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.Comment: GMP14 (Geometric Modeling and Processing

AMAT: Medial Axis Transform for Natural Images

We introduce Appearance-MAT (AMAT), a generalization of the medial axis transform for natural images, that is framed as a weighted geometric set cover problem. We make the following contributions: i) we extend previous medial point detection methods for color images, by associating each medial point with a local scale; ii) inspired by the invertibility property of the binary MAT, we also associate each medial point with a local encoding that allows us to invert the AMAT, reconstructing the input image; iii) we describe a clustering scheme that takes advantage of the additional scale and appearance information to group individual points into medial branches, providing a shape decomposition of the underlying image regions. In our experiments, we show state-of-the-art performance in medial point detection on Berkeley Medial AXes (BMAX500), a new dataset of medial axes based on the BSDS500 database, and good generalization on the SK506 and WH-SYMMAX datasets. We also measure the quality of reconstructed images from BMAX500, obtained by inverting their computed AMAT. Our approach delivers significantly better reconstruction quality with respect to three baselines, using just 10% of the image pixels. Our code and annotations are available at https://github.com/tsogkas/amat .Comment: 10 pages (including references), 5 figures, accepted at ICCV 201

A Scale-Space Medialness Transform Based on Boundary Concordance Voting

The Concordance-based Medial Axis Transform (CMAT) presented in this paper is a multiscale medial axis (MMA) algorithm that computes the medial response from grey-level boundary measures. This non-linear operator responds only to symmetric structures, overcoming the limitations of linear medial operators which create “side-lobe” responses for symmetric structures and respond to edge structures. In addition, the spatial localisation of the medial axis and the identification of object width is improved in the CMAT algorithm compared with linear algorithms. The robustness of linear medial operators to noise is preserved in our algorithm. The effectiveness of the CMAT is accredited to the concordance property described in this paper. We demonstrate the performance of this method with test figures used by other authors and medical images that are relatively complex in structure. In these complex images the benefit of the improved response of our non-linear operator is clearly visible

Using heterogeneity for obtaining the Medial Axis Transform of natural images

From Blum's Medial Axis Transform, that obtains the skeleton of binary images by joining the centers of all maximal inscribed disks in the shape, the AMAT was developed. The Appearance-MAT is a method that aims to compute the medial axes for natural images, as well as an appearance-based reconstruction. However, when considering the appearance, it deals only with color, and requires image smoothing to obtain a better result. As an approach to solving this problem, while abandoning the goal of reconstruction, we present and evaluate ten different methods for obtaining the medial axis transform, all based on the concept of heterogeneity. The idea behind this ensemble of methods is that in any natural image, any maximal inscribed disk should be homogeneous both in color and texture, while the disks that are not completely contained in an object should exhibit a much higher heterogeneity.Outgoin

Photo-consistent surface reconstruction from noisy point clouds

International audienceExisting algorithms for surface reconstruction from point sets are defeated by moderate amounts of noise and outliers, which makes them unapplicable to point clouds originating from multi-view image data. In this paper, we present a novel method which incorporates the input images in the surface reconstruction process for a better accuracy and robustness. Our approach is based on the medial axis transform of the scene, which our algorithm estimates through a global photo-consistency optimization by simulated annealing. A faithful polyhedral representation of the scene is then obtained by inversion of the medial axis transform

Complexity as a Sclae-Space for the Medial Axis Transform

The medial axis skeleton is a thin line graph that preserves the topology of a region. The skeleton has often been cited as a useful representation for shape description, region interpretation, and object recognition. Unfortunately, the computation of the skeleton is extremely sensitive to variations in the bounding contour. In this paper, we describe a robust method for computing the medial axis skeleton across a variety of scales. The resulting scale-space is parametric with the complexity of the skeleton, where the complexity is defined as the number of branches in the skeleton

Compensated convexity, multiscale medial axis maps and sharp regularity of the squared-distance function

In this paper we introduce a new stable mathematical model for locating and measuring the medial axis of geometric objects, called the quadratic multiscale medial axis map of scale λ, and provide a sharp regularity result for the squared-distance function to any closed nonempty subset K of Rn. Our results exploit properties of the function Clλ (dist2(・; K)) obtained by applying the quadratic lower compensated convex transform of parameter λ [K. Zhang, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 25 (2008), pp. 743–771] to dist2(・; K), the Euclidean squared-distance function to K. Using a quantitative estimate for the tight approximation of dist2(・; K) by Clλ (dist2(・; K)), we prove the C1,1-regularity of dist2(・; K) outside a neighborhood of the closure of the medial axis MK of K, which can be viewed as a weak Lusin-type theorem for dist2(・; K), and give an asymptotic expansion formula for Clλ (dist2(・; K)) in terms of the scaled squared-distance transform to the set and to the convex hull of the set of points that realize the minimum distance to K. The multiscale medial axis map, denoted by Mλ(・; K), is a family of nonnegative functions, parametrized by λ > 0, whose limit as λ→∞exists and is called the multiscale medial axis landscape map, M∞(・; K). We show that M∞(・; K) is strictly positive on the medial axis MK and zero elsewhere. We give conditions that ensure Mλ(・; K) keeps a constant height along the parts of MK generated by two-point subsets with the value of the height dependent on the scale of the distance between the generating points, thus providing a hierarchy of heights (hence, the word “multiscale”) between different parts of MK that enables subsets of MK to be selected by simple thresholding. Asymptotically, further understanding of the multiscale effect is provided by our exact representation of M∞(・; K). Moreover, given a compact subset K of Rn, while it is well known that MK is not Hausdorff stable, we prove that in contrast, Mλ(・; K) is stable under the Hausdorff distance, and deduce implications for the localization of the stable parts of MK. Explicitly calculated prototype examples of medial axis maps are also presented and used to illustrate the theoretical findings