Characterizing volume and surface deformations in an atlas framework: theory, applications, and implementation

Abstract

Given deformations for mapping images or surfaces into an atlas configuration, methods are described for characterizing the mean deformation and deviations from this mean. Jacobian matrices are used to characterize the deformations locally, and the method can be applied to any image warping method for which Jacobian matrices can be computed. The method makes use of the fact that each matrix descriptor of the local deformation required to match an image to the atlas corresponds to a point on a semi-Riemannian manifold. By assuring that the mean matrix lies within this manifold, fundamental geometric properties common to all of the images can be preserved. Local deviations from the mean can be characterized in a euclidean space tangent to the semi-Riemannian manifold at the mean and can be accumulated globally across multiple sampling locations within the atlas to generate a global multivariate characterization of how each image deviates from the mean.

Introduction

The field of morphometrics concerns itself with characterizing variations in biological size or shape. Using modern tomographic imaging techniques, it is now possible to noninvasively collect very large data sets rich in morphometric information from living human subjects (Mazziotta et al., 2001). While much can be learned from such data sets by focusing attention solely on the morphometry of explicitly identifiable anatomic point landmarks, interest is rapidly growing in morphometric approaches that are based on analyzing deformations of complete surfaces or deformations of entire volumes that bring the anatomy of different subjects into some standardized atlas configuration. This interest is fueled both by a desire for more comprehensive anatomic descriptors and by the availability of automated or semiautomated methods for efficiently generating such deformations, potentially without the need for anatomic expertise. Implicit in such an approach is the realization that the deformations of the individual images to the atlas configuration are almost certain to be imperfect owing either to fundamental anatomic differences (e.g., variations in the number of gyri in a certain part of the brain (Ono et al., 1990)) or to discrepancies between macroscopically identifiable homologies and more fundamental microscopic homologies that cannot currently be identified from in vivo imaging data (e.g., discrepancies between sulcal and gyral landmarks in the brain compared to cytoarchitectonic landmarks Geyer et al 2001, Rademacher et al 2001). Automatically derived deformations may also include explicit anatomic errors when evaluated by expert anatomists. The potential for all such imperfections must be kept in mind whenever deformations are used in a morphometric context and pose issues that extend beyond the current context where it will be assumed that the deformations represent an acceptably valid description of anatomic reality.

The primary objective of this article is to describe a general method for analyzing deformations that bring a group of images into some standardized atlas configuration. From a morphometric standpoint, the primary issues of interest are to establish whether two or more samples differ from one another in anatomic configuration and, if so, to describe the nature of the differences. Current automated and semiautomated methods for deriving such deformations vary enormously in their methods of parameterizing the requisite anatomic distortions, and one of the cardinal features of the proposed analysis method is that it uses localized descriptors of deformation that should be easily derived using any of a variety of methods. The generality of the approach also means that it should be useful in comparing different methods for estimating deformations to one another. The approach uses the Jacobian matrix that describes the local deformation needed to match the original images to each sampled point in the atlas as a descriptor of local morphometric change. Consequently, the method can be applied to any deformation data for which the local Jacobian matrices can be estimated from the global deformation by differentiation either analytically or numerically. The only additional requirements imposed by the method are that the determinants of these local Jacobian matrices must be positive in all cases and that the Jacobian matrices must be sufficiently similar to allow a meaningful definition of a mean Jacobian matrix. For analyzing volume deformations, the general approach will be to compute the mean Jacobian matrix at each sampled point in the atlas and to use this mean to characterize how each individual image locally deviates from the mean. These mean deviations at each point will then be combined across all sampled points to generate a multivariate description of morphometric deviation from the mean that can be analyzed using standard multivariate statistical procedures. For the analysis of surface deformations, a related but somewhat more complicated approach will be needed to characterize both the orientation of the vector normal to the surface and the deformations within the surface.

The novel aspects of the approach described here are the methods for computing the mean Jacobian matrices and deviations from these means and for computing analogous quantities for surface deformations. It is not appropriate to simply average individual matrix elements to define the mean Jacobian matrix. This can be illustrated by a simple two-dimensional example. Consider the Jacobian matrices associated with two-dimensional rotations of −45° and +45°. These matrices are

$$\text{cos(−45°)}\text{−sin(−45°)}\text{sin(−45°)}\text{cos(−45°)}$$

and

$$\text{cos(45°)}\text{−sin(45°)}\text{sin(45°)}\text{cos(45°)}\text{.}$$

Simple averaging of the matrix elements gives the unsatisfying result

$$\text{cos(−45°) + cos(45°)}\text{2}\text{−sin(−45°) − sin(45°)}\text{2}\text{sin(−45°) + sin(45°)}\text{2}\text{cos(−45°) + cos(45°)}\text{2}\text{=}\text{2}\text{2}\text{0}\text{0}\text{2}\text{2}\text{,}$$

rather than the intuitively obvious identity matrix (i.e., a rotation of 0°). Indeed, this result not only fails to preserve the rigid-body character of the two original Jacobian matrices but even produces a matrix that describes a twofold volume reduction as the “average” of matrices that are both volume preserving. The alternative definition of the mean Jacobian matrix that will be used here is based on principles of differential geometry that make it possible to preserve certain fundamental geometric properties that interrelate the various Jacobian matrices being averaged. This alternative definition does produce an identity matix when applied to this simple two-dimensional case. It preserves geometric properties by first mapping the Jacobian matrices onto a semi-Riemannian manifold and then computing a mean that is constrained to lie within this manifold. Results from Lie group theory will be used to show that this approach does indeed preserve the geometric properties associated with any matrix group that interrelates the various Jacobian matrices being averaged. Fig. 1 gives a schematic overview of the analytic approach to be applied to volume deformations. The approach to surface deformations will be outlined after the volume deformation approach has been fully described.

The need for special averaging procedures based on the differential geometry of curved manifolds is not a new concept in morphometrics. Indeed, the morphometrics literature concerned with the analysis of shapes defined by point landmarks has long recognized the inappropriateness of simple averaging of landmark coordinates in defining average shape. Mapping the landmark positions onto a Riemannian manifold to find a mean shape constrained to lie within the manifold is a well-known analytic procedure Bookstein 1996, Dryden and Mardia 1998, Kendall 1990, Kendall et al 1999, Small 1996. Such analyses also often include the use of a euclidean space tangent to the manifold at the point representing the mean shape to characterize deviations from the mean, and an analogous approach will be employed here. The novel aspects of the current approach compared to those previously employed include the need to extend the definition of the mean from a Riemannian to a semi-Riemannian context to accommodate the averaging of matrices rather than collections of coordinate positions and the use of a relatively low-dimensional manifold to separately average Jacobian matrices from each sampled point in the atlas rather than a very high-dimensional manifold to simultaneously characterize all aspects of mean shape associated with a set of point landmarks. Nonetheless, the geometric foundations of these two approaches are remarkably similar, with differences that are motivated by fundamental differences in the underlying nature of the morphometric descriptors.