Journal of Mathematical Imaging and Vision - Special issue on mathematical imaging
Texture mapping progressive meshes
Proceedings of the 28th annual conference on Computer graphics and interactive techniques
Newton's Method for Large Bound-Constrained Optimization Problems
SIAM Journal on Optimization
Proceedings of the 2003 Eurographics/ACM SIGGRAPH symposium on Geometry processing
Discrete conformal mappings via circle patterns
ACM Transactions on Graphics (TOG)
Applied Numerical Mathematics
Designing quadrangulations with discrete harmonic forms
SGP '06 Proceedings of the fourth Eurographics symposium on Geometry processing
Conformal equivalence of triangle meshes
ACM SIGGRAPH 2008 papers
Globally Optimal Surface Mapping for Surfaces with Arbitrary Topology
IEEE Transactions on Visualization and Computer Graphics
Conformal Slit Mapping and Its Applications to Brain Surface Parameterization
MICCAI '08 Proceedings of the 11th international conference on Medical Image Computing and Computer-Assisted Intervention - Part I
Model-driven harmonic parameterization of the cortical surface
MICCAI'11 Proceedings of the 14th international conference on Medical image computing and computer-assisted intervention - Volume Part II
Computing Extremal Quasiconformal Maps
Computer Graphics Forum
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We present a method for establishing correspondences between human cortical surfaces that exactly matches the positions of given point landmarks, while attaining the global minimum of an objective function that quantifies how far the mapping deviates from conformality. On each surface, a conformal transformation is applied to the Euclidean distance metric, resulting in a hyperbolic metric with isolated cone point singularities at the landmarks. Equivalently, each surface is mapped to a hyperbolic orbifold: a pillow-like surface with each point landmark corresponding to a pillow corner. An initial surface-to-surface mapping exactly aligns the landmarks, and gradient descent is used to find the single, global minimum of the Dirichlet energy of the remainder of the mapping. Using a population of real MRI-based cortical surfaces with manually labeled sulcus endpoints as landmarks, we evaluate the approach by how much it distorts surfaces and by its biological plausibility: how well it aligns previously-unseen anatomical landmarks and by how well it promotes expected associations between cortical thickness and age. We show that, compared to a painstakingly-tuned approach that balances a tradeoff between minimizing landmark mismatch and Dirichlet energy, our method has similar biological plausibility, superior surface distortion, a better theoretical foundation, and fewer arbitrary parameters to tune. We also compare to conformal mapper in the spherical domain to show that sacrificing exact conformality of the mapping does not cause noticeable reductions in biological plausibility.