Symmetric diffeomorphic modeling of longitudinal structural MRI

Abstract

This technology report describes the longitudinal registration approach that we intend to incorporate into SPM12. It essentially describes a group-wise intra-subject modeling framework, which combines diffeomorphic and rigid-body registration, incorporating a correction for the intensity inhomogeneity artifact usually seen in MRI data. Emphasis is placed on achieving internal consistency and accounting for many of the mathematical subtleties that most implementations overlook. The implementation was evaluated using examples from the OASIS Longitudinal MRI Data in Non-demented and Demented Older Adults.

Keywords: diffeomorphisms; geodesic shooting; inverse consistency; longitudinal registration; non-linear registration; shape modeling; symmetry; transitivity.

Figure 1

A graphical representation of the full model. Each of the N images (f) is assumed to be a deformed version of the template (μ) scaled by a multiplicative inhomogeneity field [exp(b)] with additive Gaussian noise (precision λ). Each deformation is modeled by the composition of a rigid-body transform (ξ) parameterized by a vector of six parameters (q), and a diffeomorphic deformation (ϕ) parameterized by its initial velocity (v).

Figure 2

A mixture of two Rician distributions fit to an MRI intensity histogram (shown dotted). The fit is shown as a continuous line, whereas dashed lines are used to show the two Rician distributions.

Figure 3

The two simulated images.

Figure 4

Warped simulated images. First column: upper image warped to match lower image (from Figure 3). Second column: deformation fields. Third column: logarithms of Jacobian determinants (color-scales are the same for all examples, and in the range of −3 to 3). First row: results from ω₁ = 0.001, ω₂ = 0.001, and ω₃ = 0.1, with an additional penalty on the square of absolute displacements of 0.0001. Second row: results from ω₁ = 0.001, ω₂ = 0.001, and ω₃ = 2.0. Third row: results from ω₁ = 0.05, ω₂ = 0.0001, and ω₃ = 0.0001. Fourth row: results from ω₁ = 0.001, ω₂ = 0.5, and ω₃ = 0.001.

Figure 5

Warped simulated images. First column: lower image warped to match upper image. Second column: deformation fields. Third column: logarithms of Jacobian determinants (color-scales are the same for all examples, and in the range of −3 to 3). First row: results from ω₁ = 0.001, ω₂ = 0.001, and ω₃ = 0.1, with an additional penalty on the square of absolute displacements of 0.0001. Second row: results from ω₁ = 0.001, ω₂ = 0.001, and ω₃ = 2.0. Third row: results from ω₁ = 0.05, ω₂ = 0.0001, and ω₃ = 0.0001. Fourth row: results from ω₁ = 0.001, ω₂ = 0.5, and ω₃ = 0.001.

Figure 6

Illustration of the results obtained from matching a pair of images of a subject with mild cognitive impairment, which were collected 1895 days apart (OAS2_0002). The three images of the residual difference shown along the bottom are all windowed the same. Black indicates a value of −500 or less, whereas white indicates values of 500 or above.

Figure 7

Detail of the results obtained from matching a pair of images of a subject with mild cognitive impairment, which were collected 1895 days apart (OAS2_0002).

Figure 8

Data from the time-points of a subject with dementia (OAS2_0048). The top row shows the original scans after rigid alignment, whereas the bottom row shows the divergence of the estimated velocity fields.

Figure 9

Plots of main eigenvector from each subject’s divergence maps within the cranium. The columns show plots from those control subjects who were scanned only twice, plots from control subjects who were scanned more than twice, plots from subjects with dementia who were scanned twice, and plots from subjects with dementia who were scanned more than twice. Dotted lines show the best linear fit. Note that the plots are sorted according to their average slope, which was done for easier visualization. Some of the eigenvectors were also rescaled by −1, such that all the gradients are positive.

Figure 10

Mean images. Left: average of all subjects’ warped mean images. Center: average of the warped expansion rate maps of the control subjects. Right: average of the warped expansion rate maps of the subjects with dementia. Mean expansion rates are shown such that values of −0.04 or below are shown as black and values of 0.04 or above are shown as white.