Fast Segmentation , Tracking , and Analysis of Deformable

We present a physically-based deformable model which can be used to track and to analyze non-rigid motion of dynamic structures in time sequences of 2D or 3D medical images. The model considers an object undergoing an elastic deformation as a set of masses linked by springs, where the natural lengths of the springs is set equal to zero, and is replaced by a set of constant equilibrium forces, which characterize the shape of the elastic structure in the absence of external forces. This model has the extremely nice property of yielding dynamic equations which are linear and de-coupled for each coordinate, whatever the amplitude of the deformation. It provides a reduced algorithmic complexity, and a sound framework for modal analysis , which allows a compact representation of a general deformation by a reduced number of parameters. The power of the approach to segment, track and analyze 2-D and 3-D images is demonstrated by a set of experimental results on various complex medical images. 1 Background Compared to the former work of Terzopoulos and his colleagues 5, 9] and Pentland and his colleagues 8], our work can be viewed as a continuation and uni-cation. Our model diiers, but is also related to the one of 6], who constrains locally the deformations to be conformal, and to the one of 1], where tracking takes into account local diierential properties of the surfaces. Finally the work of 3] and 4] provides an alternative way to segment volumetric images, but without explicitly trying to model the deformations. 2 The model We consider both the surface and volumetric properties of the objects at hand. We restrict ourselves to elastic deformations, i.e. we assume the object recovers its reference connguration as soon as all applied forces causing deformation are removed. Modelling an elastic boundary M can be achieved by a mesh of n virtual masses on the contour, each mass being attached to its neighbors by springs of stiiness k and natural length l 0. Similar discrete mass-spring models have been used in 9]. These springs model the elastic surface properties of the object. We can improve the modelling by attaching extra springs between non-neighbor nodes in order to model some volumetric elastic properties inside the object. These springs constrain the general form of the object within its deformation. The boundary M modelled as above will also be called structure. Such a structure can be easily …