Laure Tougne

Optimal Time Computation of the Tangent of a Discrete Curve: Application to the Curvature

With the definition of discrete lines introduced by REveilles [REV91], there has been a wide range of research in discrete geometry and more precisely on the study of discrete lines. By the use of the linear time segment recognition algorithm of Debled and REveilles [DR94], Vialard [VIA96a] has proposed a O(l) algorithm for computing the tangent in one point of a discrete curve where l is the average length of the tangent. By applying her algorithm to n points of a discrete curve, the complexity becomes O(n.l). This paper proposes a new approach for computing the tangent. It is based on a precise study of the tangent evolution along a discrete curve. The resulting algorithm has a O(n) complexity and is thus optimal. Some applications in curvature computation and a tombstones contours study are also presented.

Discrete Curvature Based on Osculating Circle Estimation

In this paper, we make an overview of the existing algorithms concerning the discrete curvature estimation. We extend the Worring and Smeulders [WS93] classification to new algorithms and we present a new and purely discrete algorithm based on discrete osculating circle estimation.

A parametric active polygon for leaf segmentation and shape estimation

In this paper we present a system for tree leaf segmentation in natural images that combines a first, unrefined segmentation step, with an estimation of descriptors depicting the general shape of a simple leaf. It is based on a light polygonal model, built to represent most of the leaf shapes, that will be deformed to fit the leaf in the image. Avoiding some classic obstacles of active contour models, this approach gives promising results, even on complex natural photographs, and constitutes a solid basis for a leaf recognition process.

Understanding leaves in natural images - A model-based approach for tree species identification

With the aim of elaborating a mobile application, accessible to anyone and with educational purposes, we present a method for tree species identification that relies on dedicated algorithms and explicit botany-inspired descriptors. Focusing on the analysis of leaves, we developed a working process to help recognize species, starting from a picture of a leaf in a complex natural background. A two-step active contour segmentation algorithm based on a polygonal leaf model processes the image to retrieve the contour of the leaf. Features we use afterwards are high-level geometrical descriptors that make a semantic interpretation possible, and prove to achieve better performance than more generic and statistical shape descriptors alone. We present the results, both in terms of segmentation and classification, considering a database of 50 European broad-leaved tree species, and an implementation of the system is available in the iPhone application Folia.

2D and 3D visibility in discrete geometry: an application to discrete geodesic paths

In this article, we present a discrete definition of the classical visibility in computational geometry based on digital straight fines. We present efficient algorithms to compute the set of pixels in a non-convex domain that are visible from a source pixel. Based on these definitions, we define discrete geodesic paths in discrete domain with obstacles. This allows us to introduce a new geodesic metric in discrete geometry.

Multigrid convergence and surface area estimation

Surface area of discrete objects is an important feature for model-based image analysis. In this article, we present a theoretical framework in order to prove multigrid convergence of surface area estimators based on discrete normal vector field integration. The paper details an algorithm which is optimal in time and multigrid convergent to estimate the surface area and a very efficient algorithm based on a local but adaptive computation.

Discrete parabolas and circles on 2D cellular automata

,andpossiblythehexagonalnetwork. Wecallcell each pair {vertex,automaton A}.Such a system can be considered as a dynamical one. At one time t,the global state of the system is given by an application Γ −→ S,whichassigns to each cell a state of A.And,startingfromaninitialconfiguration,the whole system synchronously evolves, at discrete times, from one config-uration to the next one. This behavior is formalized through asequenceofconfigurations (Γ

A model-based approach for compound leaves understanding and identification

In this paper, we propose a specific method for the identification of compound-leaved tree species, with the aim of integrating it in an educational smartphone application. Our work is based on dedicated shape models for compound leaves, designed to estimate the number and shape of leaflets. A deformable template approach is used to fit these models and produce a high-level interpretation of the image content. The resulting models are later used for the segmentation of leaves in both plain and natural background images, by the use of multiple region-based active contours. Combined with other botany-inspired descriptors accounting for the morphological properties of the leaves, we propose a classification method that makes a semantic interpretation possible. Results are presented over a set of more than 1000 images from 17 European tree species, and an integration in the existing mobile application Folia1 is considered.

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.