Fast Methods for Shape Extraction in Medical and Biomedical Imaging
We present a fast shape recovery technique in 2D and 3D with specific applications in modeling shapes from medical and biomedical imagery. This approach and the algorithms described is similar in spirit to our previous work in [16 ,18 ], is topologically
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Abstract. We present a fast shape recovery technique in 2D and 3D with specific applications in modeling shapes from medical and biomedical imagery. This approach and the algorithms described is similar in spirit to our previous work in [16,18], is topologically adaptable, and runs in O(N log N ) time where N is the total number of points visited in the domain. Our technique is based on the level set shape recovery scheme introduced in [16,3] and the fast marching method in [27] for computing solutions to static Hamilton-Jacobi equations.
1.1
Introduction
In many medical applications such as cardiac boundary detection and tracking, tumor volume quantification etc., accurately extracting shapes in 2D and 3D from medical images becomes an important task. These shapes are implicitly present in noisy images and the idea is to construct their boundary descriptions. Visualization and further processing like volume computation is then possible. Although many techniques exist for the aforementioned, computational time has often been a concern. In this paper, we present a shape modeling technique with specific applications in medical and biomedical image analysis that retains all the advantages and sophistication of existing techniques and executes in real-time. Active contour [9] and surface models [33] have been used by many researchers to segment objects from medical image data. These models are based on deforming a trial shape towards the boundary of the desired object. The deformation is achieved via solving Euler-Lagrange equations which attempt to minimize an energy functional. As an alternative, implicit surface evolution models have been introduced by Malladi et al. [16,18] and Caselles et al. [3]. In these models, the curve and surface models evolve under an implicit speed law containing two terms, one that attracts it to the object boundary and the other that is closely related to the regularity of the shape.
Supported in part by the Applied Mathematical Sciences Subprogram of the Office of Energy Research, U.S. Dept. of Energy under Contract DE-AC0376SD00098 and by the NSF ARPA under grant DMS-8919074.
R. Malladi (ed.), Geometric Methods in Bio-Medical Image Processing © Springer-Verlag Berlin Heidelberg 2002
2
Malladi, Sethian
One of the challenges in shape recovery is to account for changes in topology as the shapes evolve. In the Lagrangian perspective, this can be done by reparameterizing the curve once every few time steps and to monitor the merge/split of various curves based on some criteria; see [21]. However, some problems still remain in 3D where the issue of monitoring topological transformations calls for an elegant solution. In [17,18], the authors have modeled anatomical shapes as propagating fronts moving under a curvature dependent speed function [26]. They adopted the level set formulation of interface motion due to Osher and Sethian [22]. The central idea here is to represent a curve as the zero level set of a higher dimensional function; the motion of the curve is then embedded within th
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