Genus Zero Surface Conformal Mapping and Its Application to Brain Surface Mapping Xianfeng Gu, Yaling Wang, Tony Chan, Paul Thompson, Shing-Tung Yau
Dec 18, 2015
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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are needed to see this picture.
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€
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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?