A Numerical Framework for Sobolev Metrics on the Space of Curves

A general automatic approach is presented for accommodating local shape variation when mapping a two-dimensional (2-D) or three-dimensional (3-D) template image into alignment with a topologically similar target image. Local shape variability is accommodated by applying a vector-field transformation to the underlying material coordinate system of the template while constraining the transformation to be smooth (globally positive definite Jacobian). Smoothness is guaranteed without specifically penalizing large-magnitude deformations of small subvolumes by constraining the transformation on the basis of a Stokesian limit of the fluid-dynamical Navier-Stokes equations. This differs fundamentally from quadratic penalty methods, such as those based on linearized elasticity or thin-plate splines, in that stress restraining the motion relaxes over time allowing large-magnitude deformations. Kinematic nonlinearities are inherently necessary to maintain continuity of structures during large-magnitude deformations, and are included in all results. After initial global registration, final mappings are obtained by numerically solving a set of nonlinear partial differential equations associated with the constrained optimization problem. Automatic regridding is performed by propagating templates as the nonlinear transformations evaluated on a finite lattice become singular. Application of the method to intersubject registration of neuroanatomical structures illustrates the ability to account for local anatomical variability.

This paper introduces a square-root velocity (SRV) representation for analyzing shapes of curves in euclidean spaces under an elastic metric. In this SRV representation, the elastic metric simplifies to the IL(2) metric, the reparameterization group acts by isometries, and the space of unit length curves becomes the unit sphere. The shape space of closed curves is the quotient space of (a submanifold of) the unit sphere, modulo rotation, and reparameterization groups, and we find geodesics in that space using a path straightening approach. These geodesics and geodesic distances provide a framework for optimally matching, deforming, and comparing shapes. These ideas are demonstrated using: 1) shape analysis of cylindrical helices for studying protein structure, 2) shape analysis of facial curves for recognizing faces, 3) a wrapped probability distribution for capturing shapes of planar closed curves, and 4) parallel transport of deformations for predicting shapes from novel poses.

Assessment of protein subcellular location is crucial to proteomics efforts since localization information provides a context for a protein's sequence, structure, and function. The work described below is the first to address the subcellular localization of proteins in a quantitative, comprehensive manner. Images for ten different subcellular patterns (including all major organelles) were collected using fluorescence microscopy. The patterns were described using a variety of numeric features, including Zernike moments, Haralick texture features, and a set of new features developed specifically for this purpose. To test the usefulness of these features, they were used to train a neural network classifier. The classifier was able to correctly recognize an average of 83% of previously unseen cells showing one of the ten patterns. The same classifier was then used to recognize previously unseen sets of homogeneously prepared cells with 98% accuracy. Algorithms were implemented using the commercial products Matlab, S-Plus, and SAS, as well as some functions written in C. The scripts and source code generated for this work are available at http://murphylab.web.cmu.edu/software. murphy@cmu.edu