We develop a computational model of shape that extends existing Riemannian models of curves to multidimensional objects of general topological type. We construct shape spaces equipped with geodesic metrics that measure how costly it is to interpolate two shapes through elastic deformations. The model employs a representation of shape based on the discrete exterior derivative of parametrizations over a finite simplicial complex. We develop algorithms to calculate geodesics and geodesic distances, as well as tools to quantify local shape similarities and contrasts, thus obtaining a formulation that accounts for regional differences and integrates them into a global measure of dissimilarity. The Riemannian shape spaces provide a common framework to treat numerous problems such as the statistical modeling of shapes, the comparison of shapes associated with different individuals or groups, and modeling and simulation of shape dynamics. We give multiple examples of geodesic interpolations and illustrations of the use of the models in brain mapping, particularly, the analysis of anatomical variation based on neuroimaging data.
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This research was supported in part by NSF grant DMS-0713012 and NIH Roadmap for Medical Research grant U54 RR021813.
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Liu, X., Shi, Y., Dinov, I. et al. A Computational Model of Multidimensional Shape. Int J Comput Vis 89, 69–83 (2010). https://doi.org/10.1007/s11263-010-0323-0
- Multidimensional shape
- Shape of surfaces
- Elastic shapes