Elsevier

Automatica

Volume 45, Issue 2, February 2009, Pages 510-516
Automatica

Brief paper
Gradient algorithms for polygonal approximation of convex contours

https://doi.org/10.1016/j.automatica.2008.08.020Get rights and content

Abstract

The subjects of this paper are descent algorithms to optimally approximate a strictly convex contour with a polygon. This classic geometric problem is relevant in interpolation theory and data compression, and has potential applications in robotic sensor networks. We design gradient descent laws for intuitive performance metrics such as the area of the inner, outer, and “outer minus inner” approximating polygons. The algorithms position the polygon vertices based on simple feedback ideas and on limited nearest-neighbor interaction.

Introduction

In this paper we investigate algorithms to compute an approximating polygon for strictly convex planar contours. We require that the approximating polygon minimizes a certain meaningful error metric. In applications such as monitoring of environmental processes it is important to be able to approximate the contour of the region of interest. Finding efficient or optimal approximating polygons is also relevant in other applications like solving interpolation problems or data compression. Constructing an optimal polygonal approximation of a contour has been a research subject for mathematicians and engineers across the last three centuries. Still interesting problems continue to remain unsolved especially for the general setting of non-convex bodies. Boundary estimation and tracking is also a relevant problem in computer vision (Kass, Witkin, & Terzopoulos, 1987). Some references on the boundary estimation problem for robotic sensor networks include Casbeer et al. (2006), Clark and Fierro (2007), Marthaler and Bertozzi (2003) and Zhang and Leonard (2005). A final motivation for this work is the interest in dynamical systems that solve optimization problems, as described for example in Helmke and Moore (1994); discrete-time gradient systems and discrete-time balancing algorithms for networks of agents are discussed in Absil, Mahony, and Andrews (2005), Lageman (2002) and in Scardovi, Sarlette, and Sepulchre (2007).

As pointed out by the authors in Johnson and Vogt (1980), in the 19th century it was known how to geometrically characterize the polygon enclosed in a convex body that minimizes the area difference between itself and the enclosing convex body. On the other hand, the geometric characterization of a polygon, enclosing a given strictly convex body, that again minimizes the difference of the areas is more complex and less intuitive. To the best of our knowledge, the earliest reference on this matter appeared only in 1949 by Trost (1949). In the 20th century it was also proved that for a convex planar body the approximation error, for various useful metrics, goes to zero as 1/N2, where N is the number of vertices of the interpolating polygon. For a detailed list of references we refer to the survey (Gruber, 1983).

Given N points (ordered in a counter-clockwise fashion) on a strictly convex contour, it is natural to define an enclosed (i.e., inscribed) polygon and an enclosing (i.e., circumscribed) polygon to the contour. Here the faces of the enclosing polygon are subsets of the tangent lines to the strictly convex contour. We adopt three geometrically-motivated error metrics that the approximating polygon can minimize. They are described as follows. The first two metrics we consider are the difference between the area enclosed in the contour and the following areas: the inner polygon area and the outer polygon area. The third metric is the difference between the area of the outer polygon and the area of the inner polygon. We derive the expressions, two of which are novel contributions of this paper, of the error metrics as functions of the vertex positions of the approximating polygon. We propose three gradient descent vector fields for N points to dynamically construct the optimal approximating polygon. The vector fields rely only on local information about the contour and about the immediate clockwise and counter-clockwise neighboring vertices. This property allows the vector fields to be implemented by a network of robots. The robots, placed around the boundary of a convex set, have to be able to sense the tangent of the set, to communicate with each other, and to move. We analyze the dynamical system behavior of these vector fields and present simulation results. We also present discrete-time versions that allow the nodes to reach locally optimal configurations for two of the metrics introduced.

The paper is organized as follows. In Section 2 we define some notation and the three performance metrics. In Sections 3 Inner polygon approximation algorithms, 4 Outer polygon approximation algorithms we present the continuous time gradient descent algorithms and their respective discrete-time algorithms to compute the best inner and outer approximating polygon, while in Section 5 we present an algorithm to construct the polygon minimizing the “outer minus inner” area. In Section 6 we present some simulation results.

Section snippets

Problem setup

We review some basic notions on the differential geometry of curves from Do Carmo (1976). Let QR2 be a compact, strictly convex body with a twice differentiable boundary Q. Let γ:[0,length(Q)]Q be the counter-clockwise arclength parametrization of Q. For s[0,length(Q)], define the tangent vector t(s)=γ(s) and the unit inward normal vector n(s) at γ(s)Q. We then define the curvature κ:[0,length(Q)]R>0 by requiring that it satisfies dt(s)ds=κ(s)n(s),dn(s)ds=κ(s)t(s). With these

Inner polygon approximation algorithms

The algorithms of this section are based on the interpolation error EI. Observe that EI(p1,,pN)=Area(Q)Area(PI(p1,,pN)). Recalling that the points p1,,pN are ordered counter-clockwise, if (xi,yi) are coordinates of pi, then one can show Area(PI(p1,,pN))=12i=1N(xiyi+1xi+1yi). We now define a dynamical system by projecting the ith component of the gradient of EI on the tangent ti: ṗi=(tiArea(PI(p1,,pN))pi)ti=(12ti(yi+1yi1xi1xi+1))ti,i{1,,N}.

Lemma 3.1

Gradient Flow for EI

Iftη(t)=(p1(t),,pN(t))denotes a

Outer polygon approximation algorithms

The algorithms of this section are based on the interpolation error EO. We begin with a geometric characterization of the partial derivative of EO and of the critical configurations for EO. Assuming the pairs (pi1,pi) and (pi,pi+1) are cc-tangent-connected, as depicted in Fig. 3, we define the points Ai=(pi)+(pi1) and Bi=(pi+1)+(pi) and the nonnegative segment lengths di=length(piAi¯) and di+=length(piBi¯). By continuity we define di+(p,p)=di(p,p)=0, for all p. Instead, if (pi,pi+1)

“Outer minus inner” polygon approximation algorithms

In this section we provide a novel expression for the partial derivative of the symmetric difference error ES (under the assumption that the outer polygon is bounded) and we design a new gradient decent algorithm.

Lemma 5.1

Partial Derivative of ES

If(pi,pi+1)is cc-tangent-connected, then the area of the triangle formed by the segmentpi+1pi¯and the rays+(pi)and(pi+1)isAi(pi,pi+1,ni,ni+1)=12(ni(pipi+1))(ni+1(pipi+1))(ni×ni+1)3,where, forni=(ni1,ni2)andni+1=(ni+11,ni+12), we let(ni×ni+1)3=ni1ni+12ni2ni+11. Furthermore,ES(p1

Simulations

Fig. 5 shows the implementation results of the three continuous time descent algorithms described in Eqs. (2), (6), (8). The eleven nodes are on the contour described by γ(θ)=(2.1+sin(2πθ))(cos(2πθ),sin(2πθ))T, for θ[0,1). Fig. 6 shows the implementation results of the discrete time Algorithm 2 described in (5).

Conclusions

We have discussed various geometric optimization problems and corresponding gradient flows. Future works will focus on nonsmooth contours such as polygons, non-convex sets, and more general algorithms for optimal interpolation of boundaries.

Acknowledgments

This material is based upon work supported by NSF Awards CMS-0626457, CMS-0643679, IIS-0712746, and ONR Award N00014-07-1-0721. An early version of this work appeared in the 2006 IEEE Conference in Decision and Control with title “Distributed algorithms for polygonal approximation of convex contours”.

Sara Susca received the Laurea degree in Aerospace Engineering from the Politecnico di Milano, Italy, in 2002, and the Ph.D. degree in Electrical and Computer Engineering from University of California Santa Barbara in 2007. Since January 2008 she is a member of the Guidance and Navigation Advanced Development group at Honeywell.

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Sara Susca received the Laurea degree in Aerospace Engineering from the Politecnico di Milano, Italy, in 2002, and the Ph.D. degree in Electrical and Computer Engineering from University of California Santa Barbara in 2007. Since January 2008 she is a member of the Guidance and Navigation Advanced Development group at Honeywell.

Francesco Bullo received the Laurea degree in Electrical Engineering from the University of Padova, Italy, in 1994, and the Ph.D. degree in Control and Dynamical Systems from the California Institute of Technology in 1999. From 1998 to 2004, he was an Assistant Professor with the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign. He is currently an Associate Professor at the Mechanical and Environmental Engineering Department, University of California, Santa Barbara. His research interests include motion planning and coordination for autonomous vehicles, motion coordination for multi-agent networks, and geometric control of mechanical systems. He is the coauthor (with Andrew D. Lewis) of the “Geometric Control of Mechanical Systems” York: Springer, 2004, 0-387-22195-6). He is currently serving on the editorial board of the IEEE Transactions on Automatic Control and of the SIAM Journal of Control and Optimization.

Sonia Martínez is an assistant professor at Mechanical and Aerospace Engineering department at UC San Diego. She received her Ph.D. degree in Engineering Mathematics from the Universidad Carlos III de Madrid, Spain, in May 2002. Following a year as a Visiting Assistant Professor of Applied Mathematics at the Technical University of Catalonia, Spain, she obtained a Postdoctoral Fulbright fellowship and held positions as a visiting researcher at UIUC and UCSB. Dr Martínez’ main research interests include nonlinear control theory, robotics, cooperative control and networked control systems. In particular, she has focused on the modeling and control of robotic sensor networks, the development of distributed coordination algorithms for groups of autonomous vehicles, and the geometric control of mechanical systems. For her work on the control of underactuated mechanical systems she received the Best Student Paper award at the 2002 IEEE Conference on Decision and Control. She was the recipient of an NSF CAREER Award in 2007.

This paper was partially presented at IEEE Conf. on Decision and Control, San Diego, CA, pages 6512–6517, December 2006. This paper was recommended for publication in revised form by Associate Editor Henri Huijberts under the direction of Editor Hassan K. Khalil.

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