Brownian Warps: A Least Committed Prior for Non-rigid Registration

Spherical demons: fast diffeomorphic landmark-free surface registration

We present the Spherical Demons algorithm for registering two spherical images. By exploiting spherical vector spline interpolation theory, we show that a large class of regularizors for the modified Demons objective function can be efficiently approximated on the sphere using iterative smoothing. Based on one parameter subgroups of diffeomorphisms, the resulting registration is diffeomorphic and fast. The Spherical Demons algorithm can also be modified to register a given spherical image to a probabilistic atlas. We demonstrate two variants of the algorithm corresponding to warping the atlas or warping the subject. Registration of a cortical surface mesh to an atlas mesh, both with more than 160 k nodes requires less than 5 min when warping the atlas and less than 3 min when warping the subject on a Xeon 3.2 GHz single processor machine. This is comparable to the fastest nondiffeomorphic landmark-free surface registration algorithms. Furthermore, the accuracy of our method compares favorably to the popular FreeSurfer registration algorithm. We validate the technique in two different applications that use registration to transfer segmentation labels onto a new image 1) parcellation of in vivo cortical surfaces and 2) Brodmann area localization in ex vivo cortical surfaces.

Effects of registration regularization and atlas sharpness on segmentation accuracy

In non-rigid registration, the tradeoff between warp regularization and image fidelity is typically determined empirically. In atlas-based segmentation, this leads to a probabilistic atlas of arbitrary sharpness: weak regularization results in well-aligned training images and a sharp atlas; strong regularization yields a "blurry" atlas. In this paper, we employ a generative model for the joint registration and segmentation of images. The atlas construction process arises naturally as estimation of the model parameters. This framework allows the computation of unbiased atlases from manually labeled data at various degrees of "sharpness", as well as the joint registration and segmentation of a novel brain in a consistent manner. We study the effects of the tradeoff of atlas sharpness and warp smoothness in the context of cortical surface parcellation. This is an important question because of the increasingly availability of atlases in public databases, and the development of registration algorithms separate from the atlas construction process. We find that the optimal segmentation (parcellation) corresponds to a unique balance of atlas sharpness and warp regularization, yielding statistically significant improvements over the FreeSurfer parcellation algorithm. Furthermore, we conclude that one can simply use a single atlas computed at an optimal sharpness for the registration-segmentation of a new subject with a pre-determined, fixed, optimal warp constraint. The optimal atlas sharpness and warp smoothness can be determined by probing the segmentation performance on available training data. Our experiments also suggest that segmentation accuracy is tolerant up to a small mismatch between atlas sharpness and warp smoothness.

Spherical demons: fast surface registration

We present the fast Spherical Demons algorithm for registering two spherical images. By exploiting spherical vector spline interpolation theory, we show that a large class of regularizers for the modified demons objective function can be efficiently implemented on the sphere using convolution. Based on the one parameter subgroups of diffeomorphisms, the resulting registration is diffeomorphic and fast - registration of two cortical mesh models with more than 100k nodes takes less than 5 minutes, comparable to the fastest surface registration algorithms. Moreover, the accuracy of our method compares favorably to the popular FreeSurfer registration algorithm. We validate the technique in two different settings: (1) parcellation in a set of in-vivo cortical surfaces and (2) Brodmann area localization in ex-vivo cortical surfaces.

Shape Modelling Using Markov Random Field Restoration of Point Correspondences

A method for building statistical point distribution models is proposed. The novelty in this paper is the adaption of Markov random field regularization of the correspondence field over the set of shapes. The new approach leads to a generative model that produces highly homogeneous polygonized shapes and improves the capability of reconstruction of the training data. Furthermore, the method leads to an overall reduction in the total variance of the point distribution model. Thus, it finds correspondence between semi-landmarks that are highly correlated in the shape tangent space. The method is demonstrated on a set of human ear canals extracted from 3D-laser scans.

Products of random matrices

We derive analytic expressions for infinite products of random 2 x 2 matrices. The determinant of the target matrix is log-normally distributed, whereas the remainder is a surprisingly complicated function of a parameter characterizing the norm of the matrix and a parameter characterizing its skewness. The distribution may have importance as an uncommitted prior in statistical image analysis.