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After a brief survey on the parametric deformable models, we develop an iterative method based on the finite difference schemes in order to obtain energy-minimizing snakes. We estimate the approximation error, the residue, and the truncature error related to the corresponding algorithm, then we discuss its convergence, consistency, and stability. Some aspects regarding the prosthetic sugical methods that implement the above numerical methods are also pointed out.

The deformable models represent a powerful researched model-based approach to computer-assisted medical image analysis, their applications in this framework including image segmentation, shape representation and motion tracking. The theory of deformable models is an interdisciplinary scientific domain, which has appeared and developed in the last two decades, in strong connection with practical problems of medicine, image processing, and physics. This theory joins methods, results, and techniques of various mathematical fields, physics and mechanics. The mathematical foundation of this theory represents the confluence of Functional Analysis, Approximation Theory, Differential Equations, Differential Geometry, Calculus of Variations, Numerical Analysis, Linear Algebra, and Probability Theory. The ancestors of the deformable models, in classical sense, are considered Fischler and Elschlager, with their spring-loaded templates, [

The theory of deformable models, in its modern form, originates from the general theory of continuous multidimensional deformable models in a Lagrangian dynamic of Terzopoulos (1987) [

Two general types of deformable models have been developed: firstly, the parametric or variational models, which originate from the papers of Kaas et al. [

A good survey on deformable models and their applications can be found in [

In this paper, we deal with the deformable parametric models. The basic goal of the theory of parametric deformable models is to determine the energy-minimizing 2D or 3D models, namely, the curves or surfaces which minimize the corresponding energy functional. Two approaches will point out in order to obtain the optimal model. The first approach is based on the Euler-Lagrange-Poisson (ELP) and Euler-Gauss-Ostrogradski (EGO) equations of Calculus of Variations in order to minimize the energy-functional. The second one (the classical approach) consists of using reconstruction methods, such as the interpolation of the sparse data extracted from the image, in order to obtain a representation of the original data. In what follows we develop methods and techniques related to the first approach. Generally, the energy-functional is not convex, so it may have many local minimum. On the other hand, the analytic solution of (ELP) equation has a complicated form or it is inaccessible explicitly. Therefore, a practical and strong approach for finding local minimum of the energy functional is to construct a dynamic system that is governed by the energy functional and allow the system to evolve to the equilibrium state. Dynamic models are valuable for medical image analysis, because most anatomical structures are deformable and continually undergo nonrigid motion “in vivo.” In fact, the user is interested to find a good 2D or 3D contour in a given area. Consequently, a rough prior estimation of the 2D or 3D model is provided, then this initial model undergoes a deformation until reaching a local minimum of the energy functional. This deformation process can be achieved in one of the following ways:

in a Hamiltonian-type approach, by performing a strictly decreasing energy path, for example, via dynamic programming methods [

in a Lagrange-type approach, by applying the mechanical principles of Lagrange [

by using a friction force, in order to constrain the displacement of the snake [

by using the (ELP) evolution equation, associated to the initial (ELP) equation [

In this paper, we shall adopt the method of the evolution equation. So, a prior estimation of the deformable surface is provided, then it is refined step by step, based on the (EGO) equation and using discretization methods.

The paper outline is as follows. The next section is devoted to present 2D and 3D energy-minimizing models, both in their static and dynamic forms. The method for reducing the 3D problem to a 2D modeling is also pointed out, in order to minimize the computational costs of the numerical methods. The third section contains the main theoretical result of the paper. Based on finite difference schemes of explicit type, we derive an (ELP) algorithm for obtaining an energy-minimizing snake in its approximated form, then we estimate its approximation error and we discuss its consistency, convergence, and stability. The last section deals with the behaviour of prosthetic surgical methods and prosthetic medical materials, based on Software tools, which implement the iterative methods developed in the previous sections.

From mathematical point of view, a 2D parametric deformable model (usually known as

In order to find the optimal position of the snake, it is necessary to characterize its state, by means of an

the

the

the

the vectorial function

The shape of the snake

The

The internal energy characterizes the deformation of a stretchy, flexible snake (contour). The values of

The

The

Denoting by

By definition, the triple

The basic goal of a deformable parametric model is to minimize its energy-functional

The scalar (ELP) equations, derived from (

If we choose in (

If we choose in (

Roughly speaking, the differential fourth-order vectorial equation (

Denote by

A dynamic snake is represented by introducing a time-varying contour

The first two terms in the left hand side of (

In this section we define briefly the notion of deformable 3D model (deformable surface), both in the static and dynamic forms, and we describe a method for reducing the problem of its optimization to a 2D modelling problem.

From mathematical point of view, a 3D variational deformable model is emphasized by a family

Denoting by

The triple

By simple calculation, we obtain from (

Similarly to the 2D model, we can suppose that a rough prior estimate of surface is accessible, namely,

The problem of finding directly energy-minimizing surfaces, that is, solutions of the p.d.e. (

Under the hypothesis that

If we consider in (

In what follows we shall restrict to the study of 2D deformable models.

In this section we suppose that the following hypotheses are satisfied: the control functions

We approximate the partial derivatives involved in the (ELP) evolution equation (

In what follows, the formulas (

Taking into account the relation (

By replacing the expansions (

Let

On the other hand, the equality

Let us consider the approximation-error

The relations (

Now, the relations (

The intuitive idea regarding the stability is that small errors in the initial conditions of a partial differential equation should cause small errors in its solution. In fact, the study of the stability is useful in connection with the theorem of Lax concerning the convergence of the discretized schemes, [

The aim of this subsection is to examine the stability of the (ELP) algorithm (

Now, we obtain from (

A combination of the relations (

In order to apply the results of the theoretical researches detailed above in the medical imaging domain, a 3D visual software environment—named MoDef—was implemented, aiming to visualize and follow up the deformation behavior of the surgical (abdominal, maxilla-facial, and orthodontic) prosthetic materials. That is performed on three distinct, but convergent, levels, as follows:

3D reconstruction visual software component, aimed to tracks the evolution of the prosthetic materials, based on processing the US images of the anatomic context of a lot of surgical patients;

deformable prosthetic material’s behavior forecasting software component, based on software tools which implements the above described mathematical methods;

quad comparative parallel tracking software component, aimed to simultaneous supervise in time both (a) and (b) levels, in comparison with the results provided by the stochastic analysis component of the 3D visual software environment MoDef.

Concerning the 3D visualizing of the prosthetic meshes by means of the MoDef software environment components, two levels of reconstruction are performed, namely

on the first level, a polynomial interpolation method is applied on each slice of the US image of the prosthetic mesh, acquired based on succeeding positions of the transducer, obtained by rotating them with a constant angle in a same preestablished direction; more exactly, the curves representing the sections of the surgical mesh acquired by the transducer are extracted from the context of the US image, based on specific image processing methods, namely, contour detection methods, that are implemented at the level of the image processing operators of the MoDef environment’s image processing library. Starting with this set of basic mesh surface definition curves, extracted from the US images acquired at pre-established moments in time, a complete and consistent collection of 3D generator curve sets is obtained, by means of 3D polynomial interpolation methods, based on Lagrange, Hermite or Birkhoff operators;

on the second level, the complete collection of the 3D generator curves obtained at the first level is processed based on Blended Interpolating Methods (BIM), as well as with 3D continuous representation techniques, in order to obtain “solid-view,” respectively, “wired-view” representations of the prosthetic mesh.

In what follows, some preliminary experiments made in 3DS Max7, followed by some relevant results obtained with the 3D reconstruction component of MoDef 3D Visual environment are presented in Figures

Preliminary experiments made in 3DS Max7.

Results obtained with the 3D reconstruction component of MoDef 3D Visual environment: (a) initial 3D representation of the deformable surface of the surgical mesh, (b) curve representing a section of the surgical mesh acquired by the transducer, extracted from the context of the US image based on specific-image processing method, (c) the surface of the prosthetic mesh after the deformations produced in time due to the anatomic assimilation process, and (d) the basic set of generating curves, used to obtain the solid-view representations of the prosthetic mesh.

In this paper we considered parametric (variational) deformable models and we developed an iterative method based on finite difference schemes in order to solve numerically the (ELP) equation of Calculus of Variations, which provides the energy minimizing snake. We derived estimates concerning the approximation error related to the corresponding (ELP) algorithm and we established conditions for its convergence and stability. Some considerations about the implementation of the above numerical methods where presented, too. As future targets, we intend to consider probabilistic models which offer an alternative approach by using the Bayes technique, as well as geometric deformable models which provide an efficient alternative to address some limitation of parametric deformable models.