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