Kang Li1 and Xiaoping Qian2
1Illinois Institute of Technology   2University of Wisconsin-Madison
Benchmark training set of 40 hand shapes from DTU. Left figure shows the 40 hand contour shapes from 4 contributors, and figure on the right displays the superimposed B-splines, which are the input of the proposed the algorithm. Control points are in red dot and knots in blue plus.
B-spline is used to represent both the training set shapes and the reparameterization functions. The example above shows how a reparameterization function manipulates the shape correspondence.
Correspondence optimization history of a training set with 40 hand shapes. The objective function is chosen to be the Description Length (DL). After 501 iterations, DL drops from 832.8 to 766.7, and the group-wise correspondence has been brought back to a reasonable state as indicated by 11 markers (only Shape 21 is shown). As a result, the statistical shape model has become a reliable one for downstream applications (only first first two modes are shown). The dotted line represents mean shape, and the thin and thick lines denote the shape at -3 and +3 times the standard deviation of each mode respetively.
Statistical Shape Model (SSM) modes before and after optimization. At iteration 1, as the initial correspondence is in a bad state, the resultant modes fail to represent the real hand shape variations: weird hand contours and impossible stretching variation. At iteration 501, after the correspondence has been optimized, the statistical model is able to capture the underlying shape variabilities: Mode 1 represents the stretching and closing of all five fingers and particularly the thumb; Mode 2 mainly reflects the moving of the little finger.
In this paper, we propose an efficient optimization approach for obtaining shape correspondence across a group of objects for statistical shape modeling. With each shape represented in a B-spline based parametric form, the correspondence across the shape population is cast as an issue of seeking a reparameterization for each shape so that a quality measure of the resulting shape correspondence across the group is optimized. The quality measure is the description length of the covariance matrix of the shape population, with landmarks sampled on each shape. The movement of landmarks on each B-spline shape is controlled by the reparameterization of the B-spline shape. The reparameterization itself is also represented with B-splines and B-spline coefficients are used as optimization parameters. We have developed formulations for ensuring the bijectivity of the reparameterization. A gradient-based optimization approach is developed, including techniques such as constraint aggregation and adjoint senstivity for efficient, direct diffeomorphic reparameterization of landmarks to improve the group-wise shape correspondence. Numerical experiments on both synthetic and real 2D and 3D data sets demonstrate the efficiency and effectiveness of the proposed approach.