Shape registration in implicit spaces using information theory and free form deformations

We present a novel, variational and statistical approach for shape registration. Shapes of interest are implicitly embedded in a higher-dimensional space of distance transforms. In this implicit embedding space, registration is formulated in a hierarchical manner: the mutual information criterion supports various transformation models and is optimized to perform global registration; then, a B-spline-based incremental free form deformations (IFFD) model is used to minimize a sum-of-squared-differences (SSD) measure and further recover a dense local nonrigid registration field. The key advantage of such framework is twofold: 1) it naturally deals with shapes of arbitrary dimension (2D, 3D, or higher) and arbitrary topology (multiple parts, closed/open) and 2) it preserves shape topology during local deformation and produces local registration fields that are smooth, continuous, and establish one-to-one correspondences. Its invariance to initial conditions is evaluated through empirical validation, and various hard 2D/3D geometric shape registration examples are used to show its robustness to noise, severe occlusion, and missing parts. We demonstrate the power of the proposed framework using two applications: one for statistical modeling of anatomical structures, another for 3D face scan registration and expression tracking. We also compare the performance of our algorithm with that of several other well-known shape registration algorithms.