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Signal Processing

Volume 86, Issue 5, May 2006, Pages 1109-1115
Signal Processing

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The use of total least squares data fitting in the shape-from-moments problem

https://doi.org/10.1016/j.sigpro.2005.09.008Get rights and content

Abstract

In this paper we discuss the problem of recovering the vertices of a planar polygon from its measured complex moments. Because the given, measured moments can be noisy, the recovered vertices are only estimates of the true ones. The literature offers many algorithms for solving such an estimation problem. We will restrict our discussion to the Total Least Squares (TLS) data fitting models HTLS and STLS and the matrix pencil method GPOF. We show the close link between the HTLS and the GPOF method. We use the HTLS method to compute starting values for the STLS method. We compare the accuracy of these three methods on simulated data.

Introduction

In this paper we discuss the problem to recover the vertices of a planar polygon from its measured complex moments. In [1], [2], [3], the importance and relevance of this problem is discussed. First of all, Milanfar et al. [1] introduced the problems of reconstructing a planar polygon from a set of its complex moments. Moreover, by exploiting the relationship of this shape-from-moments problem with similar problems in signal and array processing, a number of algorithms based on Prony's method were obtained, which could be applied to the reconstruction problem. Then, better numerical procedures, based upon matrix pencils, were proposed for the shape-from-moments reconstruction problem, described in [2]. Also, an analysis of the sensitivity of the shape-from-moments problem is presented in this reference. Later on, instead of concentrating mainly on the numerical aspects of the noiseless case, in [4] the treatment of the reconstruction problem is extended to a given noisy but longer set of moments. Because the given, measured moments can be noisy, the recovered vertices are only estimations of the true ones. The literature offers many algorithms for solving such an estimation problem. We will discuss the Hankel Total Least Squares (HTLS) method [5], the Structured TLS (STLS) method [6], [7], [8] and the Generalized Pencil of Function (GPOF) method [9] for the reconstruction of binary polygons from their given, estimated, noisy complex moments. Many applications from diverse areas can be cited, including computed tomography [1] and inverse potential theory [10].

The outline of the paper is as follows. In Section 2, the shape-from-moments problem is formulated. In Section 3 a short description is given of the GPOF method, the HTLS method and the STLS method. These methods can be used to solve the shape-from-moments problem. In this paper we will show the link between the HTLS and the GPOF method. We will compare the HTLS, the STLS and the GPOF method on simulated data and discuss their accuracy. This will be the topic of Section 4. It will become clear that it is useful to compute the HTLS solution in order to get starting values for the STLS method. The STLS method is an optimal method and we expect to get a better accuracy than with the GPOF method. Conclusions are drawn in Section 5.

Section snippets

The shape-from-moments problem

The reconstruction problem of a closed N-sided planar polygon from a set of its complex moments is defined as follows. Assume that M+1 complex moments τk with k=0,…,M are measured. We want to recover the polygon vertices zn with n=1,…,N using the following relationship between moments and vertices:τk=∑n=1Nanznk,with an for n=1,…,N coefficients that only depend on the vertices. For more information about the derivation of Eq. (1), we refer to papers [1], [2], [4]. In [2], more details about the

Methods to solve the shape-from-moments problem

We briefly describe the GPOF, the HTLS and the STLS methods in this section. The literature offers many other algorithms to solve the shape-from-moments problem. We refer the interested reader to related work presented in [11], [12], [13], [14]. In this paper, we will discuss the GPOF, the HTLS and the STLS methods. The GPOF and the HTLS methods are both non-iterative subspace-based parameter estimation methods. In this section, we will show that their approaches differ in the way of reducing

Comparison in performance between HTLS, STLS and GPOF

In this section we compare the performance of the GPOF-method and the methods HTLS and STLS from the TLS-family. For this comparison we use experiments that are presented in [4]. Here, we will not explain the setup of the experiments in detail. The reader is referred for more details to the paper of Elad et al. [4]. Put shortly, the experiments are constructed as follows: Firstly, a polygon is created and its complex moments are computed. Secondly, complex Gaussian white noise is added to the

Conclusions

In this paper we discussed the problem of recovering the vertices of a planar polygon from its measured complex moments. Because the given moments can be noisy, the recovered vertices are estimations of the true ones. The literature offers many algorithms for solving such an estimation problem. We restricted our discussion to the HTLS, the STLS and the GPOF methods. We showed the close link between the HTLS and the GPOF methods. Through simulated data we compared the accuracy of the three

Acknowledgements

Prof. Dr. Sabine Van Huffel is a full professor, Mieke Schuermans is a research assistant and Dr. Philippe Lemmerling was a postdoctoral researcher of the FWO (Fund for Scientific Research—Flanders) at the Katholieke Universiteit Leuven, Belgium. Dr. Lieven De Lathauwer holds a permanent research position with the French Centre National de la Reserche Scientifique (C.N.R.S.); he also holds a honorary research position with the Katholieke Universiteit Leuven, Belgium.

Our research is supported by

References (25)

  • Y. Hua et al.

    Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise

    IEEE Trans. Acoust. Speech Signal Process.

    (May 1990)
  • V. Strakhov et al.

    On the uniqueness of the inverse logarithmic potential problem

    SIAM J. Appl. Math.

    (1986)

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