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deformable surfaces

Deformable surfaces are flexible three-dimensional computational models that dynamically adjust their shape to extract, segment, and reconstruct geometric structures from volumetric data or imagery. Extending two-dimensional active contour models to three dimensions, these surfaces evolve under the influence of internal forces that maintain geometric smoothness and continuity, balanced against external forces that pull the boundary toward prominent features such as edges, intensity gradients, or detected surface points. The deformation is formulated as an energy-minimization problem, typically solved through numerical techniques such as the finite element method or finite difference methods. By combining elasticity constraints with data-driven attraction, deformable surfaces reliably capture complex target boundaries, making them widely used in medical image segmentation, computer vision, and computer graphics.

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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