Groupwise surface correspondence by optimization: representation and regularization

Groupwise optimization of correspondence across a set of unlabelled examples of shapes or images is a well-established technique that has been shown to produce quantitatively better models than other approaches. However, the computational cost of the optimization is high, leading to long convergence times. In this paper, we show how topologically non-trivial shapes can be mapped to regular grids, hence represented in terms of vector-valued functions defined on these grids (the shape image representation). This leads to an initial reduction in computational complexity. We also consider the question of regularization, and show that by borrowing ideas from image registration, it is possible to build a non-parametric, fluid regularizer for shapes, without losing the computational gain made by the use of shape images. We show that this non-parametric regularization leads to a further considerable gain, when compared to parametric regularization methods. Quantitative evaluation is performed on biological datasets, and shown to yield a substantial decrease in convergence time, with no loss of model quality.