Matching 3D Shapes Using 2D Conformal Representations

1
Matching 3D Shapes Using 2D Conformal
Representations Xianfeng Gu1, Baba Vemuri2
Computer and Information Science and Engineering,
Gainesville, FL 32611-6120, USAgu_at_cise.ufl.edu,
vemuri_at_cise.ufl.edu.
Abstract Matching 3D shapes is a fundamental
problem in Medical Imaging with many applications
including, but not limited to, shape deformation
analysis, tracking etc. Matching 3D shapes poses
a computationally challenging task. The problem
is especially hard when the transformation sought
is diffeomorphic and non-rigid between the shapes
being matched. In this paper, we propose a novel
and computationally efficient matching technique
which guarantees that the estimated non-rigid
transformation between the two shapes being
matched is a diffeomorphism. Specifically, we
propose to conformally map each of the two 3D
shapes onto the canonical domain and then match
these 2D representations over the class of
diffeomorphisms. The representation consists of a
two tuple , where, is the conformal
factor required to map the given 3D surface to
the canonical domain (a sphere for genus zero
surfaces) and H is the mean curvature of the 3D
surface. Given this two tuple, it is possible to
uniquely determine the corresponding 3D surface.
This representation is one of the most salient
features of the work presented here. The second
salient feature is the fact that 3D non-rigid
registration is achieved by matching the
aforementioned 2D representations. We present
convincing results on real data with synthesized
deformations and real data with real
deformations. Mathematical
Framework for the Proposed Model Surfaces can be
represented as functions defined on their
conformal coordinate systems. Suppose and
are two surfaces we want to match. Suppose the
conformal coordinate domains of and
are and respectively. The conformal
mapping from to is , the one from
to is . Instead of finding the
mapping from to directly, we want to
find a map , such that
Thus, finding a diffeomorphic between
and can be achieved by finding a
diffeomorphism from to and
then using the commutative diagram. 2D
Conformal Representation for surfaces in 3D We
can conformal map surface S to a canonical
domain, such as the sphere, then
stereographically project the sphere to the
complex plane, and use these conformal
coordinates (u,v) to parameterize S. Then, We
can compute the following two maps directly
from the position vector S(u,v). First, the
conformal factor map is the conformal factor
function defined on (u,v), and
conceptually represents the scaling factor at
each point. Secondly, the mean curvature
function, where is the
Laplacian-Beltrami operator defined on S, n(u,v)
is the normal function, H(u,v) is the mean
curvature, conformal factor
function. The tuple is the
conformal representation of S(u,v). Theorem 1
Conformal Representation surface S(u,v) is
parameterized by some conformal parameter (u,v)
on a domain D, then the conformal factor
function and mean
curvature function H(u,v) defined on D satisfy
the Gauss and Codazzi equation. If
and H(u,v) are given, also the boundary condition
is given, then S(u,v)
can be uniquely reconstructed. 3D Shape
matching Suppose we have two genus zero surfaces
and embedded in . Our goal is to
find a diffeomorphism ,
such that minimizes the following
functional If we want to find a conformal map
between and , we can restrict
to be a Mobius transformation if and
are spheres. The position map S(u,v) is
variant under rigid motion, and the conformal
factor and mean curvature H(u,v)
are invariant under rigid motion. Therefore, it
is most efficient to use and H(u,v)
for matching surfaces. The matching energy can be
also defined as This energy is minimized
numerically to obtain the optimal and then
obtain the corresponding from the
commutative diagram shown above.

• 
  Experimental Results
• We define non-rigid motion denoted by ,using
  affine maps.
  
  is represented as
  
• The brain surface S is deformed by a randomly
  generated , then we match S with
  using conformal representation, and estimate
  using the algorithms described in 3D shape
  matching, denoting the estimated transformation
  as .We computed the reconstruction error by
  comparing the original coefficients of and
  the estimated coefficients of . The result
  is illustrated in the following table
• Numerical experiments results. The deformation
  coefficients of are generated by a random
  variable with Gaussian distribution. The
  difference between and the reconstructed
  deformation is measured, the mean is close to
  0 and the variance is very small. This
  demonstrates the accuracy of the algorithm.

of Tests
20 0.3 0.47 33.33 47.14 4.02e-4 0.538e-4 7.27e-4 4.69e-4
20 1.9 2.21 10.00 4.08 2.7333e-2 1.0607e-2 3.2645e-2 1.6135e-2
20 3.9 4.36 20.00 8.16 1.55339e-1 9.3372e-2 1.09864e-1 7.1456e-2
20 1.8 2.21 10.00 4.08 2.7333e-2 1.0607e-2 3.2645e-2 1.6135e-2
20 0.34 0.47 3.33e-3 4.714e-3 1.079e-3 9.83e-4 1.595e-4 3.05e-4