Globally optimal cortical surface matching with exact landmark correspondence

Area of Science:

• Neuroimaging and Computational Anatomy
• Medical Image Analysis
• Geometric Deep Learning

Background:

• Establishing accurate correspondences between human cortical surfaces is crucial for understanding brain structure and function.
• Existing methods often involve trade-offs between landmark alignment accuracy and surface distortion, or rely on complex parameter tuning.
• The development of robust and theoretically grounded surface mapping techniques is essential for large-scale neuroimaging studies.

Purpose of the Study:

• To present a new method for establishing correspondences between human cortical surfaces that precisely matches point landmarks.
• To minimize the deviation from conformality and achieve the global minimum of an objective function quantifying mapping distortion.
• To evaluate the biological plausibility and surface distortion of the proposed method using MRI-based cortical surface data.

Main Methods:

• A conformal transformation is applied to the Euclidean distance metric, creating a hyperbolic metric with cone point singularities at landmarks.
• Each cortical surface is mapped to a hyperbolic orbifold, with landmarks corresponding to pillow corners.
• Gradient descent is employed to find the global minimum of the Dirichlet energy for the surface-to-surface mapping, after initial landmark alignment.

Main Results:

• The proposed method achieves exact landmark matching while minimizing mapping distortion.
• Compared to a tuned approach, the new method demonstrates superior surface distortion and similar biological plausibility.
• The method shows a better theoretical foundation and requires fewer arbitrary parameters for tuning.

Conclusions:

• The novel hyperbolic orbifold mapping method provides an effective and theoretically sound approach for human cortical surface correspondence.
• This technique offers advantages in terms of reduced surface distortion and simplified parameterization compared to existing methods.
• The findings suggest that sacrificing exact conformality does not significantly reduce biological plausibility in neuroimaging applications.