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Finite-Element Methods

Finite-element methods are numerical techniques used to approximate solutions to complex differential equations and boundary value problems by discretizing a continuous domain. The approach works by subdividing an intricate geometric shape or space into a mesh of smaller, simpler interconnected subregions known as finite elements. Over each element, the governing equations are approximated using local basis functions, typically polynomials, which are then assembled into a comprehensive system of algebraic equations. Widely applied across computational mechanics, physics, and image processing, these methods allow for the precise simulation and analysis of physical phenomena, structural deformations, and energy-minimization models across complex geometries and non-uniform boundary conditions.

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Finite-Element Methods for Active Contour Models and Balloons for 2-D and 3-D Images

Finite-Element Methods for Active Contour Models and Balloons for 2-D and 3-D Images

L. Cohen, I. Cohen

OrganizationsCEREMADEINRIAParis Dauphine University

Why you should read this

Presents a three-dimensional generalization of the balloon deformable surface model and implements a finite element framework that achieves faster convergence and superior numerical stability for volumetric medical image segmentation.

The use of energy-minimizing curves, known as "snakes" to extract features of interest in images has been introduced by Kass, Witkin and Terzopoulos [23]. A balloon model was introduced in [12] as a way to generalize and solve some of the problems encountered with the original method. We present a 3D generalization of the balloon model as a 3D deformable surface, which evolves in 3D images. It is deformed under the action of internal and external forces attracting the surface toward detected edgels by means of an attraction potential. We also show properties of energy-minimizing surfaces concerning their relationship with 3D edge points. To solve the minimization problem for a surface, two simplified approaches are shown first, defining a 3D surface as a series of 2D planar curves. Then, after comparing Finite Element Method and Finite Difference Method in the 2D problem, we solve the 3D model using the Finite Element Method yielding greater stability and faster convergence. We have applied this model for segmenting magnetic resonance images.

Added

2026-09-24