1. Xianfeng Gu, Yaling Wang, Tony Chan, Paul Thompson, Shing-Tung Yau
Genus Zero Surface Conformal Mapping and Its Application to Brain Surface Mapping
Xianfeng Gu, Yaling Wang, Tony Chan, Paul Thompson, Shing-Tung Yau

2. 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

3. Example of Conformal Mapping

4. Overview Quick overview of conformal parameterization methods
Harmonic Parameterization
Optimizing using landmarks
Spherical Harmonic Analysis
Experimental results
Conclusion

5. Conformal Parameterization Methods
Harmonic Energy Minimization
Cauchy-Riemann equation approximation
Laplacian operator linearization
Angle based method
Circle packing

6. 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

7. Cauchy-Riemann example

8. 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

9. Laplacian operator linearization

10. 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

11. Angle Based method example

12. Circle packing
Classical analytical functions can be approximated using circle packing
Does not consider geometry, only connectivity

13. Circle Packing example

14. 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

15. Example of spherical mapping

16. 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

17. 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

18. Math overview Have a space Cpl, piecewise linear functions
String constants kuv for each edge
String energy is a piecewise linear function between the two vertices

19. Math II If kuv=1, tuette energy
The parameters come out to be 1/2 cotangent of the angle
Cotangent is minimized at 90 degrees, hence minimizing the harmonic energy will tend to conformal mappings

20. Math III
#Harmonic energy is the sum of energies of the vector functions
Only harmonic if no tangent and has a normal

21. Steepest Descent Algorithm
T is the step size
Steepest descent is going in the opposite direction of greatest gradient change

22. 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

23. Mobius group example
Both a and c are conformal mappings of the face, but the locations of the poles are different, giving a different parameterization

24. Mobius group
In order to uniquely parameterize the surface constraints must be added
Use zero mass-center condition and landmarks
In order to obtain a unique parameterization, more has to be done

25. 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
F arrow is the vector value piecewise linear function, sigma M1 is the area element on M1

26. Algorithm

27. Algorithm II

28. Algorithm IIb

29. 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

30. 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
U is a member of omega, the group of mobius transformations

31. Landmark Example

32. 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

33. 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

34. Geometry compression example

35. 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

36. Shape Descriptor Example

37. 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

38. 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

39. Conformal parameterization of brain meshes

40. Different triangulation results

41. Results continued
Their method is also robust enough to allow parameterization of meshes other than brains

42. 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

43. 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

44. Questions?