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

1. 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

19. Math II

20. Math III

21. Steepest Descent Algorithm

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

24. Mobius group • In order to uniquely parameterize the surface constraints must be added • Use zero mass-center condition and landmarks

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

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

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?