Genus Zero Surface Conformal Mapping and Its Application to Brain

Surface Mapping

Xianfeng Gu, Yaling Wang, Tony Chan, Paul Thompson,

Shing-Tung Yau

Conformal Mapping Overview

Map meshes onto simple geometric primitives

Map genus zero surfaces onto spheres

Conformal mappings preserve angles of the mapping

Conformally map a brain scan onto a sphere

Example of Conformal Mapping

Overview

Quick overview of conformal parameterization methods

Harmonic ParameterizationOptimizing using landmarksSpherical Harmonic AnalysisExperimental resultsConclusion

Conformal Parameterization Methods

Harmonic Energy MinimizationCauchy-Riemann equation

approximationLaplacian operator linearizationAngle based methodCircle packing

Cauchy-Riemann equation approximation

Compute a quasi-conformal parameterization of topological disks

Create a unique parameterization of surfaces

Parameterization is invariant to similarity transformations, independent to resolution and it is orientation preserving

Cauchy-Riemann example

Laplacian operator linearization

Use a method to compute a conformal mapping for genus zero surfaces by representing the Laplace-Beltrami operator as a linear system

Laplacian operator linearization

Angle based method

Angle based flattening method, flattens a mesh to a 2D plane

Minimizes the relative distortion of the planar angles with respect to their counterparts in the three-dimensional space

Angle Based method example

Circle packing

Classical analytical functions can be approximated using circle packing

Does not consider geometry, only connectivity

Circle Packing example

Harmonic energy minimization

Mesh is composed of thin rubber triangles

Stretch them onto the target meshParameterize the mesh by

minimizing harmonic energy of the embedding

The result can be also used for harmonic analysis operations such as compression

Example of spherical mapping

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Harmonic Parameterization

Find a homeomorphism h between the two surfaces

Deform h such that it minimizes the harmonic energy

Ensure a unique mapping by adding constraints

Definitions

K is the simplicial complexu,v are the vertices{u,v} is the edge connecting two verticesf, g represent the piecewise linear

functions on K represents vector value functions represents the discrete Laplacian

operator

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rf

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ΔPL

Math overview

Math II

Math III

Steepest Descent Algorithm

Conformal Spherical Mapping

By using the steepest descent algorithm a conformal spherical mapping can be constructed

The mapping constructed is not unique; it forms a Mobius group

Mobius group example

Mobius group

In order to uniquely parameterize the surface constraints must be added

Use zero mass-center condition and landmarks

Zero mass-center constraint

The mapping satisfies the zero mass-center constraint only if

All conformal mappings satisfying the zero mass-center constraint are unique up to the rotation group

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vf dσM1 = 0M 2

∫

Algorithm

Algorithm II

Algorithm IIb

Landmarks

Landmarks are manually labeled on the brain as a set of uniformly parameterized sulcal curves

The mesh is first conformally mapped onto a sphere

An optimal Mobius transformation is calculated by minimizing Euclidean distances between corresponding landmarks

Landmark Matching

Landmarks are discrete point sets, which mach one to one between the surfaces

Landmark mismatch functional is

Point sets must have equal number of points, one to one correspondence

Landmark Example

Spherical Harmonic Analysis

Once the brain surface is conformally mapped to , the surface can be represented as three spherical functions:

This allows us to compress the geometry and create a rotation invariant shape descriptor€

S2

Geometry Compression

Global geometric information is concentrated in the lower frequency components

By using a low pass filter the major geometric features are kept, and the detail removed, lowering the amount of data to store

Geometry compression example

Shape descriptor

The original geometric representation depends on the orientation

A rotationally invariant shape descriptor can be computed by

Only the first 30 degrees make a significant impact on the shape matching

Shape Descriptor Example

Experimental Results

The brain models are constructed from 3D MRI scans (256x256x124)

The actual surface is constructed by deforming a triangulated mesh onto the brain surface

Results

By using their method the brain meshes can be reliably parameterized and mapped to similar orientations

The parameterization is also conformalThe conformal mappings are

dependant on geometry, not the triangulation

Conformal parameterization of brain

meshes

Different triangulation results

Results continuedTheir method is also robust

enough to allow parameterization of meshes other than brains

Conclusion

Presented a method to reliably parameterize a genus zero mesh

Perform frequency based compression of the model

Create a rotation invariant shape descriptor of the model

Conclusion continued

Shape descriptor is rotationally invariantCan be normalized to be scale invariant1D vector, fairly efficient to calculateThe authors show it to be triangulation

invariantRequires a connected mesh - no polygon

soup or point modelsRequires manual labeling of landmarks

Questions?