Tristan Roussillon

Digital circles, spheres and hyperspheres: From morphological models to analytical characterizations and topological properties

In this paper we provide an analytical description of various classes of digital circles, spheres and in some cases hyperspheres, defined in a morphological framework. The topological properties of these objects, especially the separation of the digital space, are discussed according to the shape of the structuring element. The proposed framework is generic enough so that it encompasses most of the digital circle definitions that appear in the literature and extends them to dimension 3 and sometimes dimension n.

Analytical description of digital circles

In this paper we propose an analytical description of different kinds of digital circles that appear in the literature and especially in digital circle recognition algorithms.

On three constrained versions of the digital circular arc recognition problem

In this paper, the problem of digital circular arcs recognition is investigated in a new way. The main contribution is a simple and linear-time algorithm for solving three subproblems: online recognition of digital circular arcs coming from the digitization of a disk having either a given radius, a boundary that is incident to a given point, or a center that is on a given straight line. Solving these subproblems is interesting in itself, but also for the recognition of digital circular arcs. The proposed algorithm can be iteratively used for this problem. Moreover, since the algorithm is online, it provides a way to segment digital curves.

Measure of circularity for parts of digital boundaries and its fast computation

This paper focuses on the design of an effective method that computes the measure of circularity of a part of a digital boundary. An existing circularity measure of a set of discrete points, which is used in computational metrology, is extended to the case of parts of digital boundaries. From a single digital boundary, two sets of points are extracted so that the circularity measure computed from these sets is representative of the circularity of the digital boundary. Therefore, the computation consists of two steps. First, the inner and outer sets of points are extracted from the input part of a digital boundary using digital geometry tools. Next, the circularity measure of these sets is computed using classical tools of computational geometry. It is proved that the algorithm is linear in time in the case of convex parts thanks to the specificity of digital data, and is in O(nlogn) otherwise. Experiments done on synthetic and real images illustrate the interest of the properties of the circularity measure.

Accurate curvature estimation along digital contours with maximal digital circular arcs

We propose in this paper a new curvature estimator based on the set of maximal digital circular arcs. For strictly convex shapes with continuous curvature fields digitized on a grid of step h, we show that this estimator is mutligrid convergent if the discrete length of the maximal digital circular arcs grows in Ω(h-1/2). We indeed observed this order of magnitude. Moreover, experiments showed that our estimator is at least as fast to compute as existing estimators and more accurate even at low resolution.

Faithful polygonal representation of the convex and concave parts of a digital curve

From results about digital convexity, we define a reversible polygon that faithfully represents the maximal convex and concave parts of a digital curve. Such a polygon always exists and is unique in the general case. It is computed from a given digital curve in linear-time using well-known routines: adding a point at the front of a digital straight segment and removing a point from the back of a digital straight segment. It may helps to extract perceptually meaningful parts of shape outlines or lines.

Multigrid Convergence of Discrete Geometric Estimators

The analysis of digital shapes requires tools to determine accurately their geometric characteristics. Their boundary is by essence discrete and is seen by continuous geometry as a jagged continuous curve, either straight or not derivable. Discrete geometric estimators are specific tools designed to determine geometric information on such curves. We present here global geometric estimators of area, length, moments, as well as local geometric estimators of tangent and curvature. We further study their multigrid convergence, a fundamental property which guarantees that the estimation tends toward the exact one as the sampling resolution gets finer and finer. Known theorems on multigrid convergence are summarized. A representative subsets of estimators have been implemented in a common framework (the library DGtal), and have been experimentally evaluated for several classes of shapes. Thus, the interested users have all the information for choosing the best adapted estimators to their applications, and readily dispose of an efficient implementation.

A combined multi-scale/irregular algorithm for the vectorization of noisy digital contours

This paper proposes and evaluates a new method for reconstructing a polygonal representation from arbitrary digital contours that are possibly damaged or coming from the segmentation of noisy data. The method consists in two stages. In the first stage, a multi-scale analysis of the contour is conducted so as to identify noisy or damaged parts of the contour as well as the intensity of the perturbation. All the identified scales are then merged so that the input data is covered by a set of pixels whose size is increased according to the local intensity of noise. The second stage consists in transforming this set of resized pixels into an irregular isothetic object composed of an ordered set of rectangular and axis-aligned cells. Its topology is stored as a Reeb graph, which allows an easy pruning of its unnecessary spurious edges. Every remaining connected part has the topology of a circle and a polygonal representation is independently computed for each of them. Four different geometrical algorithms, including a new one, are reviewed for the latter task. These vectorization algorithms are experimentally evaluated and the whole method is also compared to previous works on both synthetic and true digital images. For fair comparisons, when possible, several error measures between the reconstruction and the ground truth are given for the different techniques.

Robust decomposition of a digital curve into convex and concave parts

We propose a linear in time and easy-to-implement algorithm that robustly decomposes a digital curve into convex and concave parts. This algorithm is based on classical tools in discrete and computational geometry: convex hull computation and Pickpsilas formula.

Computation of Binary Objects Sides Number using Discrete Geometry, Application to Automatic Pebbles Shape Analysis

We are working in collaboration with geographers to study a set of pebbles digital images. Among the features geographers want to get automatically, pebble sides number is intuitively defined. It is merely computed after a thick polygonalisation of pebble contour. We propose an incremental and linear algorithm, free of restrictive hypothesis, which relies on previous works of Debled et al. (2006). We give some results on synthetical and real images.