Brain Surface Conformal Mapping
Brain Surface Conformal Mapping
Xianfeng
Xianfeng Gu
Gu
1
1
,
, Yalin
Yalin Wang
Wang
2
2
, Tony F. Chan
, Tony F. Chan
2
2
, Paul M. Thompson
, Paul M. Thompson
3
3
,
, Shing
Shing-
-Tung
Tung Yau
Yau
4
4
1.
1.
Division of Engineering and Applied Science, Harvard University
Division of Engineering and Applied Science, Harvard University
2.
2.
Mathematics Department, UCLA,
Mathematics Department, UCLA,
3.
3.
Lab. Of
Lab. Of Neuro
Neuro Imaging and Brain Research Institute, UCLA School of Medicine,
Imaging and Brain Research Institute, UCLA School of Medicine,
4.
4.
Mathematics Department, Harvard University
Mathematics Department, Harvard University
, {ylwang,chan}@math.ucla.edu
ylwang,chan}@math.ucla.edu,
.
Abstract
We propose a new variational method which can find
a unique mapping between any two genus zero
manifolds by minimizing the harmonic energy of the
map. We demonstrate the feasibility of our
algorithm by applying it to the cortical surface
matching problem. We use a mesh structure to
represent the brain surface. Further constraints are
added to ensure that the conformal map is unique.
Empirical tests on MRI data show that the
mappings preserve angular relationships, are
stable in MRIs acquired at different times, and are
robust to differences in data triangulation, and
resolution. Compared with other brain surface
conformal mapping algorithms, our algorithm is
more stable and has good extensibility.
Abstract
We propose a new
We propose a new variational
variational method which can find
method which can find
a unique mapping between any two genus zero
a unique mapping between any two genus zero
manifolds by minimizing the harmonic energy of the
manifolds by minimizing the harmonic energy of the
map. We demonstrate the feasibility of our
map. We demonstrate the feasibility of our
algorithm by applying it to the cortical surface
algorithm by applying it to the cortical surface
matching problem. We use a mesh structure to
matching problem. We use a mesh structure to
represent the brain surface. Further constraints are
represent the brain surface. Further constraints are
added to ensure that the conformal map is unique.
added to ensure that the conformal map is unique.
Empirical tests on MRI data show that the
Empirical tests on MRI data show that the
mappings preserve angular relationships, are
mappings preserve angular relationships, are
stable in
stable in MRIs
MRIs acquired at different times, and are
acquired at different times, and are
robust to differences in data triangulation, and
robust to differences in data triangulation, and
resolution. Compared with other brain surface
resolution. Compared with other brain surface
conformal mapping algorithms, our algorithm is
conformal mapping algorithms, our algorithm is
more stable and has good extensibility.
more stable and has good extensibility.
Conformal Mapping
O
Any surface without holes or self-intersections can be
mapped conformally onto the sphere
O
This mapping, conformal equivalence, is one-to-one,
onto, and angle preserving
O
Locally, shape is preserved and distances and areas
are only changed by a scaling factor
O
A canonical space is useful for subsequent work
Conformal Mapping
O
Any surface without holes or self-intersections can be
mapped conformally onto the sphere
O
This mapping, conformal equivalence, is one-to-one,
onto, and angle preserving
O
Locally, shape is preserved and distances and areas
are only changed by a scaling factor
O
O
A canonical space is useful for subsequent work
A canonical space is useful for subsequent work
Conformal Mapping Properties
O
Intrinsic to geometry
O
Independent of triangulation and resolution
O
Depends on metric continuously
Conformal Mapping Properties
O
Intrinsic to geometry
O
Independent of triangulation and resolution
O
Depends on metric continuously
Genus Zero Conformal Mapping
Properties
O
Harmonic is equivalent to conformal
O
All conformal are equivalent
O
All the conformal construct a automorphism group:
Möbius group which is a linear rational group on
complex plane and a 6 dimensional group.
Genus Zero Conformal Mapping
Properties
O
Harmonic is equivalent to conformal
O
All conformal are equivalent
O
All the conformal construct a automorphism group:
Möbius group which is a linear rational group on
complex plane and a 6 dimensional group.
Algorithm Deatails
O
Harmonic energy
O
Discrete harmonic energy
O
Discrete Laplacian
Algorithm Deatails
O
Harmonic energy
O
Discrete harmonic energy
O
Discrete Laplacian
Algorithm at a Glance
O
Minimize Harmonic Energy
O
Use absolute derivative
O
All computation are on the target surface, without
projecting to complex plane
Algorithm at a Glance
O
Minimize Harmonic Energy
O
Use absolute derivative
O
All computation are on the target surface, without
projecting to complex plane
2
:SMf →
M
M
dffE
σ
2
)(
∫
∇=
∑
∈
−=
Mvu uv
vfufkfE
],[
2
)()()(
)cot(cot
2
1
βα
+=
uv
k
∑
∈
−=∆
Mvu uv
vfufkuf
],[
))()(()(
Spherical parameterization algorithm
for genus zero surface
O
Use Gauss map as the initial degree one map
O
Compute the gradient vector of harmonic energy on
each vertex
O
Project the gradient vector to the tangent space on S
2
at each vertex
O
Update the image of each vertex along the tangential
gradient direction
O
Normalize the mapping by shifting the center of the
mass to the sphere center
Spherical parameterization algorithm
Spherical parameterization algorithm
for genus zero surface
for genus zero surface
O
Use Gauss map as the initial degree one map
O
Compute the gradient vector of harmonic energy on
each vertex
O
Project the gradient vector to the tangent space on S
2
at each vertex
O
Update the image of each vertex along the tangential
gradient direction
O
Normalize the mapping by shifting the center of the
mass to the sphere center
Optimize the Conformal
Parameterization by Landmarks
O
We define a metric to measure the quality of the
parameterization.
O
Suppose two brain surfaces S
1
,S
2
, two conformal
parameterizations are denoted as f1: S
2
→S
1
and
f2: S
2
→S
2
, the matching metric is defined as
Optimize the Conformal
Optimize the Conformal
Parameterization by Landmarks
Parameterization by Landmarks
O
We define a metric to measure the quality of the
parameterization.
O
Suppose two brain surfaces S
1
,S
2
, two conformal
parameterizations are denoted as f1: S
2
→S
1
and
f2: S
2
→S
2
, the matching metric is defined as
∫
−=
2
2
2121
),(),(),(
S
dudvvufvufffE
O
Let
Ω
be the group of Möbius transformations. We can
compose a Möbius transformation such that
O
Landmarks are commonly used in brain mapping.
They are a set of sulcal curves manually drawn on the
brain surfaces.
O
We can use landmarks to obtain such a Möbius
transformation.
O
Let
Ω
Ω
be the group of Möbius transformations. We can
compose a Möbius transformation such that
O
Landmarks are commonly used in brain mapping.
They are a set of sulcal curves manually drawn on the
brain surfaces.
O
We can use landmarks to obtain such a Möbius
transformation.
),(min),(
2121
ζτ
ζ
DD ffEffE
Ω∈
=
the Conformal Parameterization by
Landmarks (Cont.)
O
Landmarks are represented as discrete point sets. We
can reduce the brain matching metric by reducing the
matching metric on landmark sets.
O
First we project the sphere onto the complex plane.
We find a Möbius transformation on the complex
plane which reduce the matching metric on landmark
sets. Then we project the results back to the sphere.
O
For a Möbius transformation on the complex plane u,
since it maps infinity to infinity, it means the north
poles of the spheres are mapped to each other.
O
Then u can be represented as a linear form az+b. Let
p
i
and q
i
, i=1 …n, are corresponding landmark points.
The functional of u can be simplified as
O
O
where z
i
is the stereo-projection of p
i
,
τ
i
is the
projection of q
i
, g is the conformal factor from the
plane to the sphere.
the Conformal Parameterization by
Landmarks (Cont.)
O
Landmarks are represented as discrete point sets. We
can reduce the brain matching metric by reducing the
matching metric on landmark sets.
O
First we project the sphere onto the complex plane.
We find a Möbius transformation on the complex
plane which reduce the matching metric on landmark
sets. Then we project the results back to the sphere.
O
For a Möbius transformation on the complex plane u,
since it maps infinity to infinity, it means the north
poles of the spheres are mapped to each other.
O
Then u can be represented as a linear form az+b. Let
p
i
and q
i
, i=1 …n, are corresponding landmark points.
The functional of u can be simplified as
O
O
where z
i
is the stereo-projection of p
i
,
τ
τ
i
i
is the
projection of q
i
, g is the conformal factor from the
plane to the sphere.
∑
=
−+=
n
iiii
bazzguE
1
2
)()(
τ
Subject
Subject Vertex #
Vertex # Face #
Face # Before
Before After
After
A
A65,538
65,538 131,072
131,072 -
--
-
B
B65,538
65,538 131,072
131,072 604.134
604.134 506.665
506.665
C
C65,538
65,538 131,072
131,072 414.803
414.803 365.325
365.325
Discussion
O
Compared with Haker’s method
[I]
, our method is more
geometric; no big distortion areas; more stable; good
extension ability (e.g. it is possible to do brain mapping
between two brains using our algorithm.)
O
Compared with Hurdal’s method
[II]
, our method preserves
angles; good mapping between brains and the canonical
space.
Discussion
O
Compared with Haker’s method
[I]
, our method is more
geometric; no big distortion areas; more stable; good
extension ability (e.g. it is possible to do brain mapping
between two brains using our algorithm.)
O
Compared with Hurdal’s method
[II]
, our method preserves
angles; good mapping between brains and the canonical
space.
Landmark Experimental Results
Landmark Experimental Results
More Genus Zero Surface Examples
More Genus Zero Surface Examples
Brief Reference
I.
S. Haker, S. Angenent, A. Tannenbaum, R. Ki kinis, g. Sapiro, and M. Halle
. “Conformal
Surface Parameterization for Texture Mapping”. IEEE Transac tions on Visualization and
Computer Graphics, 6(2):181-189, April-June 2000
II.
M.K. Hurdal, K. Stephenson, P.L. Bowers, D.W.L. Summers, and D.A. Rottenberg.
Coordinate Systems For Conformal Cerebellar Flat Maps. In NeuroImage, Vol. 11: S 467,
2000
Brief Reference
I.
S. Haker, S. Angenent, A. Tannenbaum, R. Ki kinis, g. Sapiro, and M. Halle
. “Conformal
Surface Parameterization for Texture Mapping”. IEEE Transac tions on Visualization and
Computer Graphics, 6(2):181-189, April-June 2000
II.
M.K. Hurdal, K. Stephenson, P.L. Bowers, D.W.L. Summers, and D.A. Rottenberg.
Coordinate Systems For Conformal Cerebellar Flat Maps. In NeuroImage, Vol. 11: S 467,
2000
Experimental Results
Experimental Results
Conformal mappings of surfaces with different resolutions. The original brain surface has 50,000 faces, and is
conformallymapped to a sphere, as shown in (a). Then the brain surface is simplified to 20,000 faces, and its
spherical conformal mapping is shown in (b).
Conformalitymeasu rement. The curves of iso-polar angle and iso-azimuthal angle are mapped to the
brain, and the intersection angles are measured on the brain. The histogram is illustrated.
(a) (b)
Reconstructed brain meshes and their spherical harmonic mappings. (a) and (c) are the
reconstructed surfaces for the same brain scanned at different times. Due to scanner noise
and inaccuracy in the reconstruction algorithm, there are visible geometric differences. (b)
and (d) are the spherical conformal mappings of (a) and (c) respectively; the normal
information is preserved. By the shading information, the correspondence is illustrated.
(a) (b) (c) (d)
Conformal texture mapping. The conformality is visualized by texture mapping of a che ckerboard image.