Absfrucf-The use of energy-minimizing curves, known as "snakes" to extract features of interest in images has been introduced by Kass, Witkin and Tenopoulos [W]. A balloon model was introduced in [12] 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.Index rem-Active contour models, attraction potential, deformable models, feature extraction, finite difference method, finite element method, regularization, segmentation, surface reconstruction
I. INTRODUCTIONE STUDY segmentation of medical 2-D and 3-D W images by making use of "deformable models" [29],[32] In order to achieve robust segmentation, we introduce a number of enhancements and modifications to the formulation of deformable models. In particular, we define new forces to control the evolution of the deformable model, we formulate the models for true 3-D data, and we develop a finite-element implementation.The class of "deformable models" originates with the method of "snakes" introduced by Kass et al. [23], which are used to locate smooth curves in 2-D imagery. Since then, deformable models have been used for many applications in 2;-D and 3-D by Terzopoulos, Witkin and Kass [31], [32] where the deformable surface is constrained to encourage axial symmetry and is evolving under the forces determined from a 2-D image or a pair of 2-D images. We also make use of deformable surfaces, but the data providing information about the force comes from true 3-D data sets. We further extend Manuscript His research interest?and teaching at university are in applications of variational method? to image processing and computer vision, like deformable models, surface reconstruction, image segmentation, and restoration
