**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