Geodesic estimation for large deformation anatomical shape averaging and interpolation
Introduction
Anatomical atlases have tremendous value in today's medical environment where large databases are mined for diagnostic, research, and pedagogical information. High-resolution atlases are an instance of anatomy upon which teaching or surgical planning is based (Kikinis et al., 1996, Miller et al., 1993, Yelnik et al., 2003). Surgical procedures may employ atlas-based image registration for planning the placement of deep-brain stimulators (Dawant et al., 2003). Average anatomical atlases provide a least-biased coordinate system for surgical planning, functional localization studies (Ashburner and Friston, 1996), or for studying structure–function relationships (Letovsky et al., 1998). They also operate as a reference frame for understanding the normal variation of anatomy (Talairach and Tournoux, 1988) and as a probabilistic space into which functional or structural features are mapped (Le Briquer and Gee, 1997). Genomics researchers currently build atlases to investigate the relationship of genotype to phenotype (Mackenzie-Graham et al., 2004), which is a major focus of the Allen Brain Institute (Allen Brain Atlas). Performance of algorithms based on manipulating empirical information, such as active shape (Cootes et al., 1995), should also benefit from use of an average model.
Computerized atlases based on magnetic resonance (MR) images may compile either average shape (Le Briquer and Gee, 1997), average intensity, or both (Guimond et al., 2000) within a single image. The Euclidean shape space, shown in Fig. 1, is often assumed for these models leading to the use of linear averaging of the transformations and intensity to produce the atlases. Deviations from the mean shape or intensity may then be captured separately by statistical models such as principal components (Cootes et al., 1995, Le Briquer and Gee, 1997). Average intensities are traditionally found by first computing transformations from an anatomical instance to a population data set. The averages of these displacement fields, which take a member of the population to the remainder of the data, represent an average in the sense of anatomical positions. This average transformation must then be inverted to gain the average shape (Guimond et al., 2000). However, positional differences are not explicitly minimized in the registration problem, typically because one promotes smoothness by using differential measures in the face of ill-posed problems (Tikhonov and Arsenin, 1977).
One difficulty with this approach is that the process of averaging transformations may destroy the optimal properties of the individual transformations. For example, the average of large deformation displacement fields, each of which satisfies the minimization of a well-defined variational energy, may no longer be a legitimate displacement field with respect to the optimized quantities. Another example is found in time-parameterized mappings. The flows defining these transformations at each time satisfy the fluid equations, allowing the maps to be interpreted as members of the diffeomorphism group (Miller and Younes, 2001). Averaging the final displacements in these maps eliminates the optimality in space and time.
Optimization-based atlases respect the underlying manifold structure of the optimization problem (Yezzi and Soatto, 2003). Large deformation image registration is important for studying anatomical variation within groups and, more importantly, for the comparison of topologically dissimilar groups such as normal vs. highly atrophied brains, interspecies comparisons, as well as anatomical growth and development. One approach for comparing very distinct groups is to base comparison on their representative atlases. Large deformation atlases may be used to compactly represent a group and its shape or intensity variations.
The large deformation image registration framework invites group theoretical population studies where one bases structural comparisons on the geodesic distances between members of the group. Thus, it is important to be able to compute atlases that are least biased within this theoretical framework, as well as in the small deformation elastic case.
This work provides a flexible algorithm for allowing shape averaging that enables properties of the physical model used in the registration to persist in the average shape transformation. The distances from the diffeomorphism group are used to illustrate the techniques. Thus, geodesic averaging, rather than linear averaging, is used to estimate mean atlas shapes. We find the average representation of anatomy by using the same optimization framework as is assumed for the image registration itself. This kind of averaging is a metaregistration algorithm operating with average forces along the shape manifold. This explicit optimization inherently respects the optimization problem and implies a basic framework for large deformation statistics. Here we compute means, but variances and principal components that are optimal in time should follow. The framework allows either large or small deformation averaging and realizes different outcomes than simply taking the naive Euclidean mean of displacements.
Section snippets
Methodology
The goal of this paper is to point out that average anatomical shape atlases should be computed with respect to the assumptions of the image registration problem. This simplifies the atlas estimation: the same algorithm used for pairwise registration should be the foundation of an algorithm for computing the transformation from an initial atlas to the optimal atlas. Given a variational registration algorithm that solves the pairwise problem, one simply invokes a meta-algorithm that
Population shape averaging algorithms
The image registration algorithm described in the previous section allows the computation of transformations between the domains in a population data set. The traditional linear averaging for atlas estimation will be discussed first, followed by the novel geodesic averaging method. Examples will illustrate the need for the geodesic optimization algorithm.
Curved case: averaging with the diffeomorphism group
The algorithm will now be extended into algorithm for an optimized estimate of the large deformation atlas.
Results
We now show an experimental comparison between the two algorithms above, results of which are summarized in Fig. 16. Note that this is an evaluation that only investigates the intensity optimization and does not use anatomical information explicitly. For each of three initial estimates to the mean anatomy, we:
- (1)
Compute the linear and the geodesic average to the database.
- (2)
Compute measures of the deformation-based distance and the intensity distance (SSD) to the database by registering all images to
Conclusion
This paper gave a novel approach for performing geodesic averaging of shape along anatomical image registration manifolds. The design involves transferring algorithmic knowledge about the pairwise registration case into a database-wide registration. The foundational algorithm, here, is an extended version of the viscous fluid framework, but the basic ideas could also be used with other algorithms, such as elastic image registration (Gee and Bajcsy, 1999). Applications of the methods in two and
Acknowledgment
The authors would like to acknowledge Tom Schoenemann of the University of Pennsylvania Department of Anthropology for providing data and motivation for this work.
References (32)
- et al.
Fully three-dimensional nonlinear spatial normalization: a new approach
- et al.
Active shape models—Their training and application
Comput. Vis. Image Underst.
(1995) - et al.
Elastic matching: continuum mechanical and probabilistic analysis (Chapter 11)
- et al.
Average brain models: a convergence study
Comput. Vis. Image Underst.
(2000) - “Allen Brain Atlas,”...
- et al.
Symmetric geodesic shape averaging and shape interpolation
- et al.
Shape averaging with diffeomorphic flows for atlas creation
IEEE Int. Symp. Biomed. Imaging
(2004) - et al.
A deformable neuroanatomy textbook based on viscous fluid mechanics
- et al.
Large deformation minimum mean squared error template estimation for computational anatomy
IEEE Int. Symp. Biomed. Imaging
(2004) - et al.
Computerized atlas-guided positioning of deep brain simulators: an feasibility study
Variational problems on flows of diffeomorphisms for image matching
Q. Appl. Math.
Gaussian distributions on Lie groups and their application to statistical shape analysis
Rapid coarse-to-fine matching using scale-specific priors
Computational anatomy: an emerging discipline
Q. Appl. Math.
Hierarchical estimation of a dense deformation field for 3d robust registration
IEEE Trans. Med. Imaging
Landmark and intensity-based, consistent thin-plate spline image registration
Cited by (361)
Space-feature measures on meshes for mapping spatial transcriptomics
2024, Medical Image AnalysisEffects of Cortisol Administration on Resting-State Functional Connectivity in Women with Depression
2024, Psychiatry Research - NeuroimagingIndividual differences in delay discounting are associated with dorsal prefrontal cortex connectivity in children, adolescents, and adults
2023, Developmental Cognitive NeuroscienceDynamic changes of resting state functional network following acute ischemic stroke
2023, Journal of Chemical Neuroanatomy