Technology Update Algorithms to Improve the Reparameterization of Spherical Mappings of Brain Surface Meshes

A spherical map of a cortical surface is often used for improved brain registration, for advanced morphometric analysis (eg, of brain shape), and for surface-based analysis of functional signals recorded from the cortex. Furthermore, for intersubject analysis, it is usually necessary to reparameterize the surface mesh into a common coordinate system. An isometric map conserves all angle and area information in the original cortical mesh; however, in practice, spherical maps contain some distortion. Here, we propose fast new algorithms to reduce the distortion of initial spherical mappings generated using one of three common spherical mapping methods. The algorithms iteratively solve a nonlinear optimization problem to reduce distortion. Our results demonstrate that our correction process is computationally inexpensive and the resulting spherical maps have improved distortion metrics. We show that our corrected spherical maps improve reparameterization of the cortical surface mesh, such that the distance error measures between the original and reparameterized surface are significantly decreased. Introduction Most brain MRI scanning protocols acquire volumetric data about the anatomy of the subject. However, it is sometimes desirable to conduct analyses that focus exclusively on the geometry of the cortical surface. This type of analysis may be conducted directly in volume space or by first generating a surface mesh from the volumetric data.1-3 A surface mesh is useful for computing average surfaces, and the surface coordinates are useful for making comparisons of data across subjects that take into account the cortical folding pattern. Furthermore, a surface mesh makes it easier to perform some types of shape analyses, for example, spherical harmonic analysis, Laplace–Beltrami (LB) eigenmaps (another form of shape analysis), gyrification indices that measure surface complexity in 3-dimensional (3D), and complexity analysis that regrids the cortex.4-6 To proceed with these analyses, a surface mesh that accurately represents cortical anatomy must first be generated. Surface analyses have special advantages that are not present using volumetric data alone. For instance, brain surface meshes have been shown to increase the accuracy of brain registration compared with linear Talairach registration.7-10 Brain surface meshes also permit new forms of analyses, such as gyrification indices that measure surface complexity in 3D,6,11,12 cortical thickness,13,14 and data compression and searching in mining databases.15 Furthermore, inflation or spherical mapping of the cortical surface mesh raises the buried sulci to the surface so that mapped functional activity in these regions can be easily visualized. However, the primary purpose behind spherical mapping of the brain surface mesh is to generate a common coordinate system for intersubject analysis. To place a subject into a common coordinate system, the surface mesh usually must be reparameterized. One valuable type of spherical map is a pseudo-isometric map, which attempts to conserve both angle and area information from the cortical surface mesh. Any mapping may retain area information by computing the Jacobian; however, a pseudo-isometric map preserves the local areas as much as possible, such that measuring relative areas in the sphere is essentially the same as doing so on the original surface, without having to know the Jacobian of the mapping. Because of the highly folded nature of the cortical surface, however, it is not possible to obtain a perfectly isometric spherical map of the cortical surface mesh. In addition, depending on the priorities of the application, either angle or area preservation must be optimized at the expense of the other. However, it has not been explored which metric most influences the accuracy of reparameterization. It is hypothesized that area distortion is more important than angular distortion. Because there are an infinite number of solutions for an equiareal mapping, it can Copyright ◦C 2010 by the American Society of Neuroimaging 1 be advantageous to choose a solution that also keeps angular distortion at a minimum.16 Previously, most efforts to minimize distortion concurred with the algorithms used to generate the initial spherical mapping. For instance, one can create a conformal, or anglepreserving, spherical mapping. There is some evidence that conformal maps may exist in the V1 visual cortex for processing visual data.17 Also, it may be easier to solve partial differential equations on a grid if the grid is conformal, such as for signal smoothing or for surface-to-surface registration applications. A conformal map of a surface mesh can be generated by either solving a partial differential equation that involves the LB operator of the surface coordinates,18,19 often using sulcal features as explicit landmark constraints,20-22 or by minimizing the harmonic energy of the mapping to the sphere using variational methods.23-26 Another quasi-conformal mapping procedure uses circle packing as a discrete approximation to the classical continuous mapping theory.27 More recently, some researchers have used a branch of differential geometry known as the exterior calculus to build conformal grids directly on surfaces, using holomorphic differential one-forms.28 Conformal grids on surfaces may also be generated using diverse methods such as the Ricci flow method,29,30 algebraic function theory,31 slit mapping,32 and cohomology theory.28 The resulting conformal grids may be used for shape analysis using Teichmuller space theory or tensor-based morphometry on the surface metric.28,33 Alternatively, specific landmark features, such as sulcal curves lying in the cortex, can be forced to map to designated locations on the sphere, using either covariate partial differential equations derived from linear elasticity,34 level set methods and implicit function theory,35 or approximately, by using mutual information defined on surfaces.24,36 Although some approaches have computed surface-to-surface registrations by intermediate mappings of both surfaces to the sphere, an important class of methods has performed direct surface-to-surface registration.37 In addition, methods using covariant partial differential equations attempt to produce surface-to-surface registrations that, in the continuous case at least, are independent of the intermediate spherical mappings used to impose grids on the surfaces.34,38 In these mappings, a flow vector field in the surface coordinates is developed in which the differential operators are made covariant to the surface metrics, leading to surface registrations that are provably independent of the way the surfaces are gridded, so long as the surfaces are sufficiently finely sampled. Within the grid generation field, there exist penalty functionals that will conserve a linear combination of length, orthogonality, and area so that semi-isometric flat maps of sections of the cortical surface can be generated. However, unless the surface is developable, it is not possible to conserve all three functionals. The brain surface mesh may be iteratively inflated by minimizing an energy functional that is related to metric distortion or geodesic lengths, creating a spherical mapping with low metric distortion.39,40 Alternately, an initial simple mapping may be optimized by finding the minimum solution to an error functional that accounts for both area and angular distortion41 or area and length distortion.42 Semi-isometric flat maps of sections of the cortical surface may also be generated using a constrained optimization problem that accounts for both angles and area.43 Such approaches are analogous to grid generation problems for surfaces using the compound functional method, and can be expanded to n dimensions in the general case.44 In fact, equiareal mappings could be considered a subcase achievable by using only one of the three penalty functionals.45 In this paper, we propose to apply three algorithms to optimize spherical maps with respect to area distortion (and, secondarily, local metric distortion). The algorithms are unique and the problem is solved as an optimization problem to arrive at the minimum (distortion) solution. The method starts with initial spherical maps generated using a variety of previously developed methods. As a first step, the area distortion is minimized by finding the minimum solution to a nonlinear optimization problem based on an algorithm (the distort algorithm, Eq 8) that is directly related to area distortion. After the minimum solution is found, the spherical map is further optimized using more selective variants of all three algorithms. The resulting spherical map has improved distortion metrics compared to the original map. With respect to reparameterization of the spherical map into a spherical coordinate system, the optimized spherical maps result in a reparameterization that has significantly lower distance error values compared to the original map. The 3D distance error may be important in the field of morphometry, because if the original surface is not accurately approximated during reparameterization, some statistical power may be lost for shape and area analysis. Methods Minimizing Distortion The definition of distortion is by no means established. Because of the nonheterogeneity of the distortion metrics used, it is feasible to select one metric to optimize. However, the optimization of one metric usually involves the degradation of another distortion metric, and this information is lost if other distortion metric values are not reported. Reporting a diverse set of distortion metrics for all resulting spherical maps can circumvent this. Three standard distortion metrics include metric distortion, area distortion, and angle distortion.44 The metric distortion metric is a universal measure that incorporates both area and angular distortion, whereas the area and angular distortion metrics selectively measure distortion in areas or angles, respectively. In the following discussion, the surface mesh is assumed to consist of triangles, but the metrics can be generalized to non-triangular meshes as well. Metric distortion is a measure based on differences in distances between vertices. Because two triangles with three congruent sides must by definition be isometric, any deviation corresponds to distortion. Assume that there is a vertex vo,i in the original brain surface and its matching vertex on the unit sphere vi . Let do,ij and dij be the distances between the ith and jth vertices in the original brain and the spherical map, respectively. 2 Journal of Neuroimaging Vol XX No X XX 2010 The metric distortion is then defined as follows: