Fast Polynomial Approximation of Heat Kernel Convolution on Manifolds and Its Application to Brain Sulcal and Gyral...

Area of Science:

• Computational geometry
• Medical imaging analysis
• Numerical methods

Background:

• Heat diffusion is crucial for brain imaging tasks like surface smoothing.
• Existing methods can be computationally expensive and numerically unstable.

Purpose of the Study:

• To present a novel, fast, and accurate numerical scheme for solving heat diffusion on surface meshes.
• To apply this method for analyzing sex differences in cortical patterns from MRI data.

Main Methods:

• Approximating heat kernel convolution using high-degree orthogonal polynomials in the spectral domain.
• Deriving a closed-form spectral decomposition of the Laplace-Beltrami operator.
• Solving heat diffusion on a manifold using this spectral decomposition.

Main Results:

• The proposed scheme avoids computationally costly eigenfunction computations and finite element method instabilities.
• Successfully applied the method to localize sex differences in cortical sulcal and gyral patterns from MRI.
• Achieved a fast and accurate solution for heat diffusion on surface meshes.

Conclusions:

• The novel numerical scheme offers a computationally efficient and stable alternative for heat diffusion on surface meshes.
• This method provides an innovative approach for analyzing neuroimaging data, particularly for identifying sex-based differences in brain morphology.