2D and 3D Shape Analysis using Conformal MappingAllyson ScottPhysics 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

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: PS. 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

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 show the 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

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.