IDiff: irrotational diffeomorphisms for computational anatomy.

Area of Science:

• Computational anatomy
• Differential geometry
• Image analysis

Background:

• Diffeomorphism groups are crucial for computational anatomy and image registration.
• Current methods rely on Riemannian geometry and Sobolev metrics, but computing log/exponential maps is complex.
• Existing computations for Riemannian log and exponential maps are implicitly defined through infinite-dimensional optimization problems.

Purpose of the Study:

• To develop a closed-form solution for the Riemannian log map on a specific submanifold of diffeomorphisms.
• To introduce a novel, efficient method for metric-based image registration.
• To leverage geometric insights for improved statistical analysis of anatomical variability.

Main Methods:

• Employed Brenier's polar factorization to decompose diffeomorphisms (h = S o psi).
• Identified and characterized a submanifold (IDiff(R(d))) generated by irrotational flows.
• Utilized the natural metric on this submanifold, demonstrating it is flat.

Main Results:

• Derived a closed-form solution for the Riemannian log map on the IDiff(R(d)) submanifold.
• Showcased the flat geometry of the IDiff(R(d)) manifold.
• Developed highly efficient metric-based image registration algorithms based on these findings.

Conclusions:

• The polar factorization approach simplifies computations within diffeomorphism groups.
• This method enables faster and more accurate image registration.
• The findings have significant implications for statistical shape analysis in computational anatomy.