Kang Li1 and Xiaoping Qian2
1Illinois Institute of Technology   2University of Wisconsin-Madison  


Discrete (top row) and continuous (bottom row) representations of shapes for computing the covariance matrix of a group of "line-bump" shape instances. Landmarks are sampled on shapes for discrete representation, and B-spline is used as an example for continuous representation.

Top row shows three different parameterizations: original, with reparameterization Ra(u) and with reparameterization Rb(u). The covariance matrix can be computed for discrete/continuous shape representations under the three parameterizations with two formulations I and II. Observations of covariance matrix Frobenius norm from bottom row: 1) discrete representation converges to associated continuous counterpart; 2) formulation I leads to three converged values with different parameterizations (parameterization-dependent); 3) formulation II results in the same converged covariance matrix with different parameterizations (parameterization-independent);


Computing the covariance matrix of a population of shapes is essential for establishing shape correspondence, identifying shape variation across the population, and building statistical shape models. The covariance matrix is usually computed from a discrete set of points (a.k.a. landmarks) sampled on each shape. The distribution and density of the sampled points thus greatly influence the covariance matrix and its spectral decomposition. In order to understand and overcome such dependency on point sampling, in this paper, we develop accurate and efficient methods for directly computing continuous formulations of the covariance matrix. We adopt B-splines both as a shape representation and as a form of reparameterization. We apply B-splines into two continuous formulations for computing the covariance matrix of shapes. We develop both analytical and efficient numerical methods for computing such matrices. In both formulations, the covariance matrix from the corresponding numerical approximation converges to the matrix from the continuous formulations when the number of sampled points in each shape becomes sufficiently large. Their applications in optimizing shape correspondence via minimizing the description length of a set of shapes is presented.