Construction of 3D Shape Models of Femoral Articular Cartilage Using Harmonic Maps

Abstract

Abstract. A previous publication has described a method of pairwise 3D surface correspondence for the automated generation of landmarks on a set of examples from a class of shape [3]. However, that method did not guarantee a diffeomorphic correspondence between examples. This affected the model compactness (the ability of the model to capture shape variation in a small number of parameters) and model specificity (the fact that the model will describe shapes only within the class used for training). In this paper we describe a method of generating the pairwise correspondences using piecewise-linear harmonic maps of the surfaces which is constrained to be diffeomorphic. In particular, we are interested in producing shape models of articular cartilage. In general these models will be close to being planar discs which makes the use of harmonic mapping particularly suitable for our application. An example statistical model built using this new correspondence method is shown for the human femoral articular cartilage; a complex biological shape which demonstrates considerable variation between individuals.

Automatic Construction of Statistical Shape Models for Vertebrae

“…Concerning the establishment of correspondences on the parameter space, approaches similar to ours are those of Brett and Taylor [2] and Meier and Fisher [14]. Both work on a parameter space but do not reduce any distortion.…”

“…A common way to solve this problem is to parameterize every surface to a common base domain and to establish the correspondences on this parameter space. Current work in this context focuses on genus 0 objects such as liver or kidney [2,5,9,14]. Visually spoken, the genus describes the number of holes of a surface.…”

Q-MAF Shape Decomposition

“…the MAF problem given in Eq. (2). It is easily shown that the MAF transform is invariant to affine transformations.…”

“…The Point Distribution Model (PDM) based on PCA of the Active Shape Model (ASM) [1,2] has been succesfully applied to the modelling of the shape of biological objects based on training sets represented by corresponding points. For use for simulation, prediction, or segmentation [3,4,1] a fair number of landmarks are necessary in order to achieve sufficiently good or realistic models.…”

Construction of 3D Dynamic Statistical Deformable Models for Complex Topological Shapes

“…The practical quality of a PDM relies on the definition of correspondence across a set of segmented samples. Brett and Taylor [3] generated correspondences using ICP while a diffeomorphism between the shapes was maintained by harmonic map. Davies et al [4] solved this problem by manipulating global parameterization subject to the Minimum Description Length (MDL) criterion.…”