined in a practical and efficient algorithm. First, it allows for a tradeoff between computation time and accuracy, making it well-suited for applications in which an approximation of the medial axis suffices, but computational efficiency is of particular concern. Second, it is output sensitive: the computation complexity of the algorithm does not depend on the size of the representation of a model, but on the size of the representation of the resulting medial axis. Third, the densities of the approximated medial axis points in different areas are adaptive to local free space volumes, based on the assumption that a coarser approximation in wide open area can still suffice the requirements of the applications. We present theoretical results, bounding the error introduced by the approximation process. The algorithm has been implemented and experimental results are presented that illustrate its computational efficiency and robustness.

Medial Axis (MA), also known as Centres of Maximal Disks, is a useful representation of a shape for image description and analysis. MA can be computed on a distance transform, where each point is labelled to its distance to the background. Recent algorithms allow to compute Squared Euclidean Distance Transform (SEDT) in linear time in any dimension. While these algorithms provide exact measures, the only known method to characterize MA on SEDT, using local tests and Look-Up Tables, is limited to 2D and small distance values [5]. We have proposed in [14] an algorithm which computes the look-up table and the neighbourhood to be tested in the case of chamfer distances. In this paper, we adapt our algorithm for SEDT in arbitrary dimension and show that results have completely different properties.

This paper proposes a novel hypergraph skeletal representation for 3D shape based on a formal derivation of the generic structure of its medial axis. By classifying each skeletal point by its order of contact, we shout that generically the medial axis consists of five types of points which are then organized into sheets, curves, and points: (i) sheets (manifolds with boundary) which are the locus of bitangent spheres with regular tangency/sup 1/ A/sub 1//sup 2/. Two types of curves (ii) the intersection curve of three sheets and the locus of centers of tritangent spheres, A/sub 1//sup 3/, and (iii) the boundary of sheets which are the locus of centers of spheres whose radius equals the larger principle curvature, i.e., higher order contact A/sub 3/ points; and two types of points (iv) centers of quad-tangent spheres, A/sub 1//sup 4/, and, (v) centers of spheres with one regular tangency and one higher order tangency, A/sub 1/A/sub 3/ The geometry of the 3D medial axis thus consists of sheets (A/sub 1//sup 2/) bounded by one type of curve (A/sub 3/) on their free end, which corresponds to ridges on the surface, and attached to two other sheets at another type of curves (A/sub 1//sup 3/), which support a generalized cylinder description. The A/sub 3/ curves can only end in A/sub 1/A/sub 3/ points where they must meet an A/sub 1//sup 3/ curve. The A/sub 1//sup 3/ curves can either meet one A/sub 3/ curve or meet three other A/sub 1//sup 3/ curve at an A/sub 1//sup 4/ point. This formal result leads to a compact representation for 3D shape, referred to as the medial axis hypergraph representation consisting of nodes (A/sub 1//sup 4/ and A/sub 1/A/sub 3/ points), links between pairs of nodes (A/sub 1//sup 3/ and A/sub 3/ curves) and hyperlinks between groups of links (A/sub 1//sup 2/ sheets). The description of the local geometry at nodes by itself is sufficient to capture qualitative aspects of shapes, in analogy to 2D. We derive a pointwise reconstruction formula to reconstruct a surface from this medial axis hypergraph. Thus, the hypergraph completely characterizes 3D shape and lays the theoretical foundation for its use in recognition, morphing, design and manipulation of shapes.

This paper proposes a novel hypergraph skeletal representation for 3D shape based on a formal derivation of the generic structure of its medial axis. By classifying each skeletal point by its order of contact, we show that, generically, the medial axis consists of five types of points, which are then organized into sheets, curves, and points: 1) sheets (manifolds with boundary) which are the locus of bitangent spheres with regular tangency A1(2) (Ak(n) notation means n distinct k-fold tangencies of the sphere of contact, as explained in the text); two types of curves, 2) the intersection curve of three sheets and the locus of centers of tritangent spheres, A1(3), and 3) the boundary of sheets, which are the locus of centers of spheres whose radius equals the larger principal curvature, i.e., higher order contact A3 points; and two types of points, 4) centers of quad-tangent spheres, A1(4), and 5) centers of spheres with one regular tangency and one higher order tangency, A1A3. The geometry of the 3D medial axis thus consists of sheets (A1(2)) bounded by one type of curve (A3) on their free end, which corresponds to ridges on the surface, and attached to two other sheets at another type of curve (A1(3)), which support a generalized cylinder description. The A3 curves can only end in A1A3 points where they must meet an A1(3) curve. The A1(3) curves meet together in fours at an A1(4) point. This formal result leads to a compact representation for 3D shape, referred to as the medial axis hypergraph representation consisting of nodes (A1(4) and A1A3 points), links between pairs of nodes (A1(3) and A3 curves) and hyperlinks between groups of links (A1(2) sheets). The description of the local geometry at nodes by itself is sufficient to capture qualitative aspects of shapes, in analogy to 2D. We derive a pointwise reconstruction formula to reconstruct a surface from this medial axis hypergraph together with the radius function. Thus, this information completely characterizes 3D shape and lays the theoretical foundation for its use in recognition, morphing, design, and manipulation of shapes.

Pore network models are useful for studying transport in porous materials in a computationally efficient way. Extraction of networks from volumetric images has evolved over the years, starting with medial axis-based approaches to more recent watershed segmentation. This paper reconsiders the classic medial axis method, which offers several advantages such as speed and topological correctness, and develops a modernized, updated, and improved version. The new method is named Medial Axis Guided Network Extraction Tool (MAGNET). It works by analyzing the skeleton of a porous material to identify pore centers at junctions and endpoints. Additional pore bodies are found on long throats using two different approaches. This work includes an efficient tool for calculating the cross-sectional area of throats with irregular shape by using walkers with an infinite mean-free path to probe the geometry orthogonal to the medial axis at the point of the throat constriction. This extra step was critical for obtaining an equivalent diameter needed to calculate the permeability. Lastly, MAGNET was written with computational efficiency in mind. The skeletonization approach was itself 4.2X faster than the SNOW watershed segmentation for a 10003 image. Additionally, a parallelized skeletonization was applied by processing the image in blocks with sufficient overlap which resulted in a 5.5X speed-up compared to the serial approach. To validate the output, MAGNET was tested on a 4003 voxel image of a Berea sandstone, and the flow and capillary properties of the extracted network were compared to the results from SNOW and the lattice-Boltzmann method. Structural information such as pore and throat size distribution and mercury intrusion curves was compared, and noticeable similarity was achieved. Crucially, the permeability predicted by MAGNET was within 5% of the lattice-Boltzmann prediction on the same image.

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.

Exploiting curvatures to compute the medial axis for domains with smooth boundary

Medial Axis, also known as Centres of Maximal Disks, is a representation of a shape, which is useful for image description and analysis. Chamfer or Weighted Distances, are discrete distances which allow to approximate the Euclidean Distance with integers. Computing medial axis with chamfer distances has been discussed in the literature for some simple cases, mainly in 2D. In this paper we give a method to compute the medial axis for any chamfer distance in 2D and 3D, by local tests using a lookup table. Our algorithm computes very efficiently the lookup tables and, very important, the neighbourhood to be tested.

The medial axis transformation is a means first proposed by Blum to describe a shape. In this paper we present a 0(n log n) algorithm for computing the medial axis of a planar shape represented by an n-edge simple polygon. The algorithm is an improvement over most previously known results interms of both efficiency and exactness and has been implemented in Fortran. Some computer-plotted output of the program are also shown in the paper.

A method towards robust and efficient medial axis transform (MAT) of arbitrary domains using distance solutions is presented. The distance field, d, is calculated by solving the hyperbolic-natured Eikonal (or Level Set) equation. The solution is obtained on Cartesian grids. Both the fast-marching method and fast-sweeping method are used to calculate d. Medial axis point clouds are then extracted based on the distance solution via a simple criteria: the Laplacian or the Hessian determinant of d. These point clouds in 2D-pixel and 3D-voxel space are further thinned to curves and surfaces through binary image thinning algorithms. This results in an overall hybrid approach. 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 3D, 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.

There has been much attention on sophisticated algorithm design to compute geometric arrangements with both time and space efficiency. The issue of robustness and reliability has also been the subject of some interest, although mostly at the level of theory rather than practice and commercial grade implementation. What seems to have received very little attention is the need to prepare the data for successful processing. It is almost universally assumed that the data are valid and well presented and the only real challenge is to come up with a clever way of computing the results with progressively smaller time and space bounds. The aim of this paper is to narrow this gap by focusing entirely on input data anomalies, how to prepare the data for error free computation and how to post process the results for dowstream computing. The medial axis computation, using VRONI (Held, 2001, “VRONI: An Engineering Approach to the Reliable and Efficient Computation of Voronoi Diagram of Points and Line Segments,” Comput. Geom.—Theory Appl., 18, pp. 95–123), is singled out as an example and it is shown that based on how the data are prepared, the results can be vastly different. We argue in this paper that the success of geometric computing depends equally on algorithm design as well as on data processing. VRONI (and most geometric algorithms) does not understand the concept of noise, gaps, or aliasing. It only sees a polygon and generates the medial axis accordingly. It is the job of the applications engineer to prepare the data so that the output is acceptable.

This paper presents a novel multi-step iterative scheme to solve system of nonlinear equations. Because the cost of calculating the Frechet derivative evaluation and its inversion is significant, in order to achieve its high computational efficiency, we have tried to calculate Frechet derivative and its inverse less per cycle by using the proposed multi-step iterative schemes. The basic iterative scheme has a convergence order of four; therefore, repetition of the second step can achieve higher convergence order. In fact, adding a new step to the base iterative scheme each time increases the convergence order by two units. The multi-step iterative schemes have convergence-order 2m, where m is the number of steps of the multi-step iterative schemes. Also, the computational efficiency of the iterative scheme is compared with other available methods. The numerical results presented confirm the theoretical results. A number of nonlinear system of equations associated with the numerical approximation of ordinary, partial, and fractional differential equations are made up and solved.

Context. In 2015, the International Astronomical Union (IAU) passed Resolution B3, which defined a set of nominal conversion constants for stellar and planetary astronomy. Resolution B3 defined a new value of the nominal solar radius (R⊙N = 695 700 km km) that is different from the canonical value used until now (695 990 km). The nominal solar radius is consistent with helioseismic estimates. Recent results obtained from ground-based instruments, balloon flights, or space-based instruments highlight solar radius values that are significantly different. These results are related to the direct measurements of the photospheric solar radius, which are mainly based on the inflection point position methods. The discrepancy between the seismic radius and the photospheric solar radius can be explained by the difference between the height at disk center and the inflection point of the intensity profile on the solar limb. At 535.7 nm (photosphere), there may be a difference of ∼330 km between the two definitions of the solar radius. Aims. The main objective of this work is to present new results of the solar radius in the near-ultraviolet, the visible, and the near-infrared from PICARD space-based and ground-based observations. Simulations show the strong influence of atmosphere effects (refraction and turbulence) on ground-based solar radius determinations and highlight the interest of space-based solar radius determinations, particularly during planet transits (Venus or Mercury), in order to obtain more realistic and accurate measurements. Methods. Solar radius observations during the 2012 Venus transit have been made with the SOlar Diameter Imager and Surface Mapper (SODISM) telescope on board the PICARD spacecraft. We used the transit of Venus as an absolute calibration to determine the solar radius accurately at several wavelengths. Our results are based on the determination of the inflection point position of the solar limb-darkening function (the most common solar radius definition). A realistic uncertainty budget is provided for each solar radius obtained with the PICARD space-based telescope during the 2012 Venus transit. The uncertainty budget considers several sources of error (detection of the centers of Venus and Sun in PICARD images, positions of Sun and Venus from ephemeris (planetary theory), PICARD on-board timing, PICARD spacecraft position, and optical distortion correction from PICARD images). Results. We obtain new values of the solar radius from the PICARD mission at several wavelengths and in different solar atmosphere regions. The PICARD spacecraft with its SODISM telescope was used to measure the radius of the Sun during the Venus transit in 2012. At 535.7 nm, the solar radius is equal to 696 134 ± 261 km (combined standard uncertainty based (ξ) on the uncertainty budget). At 607.1 nm, the solar radius is equal to 696 156 ± 145 km (ξ), and the standard deviation of the solar radius mean value is ±22 km. At 782.2 nm, the solar radius is equal to 696 192 ± 247 km (ξ). The PICARD space-based results as well as PICARD ground-based results show that the solar radius wavelength dependence in the visible and the near-infrared is extremely weak. The differences in inflection point position of the solar radius at 607.1 nm, 782.2 nm, and 1025.0 nm from a reference at 535.7 nm are less than 60 km for the different PICARD measurements.

Average curve of [formula omitted] smooth planar curves

As a typical feature of traditional Chinese timber structures, columns with floating foot joints are prone to rocking under horizontal force. The column rocking behavior plays an important role in resisting lateral loads. This paper proposes modelling strategies for the rocking behavior of traditional timber columns. The proposed model can capture the contact state of the column foot via a series of distributed axial springs and the possible sliding via a horizontal spring. It has a lower computational cost than refined finite element (FE) models and is independent of numerical calibration compared to rotational spring models. The modelling strategies were validated against refined FE models and experimental data available in the literature. The principle behavior of the approach was carefully investigated including contact stiffness, distribution modes, spring numbers, column bending effects, and computational efficiency. Results show that the effective compression height of the column foot can take the value of the column radius to determine spring stiffness. Column bending needs to be considered in the mechanical analysis of column foot joints for slender columns. The case study on the lateral stiffness of traditional timber frames confirms the applicability of the proposed model in the structural analysis of ancient timber buildings.