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
 Conformal mappings preserve angles of
the mapping
 Conformally map a brain scan onto a
sphere
Example of Conformal
Mapping
Overview
 Quick overview of conformal
parameterization methods
 Harmonic Parameterization
 Optimizing using landmarks
 Spherical Harmonic Analysis
 Experimental results
 Conclusion
Conformal Parameterization
Methods
 Harmonic Energy Minimization
 Cauchy-Riemann equation
approximation
 Laplacian operator linearization
 Angle based method
 Circle packing
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
Cauchy-Riemann example
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
Laplacian operator
linearization
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
Angle Based method example
Circle packing
 Classical analytical functions can be
approximated using circle packing
 Does not consider geometry, only
connectivity
Circle Packing example
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
Example of spherical mapping
QuickTime™ and a
decompressor
are needed to see this picture.
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
 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
 f represents vector value functions
 PL represents the discrete Laplacian
operator
Math overview
Math II
Math III
Steepest Descent Algorithm
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 example
Mobius group
 In order to uniquely parameterize the
surface constraints must be added
 Use zero mass-center condition and
landmarks
Zero mass-center constraint
 The mapping satisfies the zero
M1  0
 M fdif
mass-center constraint only
 All conformal mappings satisfying
the zero mass-center constraint are

unique up to the rotation group
2
Algorithm
Algorithm II
Algorithm IIb
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
 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
Landmark Example
Spherical Harmonic Analysis
 Once the brain surface is conformally
2
mapped to S , 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
 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
Geometry compression
example
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
Shape Descriptor Example
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
 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
Conformal parameterization
of brain meshes
Different triangulation
results
Results continued
 Their method is also robust enough to
allow parameterization of meshes
other than brains
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
 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
Questions?