Unbiased diffeomorphic atlas construction for computational anatomy

Construction of population atlases is a key issue in medical image analysis, and particularly in brain mapping. Large sets of images are mapped into a common coordinate system to study intra-population variability and inter-population differences, to provide voxel-wise mapping of functional sites, and help tissue and object segmentation via registration of anatomical labels. Common techniques often include the choice of a template image, which inherently introduces a bias. This paper describes a new method for unbiased construction of atlases in the large deformation diffeomorphic setting.

A child neuroimaging autism study serves as a driving application. There is lack of normative data that explains average brain shape and variability at this early stage of development. We present work in progress toward constructing an unbiased MRI atlas of 2 years of children and the building of a probabilistic atlas of anatomical structures, here the caudate nucleus. Further, we demonstrate the segmentation of new subjects via atlas mapping. Validation of the methodology is performed by comparing the deformed probabilistic atlas with existing manual segmentations.

Introduction

Since Broadman 1909, the construction of brain atlases has been central to the understanding of the variabilities of brain anatomy. More recently, since the advent of modern computing and digital imaging techniques intense research has been directed toward the development of digital three-dimensional atlases of the brain. Most digital brain atlases so far are based on a single subject's anatomy (Ho et al., 2002, Warfield et al., 2002). Although these atlases provide a standard coordinate system, they are limited because a single anatomy cannot faithfully represent the complex structural variability between individuals. A major focus of computational anatomy has been the development of image mapping algorithms (Gee et al., 1993, Miller and Younes, 2001, Rohlfing et al., 2003b, Thompson and Toga, 2002) that can map and transform a single brain atlas on to a population. In this paradigm, the atlas serves as a deformable template (Grenander, 1994). The deformable template can project detailed atlas data such as structural, biochemical, functional as well as vascular information on to the individual or an entire population of brain images. The transformations encode the variability of the population under study. A statistical analysis of the transformations can also be used to characterize different populations (Csernansky et al., 1998, Hohne et al., 1992, Talairach et al., 1988). For a detailed review of deformable atlas mapping and the general framework for computational anatomy, see Grenander and Miller (1998) and Thompson and Toga (1997). One of the fundamental limitations of using a single anatomy as a template is the introduction of a bias based on the arbitrary choice of the template anatomy.

Thompson and Toga (1997) very elegantly address this bias in their work by mapping a new data set on to every scan in a brain image database. This approach addresses the bias by in effect forgoing the formal construction of a representative template image. Although this framework is mathematically elegant and powerful, it results in a computationally prohibitive approach in which each new scan has to be mapped independently to all the data sets in a database. This is analogous to comparing each subject under study to every previously analyzed image. As brain image databases grow the analysis problem grows combinatorially.

In more recent and related work, Avants and Gee (2004) developed an algorithm in the large deformation diffeomorphic setting for template estimation by averaging velocity fields.

Most other previous work (Bhatia et al., 2004) in atlas formation has focused on the small deformation setting in which arithmetic averaging of displacement fields is well defined. Guimond et al. (2000) develop an iterative averaging algorithm to reduce the bias. In the latest work of Bhatia et al. (2004), explicit constraints requiring that the sum of the displacement fields add to zero is enforced in the proposed atlas construction methodology. These small deformation approaches are based on the assumption that a transformations of the form h(x) = x + u(x), parameterized via a displacement field, u(x), are close enough to the identity transformation such that composition of two transformations can be approximated via the addition of their displacement fields:

$$h_1\circ h_2\left(x\right)\approx x+u_1\left(x\right)+u_2\left(x\right)\text{.}$$

The focus of this paper is a development of a methodology that simultaneously estimates the transformations and an unbiased template, in the large deformation setting. The method developed herein does not assume the above approximation of Eq. (1) and build atlases of populations with large geometric variability.