2D Image-based reconstruction of shape deformation of biological structures using a level-set representation
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 such that the shape of the considered biological structure within a given observation plane at time t is given by the level-set of U:where f is a strictly monotonic continuous function. Given U, the sequence of level-sets 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 . As detailed previously, orientation field computed from the image gradient is first investigated. The second solution comes to computing 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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