Symplectomorphic registration with phase space regularization by entropy spectrum pathways

Abstract

Purpose: The ability to register image data to a common coordinate system is a critical feature of virtually all imaging studies. However, in spite of the abundance of literature on the subject and the existence of several variants of registration algorithms, their practical utility remains problematic, as commonly acknowledged even by developers of these methods.

Methods: A new registration method is presented that utilizes a Hamiltonian formalism and constructs registration as a sequence of symplectomorphic maps in conjunction with a novel phase space regularization. For validation of the framework a panel of deformations expressed in analytical form is developed that includes deformations based on known physical processes in MRI and reproduces various distortions and artifacts typically present in images collected using these different MRI modalities.

Results: The method is demonstrated on the three different magnetic resonance imaging (MRI) modalities by mapping between high resolution anatomical (HRA) volumes, medium resolution diffusion weighted MRI (DW-MRI) and HRA volumes, and low resolution functional MRI (fMRI) and HRA volumes.

Conclusions: The method has shown an excellent performance and the panel of deformations was instrumental to quantify its repeatability and reproducibility in comparison to several available alternative approaches.

Keywords: diffeomorphic mapping; diffusion tensor imaging; functional MRI; non-linear registration; symplectomorphic mapping.

FIGURE 1

3D extension of the classical “toy” example used for benchmarking of diffeomorphic registration: fitting “circle”—sphere in (B)—to “C”—spherical shell with a hole in (A). Results of direct (C) and inverse (D) maps obtained in 8 embedded energy shells. Subset of curvilinear grid lines plotted for three neighboring layers selected from three orthogonal planes for direct (E) and inverse (F) maps. The different colors were used to distinguish between the anterior-posterior grid lines (blue), the dorsal-ventral grid lines (green) and the right-left lateral grid lines (red). Both inverse and direct maps were obtained in a single run, the processing time for 200 × 200 × 200 volumetric datasets was just above 30 s on 12 cores Intel ® Core^TM i7–4930K CPU 3.40 GHz

FIGURE 2

Results of high resolution anatomical (HRA) mapping to the same anatomical reference volume (shown in A). (B) Central planes for four volumes out of ten subjects used for mapping. (D) Residual images of SWD preconditioning (fitted with orthogonal transform) for the same four volumes, (C) all ten volumes averaged. Residual images of symplectomorphic transforms using 5 (F) and 15 (H) embedded shells with all ten subject averages in (E) and (G) respectively

FIGURE 3

Different nonlinear warpage types applied to 5 different subjects (only one subject is shown for each warpage type—MNI152 T1 2 mm with 91 × 109 × 91 voxels): (A) an original subject; (B) differential rotation with the amount of rotation proportional to the distance from the center in the axial plane (whirl); (C) differential stretch in the anterior direction; (D) differential rotation with the amount of rotation proportional to the distance from the axial plane in the longitudinal direction (twist); (E) nonuniform compression in the axial plane proportional to the the longitudinal distance; (F) nonuniform compression in the longitudinal direction relative to the position in the axial plane

FIGURE 4

Medium resolution (100 × 100 × 72) diffusion weighted (DWI) volume registration to high resolution (168 × 256 × 256) T1 reference. A, Reference T1 MRI image (2D center slice top and 3D view bottom), B, DWI b0 MRI image, C, equilibrium probability DWI image (same resolution as b0 image), D, DWI image SWD preconditioned and registered to T1 image (same resolution as T1 image). Side by side comparison of reference E, and symplectomorphic registration of DWI volume F (enlarged versions of A and D)

FIGURE 5

Comparison of registration and diffusion weighted tractography using the original DWI volume space (100 × 100 × 72) (top row) and the warped HRA space (168 × 256 × 256) using the symplectomorphic registration (second row), the ANTs SyN method (third row) and the AFNI’s 3dQwarp (bottom row). The SYM-REG requires less time and results in largest overall quality improvements both in term of equilibrium probability (second column) and the details of tracts (third column). The latero–lateral (left to right and right to left), anterior–posterior, and dorsal–ventral tracts are shown in cyan, magenta and yellow respectively

FIGURE 6

3D view of low resolution (64 × 64 × 30) rs-FMRI volume A vs T1 high resolution (290 × 262 × 262) anatomical volume B. SWD preconditioned rs-FMRI volume after registration to high resolution T1 template C. The final mapping grid used 30 shells D and took about 5 minutes on 12 cores Intel ® Core^TM i7–4930K CPU 3.40 GHz. The same color scheme is used for the displacement field with blue corresponding to the anterior-posterior grid lines, green to the dorsal-ventral grid lines, and red to the right-left lateral grid lines

FIGURE 7

Several randomly selected resting state modes obtained using low resolution (64 × 64 × 30) rs-FMRI volume registered to T1 high resolution (290 × 262 × 262) anatomical volume—default mode A and D, visual lateral B and E, and visual occipital C and F—for some of the subjects from Figure 2. The upper panels A, B, and C show the original low resolution rs-FMRI modes. The symplectomorphic maps in lower panels D–F show accurate localizations of functional modes in the appropriate regions of HRA volumes