Discrete surface Ricci flow can conformally map a brain surface, cut along anatomical landmark curves, to a planar multi-hole disk whose Gaussian curvature arrangement is a free design parameter constrained only by the Gauss-Bonnet theorem. The source paper (Wang et al., IEEE Transactions on Medical Imaging, 2012) illustrates several hand-designed canonical target curvature arrangements and uses the resulting canonical space for landmark-matching registration and statistical morphometry, but explicitly leaves open which curvature design maximizes the sensitivity and specificity of downstream statistical detection, and whether the design can be optimized jointly with the registration step. Later work optimized conformal parameterizations only for geometric objectives (area-distortion uniformity), fixed the canonical space and optimized registration within it, added complementary intrinsic feature families, and validated related conformal pipelines at scale on hippocampal and ventricular surfaces, but has not formulated or solved the task-driven selection of the target curvature arrangement, its joint optimization with registration, or a controlled comparison of curvature-designed canonical spaces against established spherical templates on the same detection tasks.

Background

Surface-based brain morphometry compares 3-D models of cortical and subcortical surfaces across subjects to locate disease, developmental, or genetic effects. It requires (i) a common coordinate system, or canonical space, onto which every subject's surface is flattened, and (ii) registration, a correspondence map between subjects, which is usually computed in that canonical space. The dominant industrial approach inflates a genus-zero cortical surface to a sphere (as in FreeSurfer-style spherical pipelines); spherical maps, however, concentrate area distortion and depend on an inflation heuristic. Conformal parameterization offers an alternative: a conformal map preserves angles and local shape exactly, distorting area only through a scalar conformal factor, and is determined by the surface's conformal structure rather than by arbitrary choices. A closed cortical surface (topologically a sphere, i.e., genus zero with no boundary) admits conformal spherical maps; a cortical surface cut along selected landmark sulcal curves becomes a multiply connected surface (a disk with several holes, one per cut) and admits conformal maps to planar multi-hole disks. Surface Ricci flow deforms a surface's Riemannian metric — the rule assigning lengths and angles on the surface — so that its Gaussian curvature evolves toward prescribed target values; on a triangulated surface the discrete Gaussian curvature at a vertex is its angle defect (the deviation of the surrounding angles' sum from 2π2\pi), and discrete Ricci flow is the negative gradient flow of a convex Ricci energy, solvable to global optimality by Newton's method. Equivalently, the method solves the discrete Yamabe equation: find a conformal metric inducing the prescribed curvature. The prescribed (target) curvatures must satisfy the Gauss-Bonnet constraint, e.g., ∑iK~i=2πχ\sum_i \tilde K_i = 2\pi\chi for a closed surface of Euler characteristic χ\chi (with boundary terms added for surfaces with boundary), but within that constraint the arrangement is otherwise free. Wang, Shi, Yin, Gu, Chan, Yau, Toga, and Thompson (IEEE Transactions on Medical Imaging, 2012; earlier version in ISBI 2007) used discrete Ricci flow to map landmark-cut cortical surfaces to multi-hole disks with designed target curvature — the canonical choice sets zero Gaussian curvature at interior vertices and constant geodesic curvature along boundaries, so each landmark curve maps to a circular boundary and landmark matching becomes exact. In the canonical space they computed a shape index from conformal invariants (quantities unchanged by conformal re-parameterization, usable as global shape descriptors), performed inter-subject matching by a constrained harmonic map (a boundary-constrained minimizer of the Dirichlet stretching energy), and ran statistical morphometry: a manifold version of Hotelling's T2T^2 test applied pointwise to the Jacobian matrices (local area-and-orientation distortion tensors) of the parameterization/registration maps, with permutation tests for global significance, demonstrating detection of brain asymmetry in 14 healthy subjects. The design freedom — the target curvature arrangement of the canonical space — is thus a free knob of the whole pipeline: it determines where area is expanded or compressed in the parameter domain, hence the signal-to-noise structure of the features and the difficulty of registration. Optimizing this knob was not new in itself: Jin, Wang, Yau, and Gu (IEEE Visualization 2004) and Yang, Kim, Luo, Hu, and Gu (IEEE Transactions on Visualization and Computer Graphics, 2008) recast conformal parameterization as optimization over vertex target curvatures — but only for geometric objectives, minimizing area distortion or uniformizing the conformal factor, not for the statistical power of downstream morphometric detection. The source paper illustrates several curvature arrangements by hand and stops there. What no principle currently supplies is the selection of the canonical curvature design for a statistical end: which admissible target curvature arrangement maximizes the sensitivity and specificity of a given detection task on a given anatomical surface class, and whether that choice can be optimized jointly with the registration step instead of being made independently before it.

Problem Statement

In discrete-Ricci-flow conformal parameterization of brain surfaces, the canonical target curvature arrangement is a free design parameter: for a cortical or subcortical surface cut along landmark curves, any admissible Gaussian curvature assignment satisfying the Gauss-Bonnet constraint (with consistent boundary geodesic curvature) yields a valid multi-hole-disk canonical space, and the choice propagates into every downstream statistic through the conformal factor, the registration, and the resulting features. The source paper illustrates a handful of hand-designed arrangements but selects none by principle. The open problem is the principled, goal-driven selection of this design freedom: (1) Given a statistical morphometric task — detecting a specified group contrast (e.g., disease vs. control, or genetic dose effects) on a specified class of anatomical surfaces with landmark constraints — determine which admissible canonical target curvature arrangement maximizes the detection power of the downstream statistical detection, that is, its sensitivity at a fixed significance level (e.g., Hotelling T2T^2 statistics on surface Jacobian or tensor features with permutation-based significance), and provide a usable selection or optimization procedure. (2) In the joint formulation, optimize the target curvature layout together with the registration step (harmonic-map or deformable registration in the canonical domain) so that registration accuracy and statistical power are directly improved, rather than choosing the canonical space first and the registration independently. Both the selection of the canonical target curvature arrangement (1) and its joint optimization with the registration step (2) are part of the open problem. The source's generality condition is to establish, or refute, the advantage across more than one anatomical structure and more than one population/disease cohort, and against both the source's hand-designed arrangements and an established canonical-space choice such as a spherical parameterization template — thereby also settling whether curvature-designed Ricci-flow canonical spaces yield systematic practical advantages over established template choices. The target setting requires publicly available data covering surfaces of at least two anatomical structures with landmark curves, and at least two population or disease cohorts with group labels. The optimization target is detection power, not a geometric proxy such as area-distortion uniformity.

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