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<p>The use of energy-minimizing curves, known as "snakes" to extract features of interest in images has been introduced by Kass, Witkin and Terzopoulos (1987). A balloon model was introduced by Cohen (1991) as a way to generalize and solve some of the problems encountered with the original method. A 3-D generalization of the balloon model as a 3-D deformable surface, which evolves in 3-D images, is presented. 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 3-D edge points. To solve the minimization problem for a surface, two simplified approaches are shown first, defining a 3-D surface as a series of 2-D planar curves. Then, after comparing finite-element method and finite-difference method in the 2-D problem, we solve the 3-D model using the finite-element method yielding greater stability and faster convergence. This model is applied for segmenting magnetic resonance images.</p>
active contour models; 3D images; energy-minimizing curves; balloon model; 3D deformable surface; attraction potential; minimization; 2D planar curves; finite-element method; magnetic resonance image segmentation; edge detection; feature extraction; finite element analysis; minimisation

I. Cohen and L. Cohen, "Finite-Element Methods for Active Contour Models and Balloons for 2-D and 3-D Images," in IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 15, no. , pp. 1131-1147, 1993.
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