brain surface conformal spherical mappingpaupert/zhangslides.pdf · 2013-10-18 · theorem maps of...
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BRAIN SURFACE CONFORMAL
SPHERICAL MAPPING
Min Zhang
𝑓:
DEFINITIONS
Conformal Map 𝑐:
• Bijective Holomorphic (or Analytic) Map
• It is Angle Preserving
Harmonic Map ℎ:
• Twice Continuously Differentiable Function with Laplacian Δℎ = 0
Gauss Map 𝑔:
• Unit normal Spherical map
Image credit to Wikipedia.org
THEOREM
Maps of genus zero Riemannian Surfaces are
conformal iff. they are harmonic
Proof can be found in Book Lectures on Harmonic
Maps, R. Schoen and S.T. Yau, International Press, Harvard
University, 1997
APPLICATION
1. Find a homeomorphism f between the two surfaces
2. Deform h such that it minimizes the harmonic energy
3. Ensure a unique mapping by adding constraints
𝑓:
3D MRI IMAGE TRIANGULATION
Marching Cube Algorithm (Patented by IBM, but expired in 2007 )
DEMO
COMPOSITE MAPPING
𝐺𝑎𝑢𝑠𝑠 𝑀𝑎𝑝 𝑔
COMPOSITE MAPPING
𝑇𝑢𝑒𝑡𝑡𝑒 𝑀𝑎𝑝 𝑡
COMPOSITE MAPPING
𝐻𝑎𝑟𝑚𝑜𝑛𝑖𝑐 𝑀𝑎𝑝 ℎ
COMPOSITE MAPPING
ℎ ∘ 𝑡 ∘ 𝑔
Algorithm Provided by: X. Gu, Y. Wang, T. Chan, P. Thompson, ST Yau, Genus Zero Surface Conformal Mapping and
Its Application to Brain Surface Mapping, IEEE Transaction on Medical Imaging, 23(8), Aug. 2004, pp. 949-958
ALGORITHM II
Algorithm Provided by: X. Gu, Y. Wang, T. Chan, P. Thompson, ST Yau, Genus Zero Surface Conformal Mapping and
Its Application to Brain Surface Mapping, IEEE Transaction on Medical Imaging, 23(8), Aug. 2004, pp. 949-958
ALGORITHM
Algorithm Provided by: X. Gu, Y. Wang, T. Chan, P. Thompson, ST Yau, Genus Zero Surface Conformal Mapping and
Its Application to Brain Surface Mapping, IEEE Transaction on Medical Imaging, 23(8), Aug. 2004, pp. 949-958
REFERENCE
X. Gu, Y. Wang, T. Chan, P. Thompson, ST Yau, Genus Zero Surface Conformal Mapping and Its Application to
Brain Surface Mapping, IEEE Transaction on Medical Imaging, 23(8), Aug. 2004, pp. 949-958
R. Schoen and S. Yau, Lectures on Harmonic Maps. Cambridge, MA: Harvard Univ., Int. Press, 1997.
WWL Chen, Introduction to Complex Analysis,
http://rutherglen.science.mq.edu.au/wchen/lnicafolder/lnica.html
John. M. Lee, Introduction to Smooth Manifolds, Springer, August 26, 2012
THANK YOU!
SUPPLEMENTS
Courtesy to Prof. Yalin Wang
CONFORMAL SPHERICAL
MAPPING
By using the steepest descent algorithm a conformal spherical
mapping can be constructed
MOBIUS GROUP
ZERO M ASS - 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 dM1 0M 2