16 April 1996 Deformation of n-dimensional digital Jordan surfaces
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Abstract
Active contour models have become popular recently especially in medical image segmentation research and in the modeling of deformable structures. There are a number of theoretical, computational, and operational advantages to treating the boundary as a digital entity and defining deformations digitally on the digital representation. Nevertheless, this does not seem to have been pursued in the literature. A significant body of knowledge exists in the literature on the theory and algorithms related to nD, oriented, closed, connected, digital surfaces. In this paper we examine how this knowledge can be utilized to extend active contour models to nD digital surfaces. We present an algorithm for deforming nD, oriented, closed, connected digital surfaces into nD, oriented, closed, connected digital surfaces. The algorithm uses a data structure which we call a random insertion/deletion list. It contains spatial elements (spels) that constitute the immediate interior (II) of the surface as well as information that establishes the relationship of the spels to the surface elements (surfels). (In our digital surface model, the surfels are one lower dimensional than the spels. For example, in 3D, the spel is a voxel and a surfel is a square representing the voxel face). All deformations are realized essentially by adding/deleting spels to/from the data structure. Our results indicate that it is possible to model surfaces digitally and that given an nD, oriented, closed, connected, digital surface, we can deform this surface using local operations only to produce an nD, oriented, closed, digital surface. We also describe an algorithm and data structures to implement this model efficiently. Furthermore, if we allow more global tests, we can also insure that connectedness of the surface is insured as well. We conclude that the extensively studied digital surface models are appropriate for representing deformable objects also. Because of the finite (and small number of) local configurations of surfels and spels, these models offer efficient deformation computation as has been repeatedly demonstrated in the past for a variety of visualization and analysis operations. We demonstrate that we can deform the surface in such ways that maintain orientation and closure using only local checks and we can also insure connectedness at the expense of global checks.
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George J. Grevera, Jayaram K. Udupa, "Deformation of n-dimensional digital Jordan surfaces", Proc. SPIE 2710, Medical Imaging 1996: Image Processing, (16 April 1996); doi: 10.1117/12.237964; https://doi.org/10.1117/12.237964
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