The medial axis of a geometric shape captures its connectivity. In spite of its inherent instability, it has found applications in a number of areas that deal with shapes. In this survey paper, we focus on results that shed light on this instability and use the new insights to generate simplified and stable modifications of the medial axis.

/pdf/stability-and-computation-of-medial-axes-a-state-of-the-art-4dxv78m8wi.pdf

Stability and Computation of Medial Axes - a State-of-the-Art Report

Definitions and basic properties of the Voronoi diagram generalizations of the Voronoi diagram algorithms for computing Voronoi diagrams poisson Voronoi diagrams spatial interpolation models of spatial processes point pattern analysis locational optimization through Voronoi diagrams.

Spatial tessellations. Concepts and Applications of Voronoi diagrams

We present a simple, efficient, and stable method for computing-with any desired precision-the medial axis of simply connected planar domains. The domain boundaries are assumed to be given as polynomial spline curves. Our approach combines known results from the field of geometric approximation theory with a new algorithm from the field of computational geometry. Challenging steps are (1) the approximation of the boundary spline such that the medial axis is geometrically stable, and (2) the efficient decomposition of the domain into base cases where the medial axis can be computed directly and exactly. We solve these problems via spiral biarc approximation and a randomized divide & conquer algorithm.

/pdf/medial-axis-computation-for-planar-free-form-shapes-3un27wwpe3.pdf

Medial axis computation for planar free-form shapes

Voronoi diagrams are fundamental data structures that have been extensively studied in Computational Geometry. A Voronoi diagram can be defined as the minimization diagram of a finite set of continuous functions. Usually, each of those functions is interpreted as the distance function to an object. The as- sociated Voronoi diagram subdivides the embedding space into regions, each region consisting of the points that are closer to a given object than to the others. We may define many variants of Voronoi diagrams depending on the class of objects, the distance functions and the embedding space. Affine di- agrams, i.e. diagrams whose cells are convex polytopes, are well understood. Their properties can be deduced from the properties of polytopes and they can be constructed efficiently. The situation is very different for Voronoi dia- grams with curved regions. Curved Voronoi diagrams arise in various contexts where the objects are not punctual or the distance is not the Euclidean dis- tance. We survey the main results on curved Voronoi diagrams. We describe in some detail two general mechanisms to obtain effective algorithms for some classes of curved Voronoi diagrams. The first one consists in linearizing the diagram and applies, in particular, to diagrams whose bisectors are algebraic hypersurfaces. The second one is a randomized incremental paradigm that can construct affine and several planar non-affine diagrams. We finally introduce the concept of Medial Axis which generalizes the concept of Voronoi diagram to infinite sets. Interestingly, it is possible to efficiently construct a certified approximation of the medial axis of a bounded set from the Voronoi diagram of a sample of points on the boundary of the set.

/pdf/curved-voronoi-diagrams-4ggvadan0t.pdf

Curved Voronoi diagrams

A key challenge underlying theories of vision is how the spatially restricted, retinotopically represented feature analysis can be integrated to form abstract, coordinate-free object models. A resolution likely depends on the use of intermediate-level representations which can on the one hand be populated by local features and on the other hand be used as atomic units underlying the formation of, and interaction with, object hypotheses. The precise structure of this intermediate representation derives from the varied requirements of a range of visual tasks which motivate a significant role for incorporating a geometry of visual form. The need to integrate input from features capturing surface properties such as texture, shading, motion, color, etc., as well as from features capturing surface discontinuities such as silhouettes, T-junctions, etc., implies a geometry which captures both regional and boundary aspects. Curves, as a geometric model of boundaries, have been extensively used as an intermediate representation in computational, perceptual, and physiological studies, while the use of the medial axis (MA) has been popular mainly in computer vision as a geometric region-based model of the interior of closed boundaries. We extend the traditional model of the MA to represent images, where each MA segment represents a region of the image which we call a visual fragment. We present a unified theory of perceptual grouping and object recognition where through various sequences of transformations of the MA representation, visual fragments are grouped in various configurations to form object hypotheses, and are related to stored models. The mechanisms underlying both the computation and the transformation of the MA is a lateral wave propagation model. Recent psychophysical experiments depicting contrast sensitivity map peaks at the medial axes of stimuli, and experiments on perceptual filling-in, and brightness induction and modulation, are consistent with both the use of an MA representation and a propagation-based scheme. Also, recent neurophysiological recordings in V1 correlate with the MA hypothesis and a horizontal propagation scheme. This evidence supports a geometric computational paradigm for processing sensory data where both dynamic in-plane propagation and feedforward-feedback connections play an integral role.

On the role of medial geometry in human vision.

http://directory.umm.ac.id/Data%20Elmu/jurnal/J-a/Journal%20of%20Computational%20And%20Applied%20Mathematics/Vol102.Issue1.1999/2666.pdf

Voronoi diagram and medial axis algorithm for planar domains with curved boundaries I. Theoretical foundations

http://directory.umm.ac.id/Data%20Elmu/jurnal/J-a/Journal%20of%20Computational%20And%20Applied%20Mathematics/Vol102.Issue2.1999/2678.pdf

Voronoi diagram and medial axis algorithm for planar domains with curved boundaries — II: Detailed algorithm description

The bisector of two plane curve segments (other than lines and circles) has, in general, no simple — i.e., rational — parameterization, and must therefore be approximated by the interpolation of discrete data. A procedure for computing ordered sequences of point/tangent/curvature data along the bisectors of polynomial or rational plane curves is described, with special emphasis on (i) the identification of singularities (tangent–discontinuities) of the bisector; (ii) capturing the exact rational form of those portions of the bisector with a terminal footpoint on one curve; and (iii) geometrical criteria the characterize extrema of the distance error for interpolants to the discretely–sample data. G1 piecewise– parabolic and G2 piecewise–cubic approximations (with O(h4) and O(h6) convergence) are described which, used in adaptive schemes governed by the exact error measure, can be made to satisfy any prescribed geometrical tolerance.

Specified–Precision Computation of Curve/Curve Bisectors

Degenerate point/curve and curve/curve bisectors arising in medial axis computations for planar domains with curved boundaries

The Voronoi diagram of the boundary of a planar domain is a unique partition of the plane, such that every point within a region formed by the partition is at least as close to one particular boundary segment as it is to all the other boundary segments. The medial or symmetric axis of that domain is the locus of centers of maximum-radius circles (touching the boundary in two points) that can be inscribed within the domain. The edges of the Voronoi diagram/medial axis are either point/curve or curve/curve bisector loci.
The a priori computations of Voronoi diagrams and medial axes of planar domains with curved boundaries are useful in a variety of application contexts. Several algorithms, for Voronoi diagram and medial axis constructions, are currently available in the computational geometry literature. However, most of these algorithms are incapable of operating on planar domains with curved boundaries.
Algorithms for constructing both Voronoi diagrams and medial axes of domains bounded by polynomial/rational curve segments are developed in this dissertation. These employ algorithms for basic curve/curve bisector constructions, which are also described. The Voronoi diagram of a domain is constructed by an incremental merging scheme, wherein one boundary segment is included at a time, and the existing Voronoi diagram is "updated" to account for the new boundary segment. The medial axis algorithm, which takes the interior Voronoi diagram as input, removes those edges of the interior Voronoi diagram whose points have a single footprint on the boundary, and then adds the self-bisectors of individual boundary segments. The algorithms are designed to (i) capture all rational segments of the Voronoi diagram and medial axis explicitly; (ii) approximate all other edges, that are high degree algebraic curves having no rational representation, as sequences of parabolic/cubic Bezier curve interpolants, within the prescribed geometric error; and (iii) to remain faithful, within the specified tolerance, to the true topology of the Voronoi diagram/medial axis. Features (i) and (ii) are currently unavailable in any of the existing algorithms.

Rajesh Ramamurthy

Papers

Voronoi diagram and medial axis algorithm for planar domains with curved boundaries I. Theoretical foundations

Voronoi diagram and medial axis algorithm for planar domains with curved boundaries — II: Detailed algorithm description

Specified–Precision Computation of Curve/Curve Bisectors

Degenerate point/curve and curve/curve bisectors arising in medial axis computations for planar domains with curved boundaries

Voronoi diagrams and medial axes of planar domains with curved boundaries