2D Image-based reconstruction of shape deformation of biological structures using a level-set representation

https://doi.org/10.1016/j.cviu.2007.12.005Get rights and content

Abstract

This paper copes with the reconstruction of accretionary growth sequence from images of biological structures depicting concentric ring patterns. Accretionary growth shapes are modeled as the level-sets of a potential function. Given an image of a biological structure, the reconstruction of the sequence of growth shapes is stated as a variational issue derived from geometric criteria. This variational setting exploits image-based information, in terms of the orientation field of relevant image structures, which leads to an original advection term. The resolution of this variational issue is discussed. Experiments on synthetic and real data are reported to validate the proposed approach.

Section snippets

Introduction and problem statement

A number of biological structures, for instance corals, seashells, fish otoliths,1 tree trunks or vertebrae grow according to an accretionary process. In other words, they can be viewed as a succession of three-dimensional concentric layers (with respect to an initial core). The composition of these layers, in terms of crystalline organization and chemical features, vary according to endogenous and exogenous factors [27]. Often,

Level-set setting

As suggested in the seminal work of Thompson [32], we adopt a level-set setting to represent the accretionary growth process (Fig. 3). It comes to introduce a potential function U defined over R2 such that the shape Γt(U) of the considered biological structure within a given observation plane at time t is given by the level-set of U:Γt(U)={pR2such thatU(p)=f(t)},where f is a strictly monotonic continuous function. Given U, the sequence of level-sets {Γt(U)}[0,T] represents the evolution of the

Numerical resolution

To solve for the above constrained minimization, we adopt a two-step iterative approach: the first step comes to project the current solution onto the set of level-representations with uniform first-order statistics. We detail below the unconstrained minimization. To improve the convergence, a multiresolution framework is adopted.

Extraction of the orientation fields

The proposed approach initially relies on the extraction of orientation field ω. As detailed previously, orientation field ω=I/|I| computed from the image gradient I is first investigated. The second solution comes to computing ω as the AMLE of the orientations of the gradient measures selected by a thresholded Canny–Deriche edge filter [11], [16]. Fig. 5 reports the orientation fields issued from these two schemes for the image depicted in Fig. 1. These orientation fields are visualized via

Discussion

We have proposed a scheme aimed at reconstructing from an image the evolution of the shape of biological structures involving accretionary growth process. Its key feature is a level-set representation, which intrinsically accounts for the major characteristics of the accretionary growth process. From purely geometric criteria on shape regularity and on local orientation coherence with respect to the observed image, a variational formulation is derived. The associated minimization is efficiently

Acknowledgments

We are grateful to A. Alvarez, A. Christensen, H. Mosegaard, M. Palmer and H. de Pontual for fruitful discussions. We also thank the anonymous referees for their helpful comments on the early version of the manuscript. This work has been carried out with the financial support from the Commission of the European Communities, specific program “Specific Support to Policies”, SSP8-044132 “AFISA”. It does not necessarily reflect its views and in no way anticipates the Commission’s future policy in

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