## FLASH: Fast Landmark Aligned Spherical Harmonic Parameterization for Genus-0 Closed Brain Surfaces

Project Description:
Surface registration between cortical surfaces is crucial in medical imaging to perform systematic comparisons between brains. Landmark-matching registration that matches anatomical features, called the sulcal landmarks, are often required, to obtain a meaningful 1-1 correspondence between brain surfaces. This is commonly done by parameterizing the surface onto a simple parameter domain, such as the unit sphere, which aligns sulcal landmarks consistently. Landmark-matching surface registration can then be obtained from the composition map of the parameterizations. For genus-0 closed brain surfaces, optimized spherical harmonic parameterization, which aligns landmarks to consistent locations on the sphere, have been widely used. This approach performs well under small deformations. However, the bijectivity is usually lost when large deformations occur or large amount of landmark constraints are enforced. Besides, the algorithm involves solving an optimization problem over the space of all diffeomorphisms from the surface onto the sphere, which is nonlinear. Hence, the computation is slow. In this paper, a fast algorithm (called {\it FLASH}) to compute the optimized landmark aligned spherical harmonic parametreization is proposed. This is done by formulating the optimization problem to the extended complex plane $\overline{\mathbb{C}}$ and thereby linearizing the problem. Error introduced near the pole (or the infinity point in $\overline{\mathbb{C}}$) is corrected using quasi-conformal theories. Also, by adjusting the Beltrami differential of the mapping, which measures the conformality distortion, a diffeomorphic (1-1, onto) spherical parameterization can be effectively obtained. Using the proposed algorithm, the computation of the optimized spherical harmonic parameterization with consistent landmark alignment can be significantly speeded up (100 times faster than the conventional method). Experiments have been carried out on 38 human brain surfaces, which demonstrate the effectiveness of the proposed algorithm.

Publication:

• K.C. Lam, P.T. Choi and L.M. Lui, FLASH: Fast landmark aligned spherical harmonic parameterization for genus-0 closed brain surfaces, UCLA CAM Report, 13-79