1. 2D and 3D Shape Analysis using Conformal Mapping
Allyson Scott
Physics 101
Abstract
Shape analysis is a technique of static code analysis, which detects qualities of
connected and dynamically distributed data structures of geometric shapes and inputs that
information into a computer program [1]. While this sort of code analysis is typically
used to search out program bugs and validate the accuracy of software, its applications
are far more widespread making it a useful tool in the fields of archaeology, architecture,
medical imaging, and computer design.
Introduction
When studying and comparing 2-
dimensional shapes, it is crucial to draw
an association to vision and how humans
classify objects from discerned
silhouettes. Interpreting distances
between 2-dimensional shapes
constructs a metric space, which is
mathematically structured and crucial to
the classification of objects [1]. A
specific metric space is derived from
conformal mappings of 2-dimensional
shapes onto one another and this can be
drawn from the theory of Teichmuller
spaces [5]. In particular, any given
object in this metric space is assigned a
unique function that is analytic, periodic,
and invertible and does not change under
transformations such as scaling and
translation. Similarly, 3-dimensional
shape analyses use geometric invariants
that exist in metric space. While
previous methods such as
parameterization have worked in the
past, the use of symmetry in conformal
mapping ensures that the image of the
mapping is symmetric and the distortion
area on the mapping is also symmetric.
Shapes morph through Conformal
Mapping
It is first important to define a
shape as a closed and smooth surface,
which is where the Cauchy-Riemann
Integration Theories come into play. The
conformal mapping theorem states that
unit discs can be conformally mapped to
the inside of another shape. There are
four main methods of conformal
mapping that include harmonic, least
square conformal, holomorphic
differentials, and Ricci Flow maps. [2]
Figure 1: Genus surfaces and their values.[6]
While harmonic maps are linear methods
of conformal mapping, they can only be
applied to spheres and disks (genus zero
surfaces) [7] and Holomorphic
Differentials can have multiple genus
surfaces [8]. The least square conformal
map can quite simply be connected to
the Cauchy Riemann Equation and
lastly, the Ricci Flow mappings have no
limitations but are nonlinear. It is
important to note that all of these
methods are extremely affected by
boundaries and if boundaries are
inconsistent, conformal mappings are

2. altered. This proves troublesome when
analyzing and differentiating shapes.
One of the most important features of
conformal mapping is its ability to
preserve symmetry on surfaces.
Figure 2: Conformal Mapping of Human Faces
with different expressions. [2]
Figure 1 provides a visualization of
mapping of humans; it takes the surface
of a face and overlays it with a
symmetric mapping, which is then
spread out as if it were a flat surface. In
order to conceptualize the symmetry of
conformal mapping, Figure 2 shows that
points can be chosen on a symmetric
plane that intersects the surface of a
shape (in this case the shape is a human
face). The relationship between the
conformal mapping over the surface of
the face and the symmetrically preserved
mapping becomes clear.
Figure 3: Preservation of symmetry through
conformal mapping. [1]
Mathematics of Conformal Mapping
Three dimensional surfaces have
a Euclidean metric (g) which can be
represented as g=exp(2u)(dx^2+dy^2) if
they constitute a conformal structure. If
two regions are overlapping on the
surface of the shape, the transition
function is analytic and is thus a
Riemann Surface. A mapping of
Riemann surfaces is conformal if it
satisfies the condition that picking
complex coordinates (x+iy) outputs a
holomorphic display of the function (f).
Cauchy- Riemann Equation
∂u/∂x = ∂v/∂y, ∂u/∂y = -∂v/∂x
z=x+iy w=u+iv where u and v are functions
of x and y
By solving this equation with boundary
conditions, you can determine a
conformal mapping.
Harmonic Mapping
A simple way to think about the
harmonic map is to consider two
materials, each whose shape is defined
by their own metrics. Say you are
mapping plastic wrap (P) onto a
Styrofoam ball (S), then the map is
labeled by Φ: PS. This map
determines how the plastic is applied to
the Styrofoam. Φ is considered harmonic
if the plastic is already in an equilibrium
position when it is released from
potential energy and thus stays intact and
wrapped around the Styrofoam.
Consider this a map that is reluctant to
expand in orthogonal directions. A
harmonic map can be achieved by using
the equation:
df/dt = -Δf

3. Where Δ is the Laplace operator. If the
domain that you are aiming for is
convex, the boundary mapping is a
homeomorphism and the harmonic map
is a diffeomorphism. If the surface is a
genus zero surface then the harmonic
map is conformal as well.
Practical Uses of Harmonic Mapping
Conformal mapping of spherical
harmonic transformations as well as
shape analysis can be utilized to produce
mappings of human brain surfaces [2].
To complete this mapping you must use
an eigenfunction of the Laplace operator
Δf. The dimensions of the subspace of
L^2 is invariant under rotations and the
expansion factor of a spherical mapping
is scaled by a Fourier coefficient. The
equation for the spherical harmonic is as
follows:
ϒ(θ,Φ)=κΡ(cosθ)exp(imΦ)
In this equation, P is the Legendre
function and k is the normalization
factor. For a more detailed definition of
the Spherical harmonic equation, see [2].
The spherical equation can be dissected
into three functions, which represent a
conformally mapped surface. These
equations are as follows:
x^0(θ,Φ),x^1(θ,Φ), and x^2(θ,Φ)
Figure 4: There are two brain surfaces, A and B,
and a chosen landmark [3] cut as boundaries.
Then with that, a hyperbolic embedding of A and
B on a Poincare disk [9] can be created.
Decomposition can be done which will take it to
a convex hexagonal shape under the Klein
model. From there, a 1-to-1 map overlaying A
and B can be outputted. In the final stage of this
diagram, a heat diffusion algorithm is used to get
a harmonic diffeomorphism and is then color
coded to showthe results of A and B.
Using what is known about
spherical harmonics, an analysis can be
performed of compressed geometric
brain data. Geometric data in low
frequency regions are concentrated so
using filters that only allow low
frequencies through, geometric features
can be recorded and the brain surface
can be compressed [3].
In order to compare two different
brain surface images, a shape descriptor
that is shape invariant and based upon
coefficients of frequency must be
contrived.
Figure 5: Shows the compression stages using
S.H.: (a) is the original brain surface and (b), (c),
and (d) are reconstructed surfaces using fractions
of the original low-frequency coefficients. [3]
Experimental Results of Brain
Mapping
The solution to comparing
geometric brain information from
conformal mappings uses covariant
differentials to solve nonlinear PDE’s. In
comparison to other algorithms that are
used to solve linear PDE’s, this

4. nonlinear algorithm proves to be useful
for brain surface mapping. A key feature
is that every point on the brain is seen in
a uniform way so no point tends to
infinity. With this condition, there aren’t
regions with distortion. This algorithm is
also quite vague because it does not
require that the target surface be a sphere
so generalization makes solving the
nonlinear PDE a simpler task [3].
Using 3-dimensional MRI
images, brain meshes are reconstructed
using an active surface method that
shapes a triangulated mesh onto the
surface of the brain [3]. Researchers
scanned the brains at different times and
due to fluctuations of scanner noises and
biological changes that occurred during
the time lapse, the geometric information
that was collected shows slight
differences in the conformal mappings.
Discussion
2-dimensional and 3-dimensional
conformal mappings prove to be
extremely useful to the medical field and
with more research can be shown to aid
in many different occupations. Using the
Cauchy Riemann formula along with
spherical equations and the LaPlace
operator, it becomes a simple task to
preserve angles of shapes through
conformal mapping.
While researching conformal
mapping, it became clear that such
unusual results were found simply
through complex analysis. Being able to
find real life examples of conformal
mapping helped to solidify my
understanding of this topic.
References
[1] M. Stegmann, D. Gomez. “A Brief
Introduction to Statistical Shape
Analysis” Informatics and
Mathematical Modeling.
Technical University of Denmark.
March 2002.
[2] W. Zeng, J. Hua, X. Gu. “Symmetric
Conformal Mapping for Surface
Matching and Registration”.
Internation Journal of CAD/CAM.
Vol. 9, No. 1, pp. 103-109.
[3] X. Gu, Y. Wang, T. Chan, P.
Thompspon, S. Yau. “Genus Zero
Surface Conformal Mapping and
Its Application to Brain Surface
Mapping” IEEE Transactions on
Medical Imaging, Vol. 23, No. 8,
August 2004
[4] R. Shi, W. Zeng, Z. Su, H. Damasio,
Z. Lu, Y. Wang, S. Yau, X. Gu.
“Hyperbolic Harmonic Mapping for
Constrained Brain Surface
Registration” Department of Comp.
Science, Stony Brook University.
[5] E. Sharon, D. Mumford. “2D-Shape
Analysis using Conformal
Mapping” Division of Applied
Mathematics. Brown University.
[6] “Genus”
[7] “Harmonic Map” Wikipedia.
Wikimedia Foundation, n.d. Web.
05 Feb. 2016.
[8] “Complex Differential Form”
[9] “Chapter 9: Poincare’s Disk Model
for Hyperbolic Geometry” Univ.
of Kentucky, Materials sciences.
2008.