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Surface Parameterization using Riemann Surface Structure
Yalin Wang
Mathematics Department
UCLA
[email protected]
Xianfeng Gu
Comp. Sci. Department
SUNY at Stony Brook
[email protected]
Kiralee M. Hayashi
Lab. of Neuro Imaging
UCLA School of Medicine
[email protected]
Tony F. Chan
Mathematics Department
UCLA
[email protected]
Paul M. Thompson
Laboratory of Neuro Imaging
UCLA School of Medicine
[email protected]
Shing-Tung Yau
Mathematics Department
Harvard University
[email protected]
Abstract
We propose a general method that parameterizes general
surfaces with complex (possible branching) topology using
Riemann surface structure. Rather than evolve the sur-
face geometry to a plane or sphere, we instead use the fact
that all orientable surfaces are Riemann surfaces and ad-
mit conformal structures, which induce special curvilinear
coordinate systems on the surfaces. We can then automat-
ically partition the surface using a critical graph that con-
nects zero points in the global conformal structure on the
surface. The trajectories of iso-parametric curves canoni-
cally partition a surface into patches. Each of these patches
is either a topological disk or a cylinder and can be confor-
mally mapped to a parallelogram by integrating a holomor-
phic 1-form defined on the surface. The resulting surface
subdivision and the parameterizations of the components
are intrinsic and stable. For surfaces with similar topology
and geometry, we show that the parameterization results
are consistent and the subdivided surfaces can be matched
to each other using constrained harmonic maps. The sur-
face similarity can be measured by direct computation of
distance between each pair of corresponding points on two
surfaces. To illustrate the technique, we computed confor-
mal structures for anatomical surfaces in MRI scans of the
brain and human face surfaces. We found that the result-
ing parameterizations were consistent across subjects, even
for branching structures such as the ventricles, which are
otherwise difficult to parameterize. Our method provides a
surface-based framework for statistical comparison of sur-
faces and for generating grids on surfaces for PDE-based
signal processing.
1. Introduction
Surface-based modeling is valuable for shape analysis,
surface matching and object recognition. For medical imag-
ing applications, it is useful to help analyze anatomical
shape, to statistically combine or compare 3D anatomi-
cal models across subjects, and to map functional imag-
ing parameters onto anatomical surfaces. Parameterization
of these surface models involves computing a smooth (dif-
ferentiable) one-to-one mapping of regular 2D coordinate
grids onto the 3D surfaces, so that numerical quantities can
be computed easily from the resulting models. Even so, it
is often difficult to smoothly deform a complex 3D surface
to a sphere or 2D plane without substantial angular or area
distortion. Here we present a new method to parameterize
general surfaces based on their Riemann surface structure.
By contrast with variational approaches based on surface in-
flation, our method can parameterize surfaces with arbitrary
complexity including branching surfaces not topologically
homeomorphic to a sphere (higher-genus objects) while for-
mally guaranteeing minimal distortion.
1.1. Previous Work
Thirion [14] uses the extremal mesh to describe 3D
smooth surfaces. The extremal mesh is the graph of a
surface whose vertices are the extremal points and whose
edges are the extremal lines. It is invariant with respect to
rigid transformations. Davies et al. [1] describe a method
for building statistical shape models by posing a correspon-
dence problem to identify a consistent parameterization for
each shape in a training set. Several recent advances in sur-
face parameterization have been based on solving a discrete
Laplace system [11, 3]. L´evy et al. [10] describe a technique
for finding conformal mappings by least squares minimiza-
tion of the conformal energy , and Desbrun et al. [2] for-
mulate a theoretically equivalent method for discrete con-
formal parameterization. Sheffer et al. [13] give an angle-
based flattening method for conformal parameterization.
Gu and Yau [6] consider construction of a global conformal
structure for a manifold of arbitrary topology by finding a
basis for holomorphic differential forms, based on Hodge
theory.
Brain surface parameterization has been studied inten-
sively. Schwartz et al. [12] compute quasi-isometric flat
maps of the cerebral cortex. Hurdal and Stephenson [8] re-
port a discrete mapping approach that uses circle packings
to produce “flattened” images of cortical surfaces. Haker
et al. [7] implement a finite element approximation for pa-
rameterizing brain surfaces via conformal mappings. Gu et
al. [4] propose a method to find a unique conformal map-
ping between any two genus zero manifolds by minimiz-
ing the harmonic energy of the map. They demonstrate this
method by conformally mapping the cortical surface to a
sphere.
1.2 Theoretical Background and Definitions
A manifold of dimension
is a connected Hausdorff
space
for which every point has a neighborhood that
is homeomorphic to an open subset of . Such a home-
omorphism is called a coordinate chart. An
atlas is a family of charts for which constitute
an open covering of
(Figure 1). Suppose and PSfrag replacements
Figure 1. The Structure of a Manifold. An
atlas is a family of charts that jointly form an
open covering of the manifold.
are two charts on a manifold
,
then the chart transition is defined as + ,#- !
./#0 !. An atlas on a manifold is called dif-
ferentiable if all chart transitions are differentiable of class 1
. A chart is called compatible with a differentiable atlas
if adding this chart to the atlas still yields a differentiable
atlas. The set of all charts compatible with a given differen-
tiable atlas yields a differentiable structure. A differentiable
manifold of dimension
is a manifold of dimension
to-
gether with a differentiable structure.
For a manifold
with an atlas 3
, if all
chart transition functions,
! : .$#; !, are holomorphic, then 3 is a con-
formal atlas for
. A chart < is compatible with an
atlas 3 , if the union 3 >=?/ is still a conformal atlas.
Two conformal atlases are compatible if their union is
still a conformal atlas. Each conformal compatible equiv-
alence class is a conformal structure. A 2-manifold with
a conformal structure is called a Riemann surface. It has
been proven that all metric orientable surfaces are Riemann
surfaces.
Holomorphic and meromorphic functions and differen-
tial forms can be generalized to Riemann surfaces by using
the notion of conformal structure. For example, a holomor-
phic 1-form @ is a complex differential form, such that in
each local frame A
CB /EDF !, the parametric representa-
tion is @
HG
A
!EI A , where
G
A! is a holomorphic func-
tion. On a different chart < , @
'G
A
A !J!FKL<M
KLON
I
A
.
For a genus P closed surface, all holomorphic 1-forms form
a real QP dimensional linear space.
At a zero point R;S
of a holomorphic 1-form @ , any
local parametric representation @
TG
A
"!EI A
G
,U VW(TXZY
According to the Riemann-Roch theorem, in general there
are Q[P]Q zero points for a holomorphic 1-form defined on
a surface of genus P.
A holomorphic 1-form induces a special system of
curves on a surface, the so-called conformal net. A curve ^5_ is called a horizontal trajectory of @ , if @ `I
^
! a
X
;
similarly,
^
is a vertical trajectory if @ `I
^
!cb
X
. The
horizontal and vertical trajectories form a web on the sur-
face. The trajectories that connect zero points, or a zero
point with the boundary are called critical trajectories. The
critical horizontal trajectories form a graph, which is called
the critical graph. In general, the behavior of a trajectory
may be very complicated, it may have infinite length and
may be dense on the surface. If the critical graph is finite,
then all the horizontal trajectories are finite. The critical
graph partitions the surface into a set of non-overlapping
patches that jointly cover the surface, and each patch is ei-
ther a topological disk or a topological cylinder. Each patch d _ can be mapped to the complex plane using the fol-
lowing formulae. Suppose we pick a base point Re;S
d ,
and any path
^
that connects Re to R. Then if we define
R!
(*fhg @ , the map is conformal, and
d ! is a par-
allelogram. We say is the conformal parameterization of induced by @. maps the vertical and the horizontal
trajectories to iso-u and iso-v curves respectively on the pa-
rameter plane. The structure of the critical graph and the
parameterizations of the patches are determined by the con-
formal structure of the surface. If two surfaces share similar
where
is the valence of vertex D.
Locating Zero Points We find the cluster of vertices with
relatively small conformal factors (the lowest μ]\·¶j¸ ).
These are candidates for zero points. We cluster all the can-
didates using the metric on the surface. For each cluster,
we pick the vertex that is closest to the center of gravity
of the cluster, using the surface metric to define geodesic
distances.
Because the triangulation is finite and the computation is
an approximation, the number of zero points may not equal
the Euler number. In this case, we refine the triangulation
of the neighborhood of the zero point candidate and refine
the holomorphic 1-form @.
2.4. Holomorphic Flow Segmentation
Tracing Horizontal Trajectories Once the zero points
are located, the horizontal trajectories through them can be
traced.
First we choose a neighborhood of a vertex D repre-
senting a zero point, is a set of neighboring faces of D,
then we map it to the parameter plane by integrating @. Sup-
pose a vertex ¹S>
, and a path composed by a sequence
of edges on the mesh is
^
, then the parameter location of ¹
is ¹!
fhg @ .
The map ¹! is a piecewise linear map. Then the hor-
izontal trajectory is mapped to the horizontal line º
nX in
the plane. We slice ! using the line º
X
by edge
splitting operations. Suppose the boundary of ! inter-
sects º
»X
at a point Dj¼ , then we choose a neighborhood of
Ds¼ and repeat the process.
Each time we extend the horizontal trajectory and en-
counter edges intersecting the trajectory, we insert new ver-
tices at the intersection points, until the trajectory reaches
another zero point or the boundary of the mesh. We repeat
the tracing process until each zero point connects ½ horizon-
tal trajectories.
Critical Graph Given a surface
and a holomorphic 1-
form @ on
, we define the graph
q
O@!
[
k <o ,
as the critical graph of @. Here is the set of zero points
of @ ,
k is the set of horizontal trajectories connecting zero
points or the boundary segments of
, and o is the set of
surface patches segmented by
k .
Given two surfaces with similar topologies and geome-
tries, by choosing appropriate holomorphic 1-forms, we can
obtain isomorphic critical graphs, which will be used for
patch-matching described in the next section.
3. Experimental Results
We tested our algorithm on various surfaces, including
a synthetic geometric example, human face surfaces and
anatomic surfaces extracted from 3D MRI scans of the
brain. Figure 2(a)-(d) shows a closed genus 2 surface. We
Figure 2. Holomorphic flow segmentation re-
sults on a synthetic surface and a surface
model of the face. With two cuts intro-
duced (e), the face surface becomes an open
boundary genus 2 surface. (a) and (f) are
conformal parameterizations of the two sur-
faces. (b) and (g) show horizontal trajecto-
ries. (d) and (h) are the two rectangles to
which two segments in (c) and (g) are con-
formally mapped, respectively.
visualized the conformal structure by projecting a checker-
board image back onto the surface (Figure 2(a)). There is a
zero point shown in Figure 2(a). Another zero point is on
the back of the ”figure-eight” shaped surface and is sym-
metric to this zero point. The traced horizontal and vertical
trajectories are shown in Figure 2(c). From the computed
conformal structure, the ”figure-eight” surface can be seg-
mented into two patches (Figure 2(c)). Each patch can then
be conformally mapped to a rectangle (Figure 2(d)). Fig-
ure 2(e)-(g) shows experimental results for a human face
surface. The surface was built with a high resolution, real-
time 3D face acquisition [15]. For a detailed studies of ge-
ometrical differences between faces (e.g. for face tracking
and recognition applications), we can optimize the confor-
mal parameterization by modifying the topology of the face
model. We introduce two cuts on the tip of the nose and
mouth (the blue lines in (e)), so a human face model be-
comes an open-boundary genus two surface. (f) shows its
conformal structure and there is a zero point between the
nose and mouth illustrated by the black dot. The horizontal
trajectory curves are shown in (g). We can conformally map
the face surface to two rectangles (h). Compared with other
face surface parameterization methods, our method can rep-
resent the surface with minimal distortion.
Shape analysis of the lateral ventricles - a structure in
the brain - is of great interest in the study of psychiatric
illnesses, including schizophrenia, and in degenerative dis-
Figure 3. Surface parameterization results
for the lateral ventricles. The upper row
shows models parameterized using holomor-
phic 1-forms, for a 65-year-old subject with
HIV/AIDS and the lower row shows the same
maps computed for a healthy 21-year-old
control subject. The left column shows that μ
cuts are introduced and they convert the lat-
eral ventricular surface into a genus 4 sur-
face. The computed conformal structure,
holomorphic flow segmentation and their as-
sociated parameter domains are also shown.
eases such as Alzheimer’s disease. These structures are of-
ten enlarged in disease and can provide sensitive measures
of disease progression. For the lateral ventricular surface in
each brain hemisphere, we introduce five cuts. Since these
cutting positions are at the ends of the frontal, occipital, and
temporal horns of the ventricles, they can potentially be lo-
cated automatically. The left column in Figure 3 shows 5
cuts introduced on two subjects ventricular surfaces. Af-
ter the cutting, the surfaces become open boundary genus
4 surfaces. The rest of Figure 3 shows parameterizations
of the lateral ventricles of the brain. The upper row shows
the results of parameterizing a ventricular surface for a 65-
year-old patient with HIV/AIDS (note the disease-related
enlargement) and the lower row shows the results for the
ventricular model of a 21-year-old control subject. The sur-
faces are initially generated by using an unsupervised tissue
classifier to isolate a binary map of the cerebrospinal fluid
in the MR image, and tiling the surface of the largest con-
nected component inside the brain. There are a total of 3
zero points on each of the ventricular surfaces. Two of them
are located at the middle part of the two ”arms” (where the
temporal and occipital horns join at the ventricular atrium),
as shown by the large black dots in the second row. The
third zero point is located in the middle of the model, where
the frontal horns are closest to each other. Based on the
computed conformal structure, we can partition the surface
into 6 patches. Each patch can be conformally mapped to a
rectangle. Although the two brain ventricle shapes are very
different, the segmentation results are consistent in that the
surfaces are partitioned into patches with the same relative
arrangement and connectivity.
Not only are our results consistent on two different ven-
tricle meshes, the bijective conformal mapping of each sur-
face patch to rectangles in the parameter domain induces
a parametric grid onto each surface, providing a way for
direct surface matching between the two ventricles. One
way to do this is to use a constrained harmonic map, ¾
, where
and
are two surfaces to be matched.
The basic pipeline is as follows: first we manually label the
corresponding feature points. Then we Delaunay triangu-
late one segment based on the feature points, and induce
the same triangulation for the corresponding segment of the
second surface. By using a piecewise affine transformation,
we map the second segment to the first one and denote the
resulting mapping by e. We improve the mapping by us-
ing a constrained harmonic map using the heat diffusion
method, ¿
y }Àr
À
\
ÂÁ? rÃE! <
X
e. After that, we re-
sample the meshes using a regular grid in the parameter do-
main and construct new meshes with the same connectivity
for the two segments.
It is difficult to find directly, but instead we can easily
find a harmonic map between the parameter domains. Sup-
pose the the conformal parameterization of
is Ä 7
, confor-
mal parameterization for
is Ä
, then Ä 7
! and Ä
are rectangles in . We want to find a harmonic map
ÄÅF Â , such that Ä 4 Ä 7
G
Ä
G
!EÄ 4 Ä
CÆ
Ä
`Æ
!<ÁÇÄ
ÈX
, where Á is the Laplacian operator de-
fined on the plane. Then the map can be obtained by
Ä
4 Ä 4 Ä^6 /
(^). Because both Ä 7
and Ä
are conformal, Ä
is harmonic, and therefore is harmonic. Once we get , we can explicitly compute the distance between two surfaces based on the surface correspondence by É
Ê |«°Ê
ÌË<Í
¦¤Î¯
6 ί
ZÏ (^) y+Ð
K
Î
ËÍ
Kί
(2)
With Equation 2, we computed a surface distance be-
tween various examples of lateral ventricle and human face
models. The upper row in Figure 4 shows three left brain
lateral ventricular surfaces. The left two are for control sub-
jects and the right one is for an HIV/AIDS patient. For each
surface, we introduce three cuts and turn them into genus Q
surface and conformally map them to two rectangles. The
distance between the left two surface is
{XZY X
μ and the dis-
tance between the right two is
hÑZY Ò μ
. It demonstrates the
intra-class distance for control subjects is far less than the
inter-class distance between control and HIV/AIDS classes.
Thus our technique is useful to combine and compare 3D
anatomical models across subjects or map functional imag-
