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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

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Genus Zero Surface Conformal Mapping and Its Application to Brain Surface Mapping


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genus zero surface conformal mapping and its application to brain surface mapping

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
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
overview
Overview
  • Quick overview of conformal parameterization methods
  • Harmonic Parameterization
  • Optimizing using landmarks
  • Spherical Harmonic Analysis
  • Experimental results
  • Conclusion
conformal parameterization methods
Conformal Parameterization Methods
  • Harmonic Energy Minimization
  • Cauchy-Riemann equation approximation
  • Laplacian operator linearization
  • Angle based method
  • Circle packing
cauchy riemann equation approximation
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
laplacian operator linearization
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
angle based method
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
circle packing
Circle packing
  • Classical analytical functions can be approximated using circle packing
  • Does not consider geometry, only connectivity
harmonic energy minimization
Harmonic energy minimization
  • Mesh is composed of thin rubber triangles
  • Stretch them onto the target mesh
  • Parameterize the mesh by minimizing harmonic energy of the embedding
  • The result can be also used for harmonic analysis operations such as compression
harmonic parameterization
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
Definitions
  • K is the simplicial complex
  • u,v are the vertices
  • {u,v} is the edge connecting two vertices
  • f, g represent the piecewise linear functions on K
  • represents vector value functions
  • represents the discrete Laplacian operator
conformal spherical mapping
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
Mobius group
  • In order to uniquely parameterize the surface constraints must be added
  • Use zero mass-center condition and landmarks
zero mass center constraint
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
landmarks
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
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
spherical harmonic analysis
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
geometry compression
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
shape descriptor
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
experimental results
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
Results
  • By using their method the brain meshes can be reliably parameterized and mapped to similar orientations
  • The parameterization is also conformal
  • The conformal mappings are dependant on geometry, not the triangulation
results continued
Results continued
  • Their method is also robust enough to allow parameterization of meshes other than brains
conclusion
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
Conclusion continued
  • Shape descriptor is rotationally invariant
  • Can be normalized to be scale invariant
  • 1D vector, fairly efficient to calculate
  • The authors show it to be triangulation invariant
  • Requires a connected mesh - no polygon soup or point models
  • Requires manual labeling of landmarks