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Section 2. Mathematics
Duka Anila, Ph D., candidates in mathematics, the Faculty of Natural Siences NdricimSadikajUniversity "Ismail Qemali", Vlora E-mail: anila.duka@univlora.edu.al, ndricim.sadikaj@univlora.edu.al
APPLICATION OF CONFORMAL AND QUASICONFORMAL MAPPINGS IN HUMAN SPLEEN
Abstract: We will use the algorithm in [2] which is a general method for global conformal pa-rameterizations based on the structure of the cohomology group of holomorphic one-forms for surfaces with or without boundaries Gu and Yau, 2002, Gu and Yau, 2003. For genus zero surfaces, this algorithm can find a unique mapping between any two genus zero manifolds by minimizing the harmonic energy of the map. In this paper, we apply the algorithm to spleen surface matching problem. We use a mesh structure to represent the spleen surface. This algorithm is more stable, compared with other genus zero surface conformal mapping algorithms.
Keywords: Tuette energy, quasiconformal mapping, Riemann surfaces.
1. Introduction and meningitis are fight by it.The spleen surface is a
Conformal mapping are mappings which pre- genus zero surface, so, we will use these algorithms
serves the angles, and quasiconformal mappings are generalization of them, in [1] you can find more information. Conformal geometry has been widely applied in medical imaging and computer graphics, such as in brain registration and texture mapping, where the mappings are constructed to be as conformal as possible to reduce geometric distortions.
Examining deformations with human eye is inappropriate and inefficient, for this reason is good to develop automatic methods to track and detect abnormalities.
The spleen is very important in human body, it plays multiple supporting roles.
It acts as a filter for blood as part of the immune system. There old red blood cells are recycled and there are stored platelets and white blood cells. Certain kinds of bacteria that cause pneumonia
only for the case of genus zero surface.
1.1 Previews work
For surfaces with arbitrary topologies, Gu and Yau [2] introduce a gen-eral conformal parameterization based on a nonlinear flow for the genus zero case, and on the structure of the cohomology group of holomorphic one-forms in the case of genus greater than one. In this paper, we apply part of these algorithms (for the genus zero case) to the spleen surface matching problem and report our experimental results. We know that all orientable surfaces are Riemann surfaces. If two surfaces can be conformally mapped to each other, they have the same conformal structure. So, computing conformal mappings is the same as computing conformal structures for surfaces. Harmonic maps are equivalent to conformal maps of genus zero closed surfaces. There are
APPLICATION OF CONFORMAI. AND QUASICONFORMAL MAPPINGS IN HUMAN SPLEEN
many algorithms for surface parameterization which are based on harmonic maps. To achieve conformal parameterizations are used Harmonic energy minimization, Cauchy-Riemann equation approximation, Laplacian operator linearization, angle based method, circle packing.
1.2 Mathematical work
Suppose S2 are two surfaces. Locally they can be represented as r1(x1, x2), r2(x1, x2) where (x1, x2) are their local coordinates, and r1,r2: R2 ^R3 are vector-valued functions. We find the first fundamental form of them and define a mapping between two surfaces. We use the pull-back metric and we call f a conformal mapping, if there exists a positive scalar function, such thatf f *ds2 = X(x\x2)sl, where X(x1,x2) is called the conformal factor. All the angles on are preserved on.
It is well known that any genus zero surface can be mapped conformally onto the sphere and any local
portion thereof onto a disk. This mapping, a conformal equivalence, is one-to-one, onto, and angle-preserving. Moreover, the elements of the first fundamental form remain unchanged, except for a scaling factor.
Conformal mappings are similarities in the small. Since the spleen surface is a genus zero surface, conformal mapping offers a convenient method to take local geometric information.
This algorithm depends only on the surface geometry and is invariant to changes in image resolution and the specifics of the data triangulation. Given two genus zero meshes M1 M2, there are many conformal mappings between them.For genus zero surfaces S1,S2 f: S1 ^ S2 is conformal if and only if f is harmonic. All conformal mappings between S1 S2 form a group, the Möbius group.
Our method is summarized as follows: we first find a homeomorphism h between S1 S2 then deform h such that h minimizes the harmonic energy.
Figure 1. Decomposition of Laplacian operator
Definition 1: All piecewise linear functions defined on form a linear space, denoted by CPL(K).
In practice, we use CPL(K) to approximate all functions defined on K. So the final result is an approximation to the conformal mapping. The higher the resolution of the mesh is, the best approximated conformal mapping is.
Definition 2: Suppose a set of string constants are as-Signed k{u v} for each edge {u, v}, the inner product on CPL is defined as the quadratic form
f,g >=11 ku (((u)-/(v))(g(u)-g(v)).
Definition 3: Suppose f eCPL, the string energy is defined as
E (f ) =< f, f k,}|| f (u )-f (v )||2.
<
{u ,v}eK
The energy is defined as the norm on CPL.
{u ,v}eK
If k{u v} = 1, is called Tuette Mapping.
We see the other definitions in [2].
2. Conformal map for genus zero closed surfaces
Genus 0 closed surfaces can be conformally parameterized over a unit sphere, and harmonic maps of these surfaces are equivalent to conformal maps. We use a Gauss map as the initial map, and then we use the heat flow method to reduce the harmonic energywith special constraints. The final harmonic
map is a global con-formal parameterization. By composing it with a Möbius map of the sphere, we can obtain all possible global conformal parameter-izations. The Möbius transformation on the complex
и (z) = (az + b) ,\,ad - bc ^ 0. v ' /(cz + d)
A genus zero open surface can be globally con-formally parameterized by the unit disk. Two such kinds of parameterizations differ by a Möbius transformation defined on the disk.
3. Mesh parametrization
In the digital geometry, surfaces are mostly represented as triangular meshes, because they are supported by graphics hardware. For display purposes all surfaces need to be converted to triangular meshes [2] In theory, any surface with C1 continuity can be triangulated, here are based theories of simplical homology and cohomologies in topology.
In practice, all surfaces in real life can be digi-talized with 3D scanners.
A triangular mesh is a two-dimensional simplical complex in algebraic topology.
Or we can say that a mesh is a set of triangular faces coherently glued together.
Connectivity gives us all topological information.
A mesh is embedded in R3, all vertices have Euclidean coordinates and all faces are planar triangles. The lengths of all edges are induced by The Euclidean metric of R3.
Except at the vertices a mesh is flat everywhere. Meshes help us to solve topological problems, but for differential forms, curvatures, geodesics and conformal mapping can only be solved by approximation.
Mesh parametrization has many applications like Detail Mapping, Texture Mapping, Normal Mapping, Detail Transfer, Morphing, Mesh Completion, Editing, Databases, Remeshing, Surface Fitting.
4. Algorithms
Algorithms used here are Spherical Tuette Mapping and Spherical Conformal Mapping.
Algorithm 1: Spherical Tuette Mapping: Input (mesh M, step length, energy difference threshold
SE ), output (t : M — S2), where t Tuette energy.
1) Compute Gauss map N: M —S2. Let t = N, compute Tuette energy E .
2) For each vertex v e M, compute absolute derivative At.
3) Update i (v) by öT(v) = -At (v)St
4) Compute Tuette energy E.
5) If \f\ E - E01| <SE, return t. Otherwise, assign E to E0 and repeat steps 2) -5).
The Tuette energy has a unique minimum, the algorithm converges rapidly and is stable. We use it as the initial condition for the conformal mapping.
Algorithm 2: Spherical Conformal Mapping: Input (mesh M, step length, energy difference threshold SE ), output (h : M—S2), where h minimizes
The harmonic energy and satisfies the zero masscenter constraint.
1) Compute Tuette embedding t. Let h = t, compute Tuette energy E0.
2) For each vertex v e M, compute absolute derivative At.
3) Update h (v) by Sh(v ) = -At (v)öt.
4) Compute Möbius transformation (p_0: S2 —S2, such that Г(ф) = J<p oh d oM, (pa Möbius(S2 ),p0 =
M
= min ||ф|| , where d o M, is the area element on M, Г(<р) is the mass center and (p minimizes the norm in the mass center condition. Möbius (S2) is the conformal automorphism group of (s2 ), and it can be analytically represented as т _10 -вот, where т : S2 C is the stereographic projection
-(p ) =
7
ip (x ,7, z )eS2
.1 - z 1 - z,
and 0 is a Möbius transformation as defined in Definition 11 in [2].
5) compute the harmonic energy E.
6) If |/| E - E0|| < SE, return t, return. Otherwise, assign E to E0, assign and repeat Step 2) through to Step 6).
This approximation method is good enough for our purpose.The resulting angle distortion is propor-
APPLICATION OF CONFORMAL AND QUASICONFORMAL MAPPINGS IN HUMAN SPLEEN
mesh, n is the number of required iterations, n mainly depends on the initial condition, so, how close it is to a conformal map, n also depends on the step length. In [2] Table II illustrates the CPU time for computing conformal maps of surfaces of different triangle numbers on a 1.9-GHz PC with the Windows XP operating system. This nonlinear algorithm has some advantages: 1) Every point on the spleen is treated in a uniform way, so, no point maps to infinity as in other algorithms. This means that, there are no specific areas with large distortion; 2) The method is general, because it does not require the target surface to be a sphere. It can be used to compute harmonic maps between any two arbitrary genus zero surfaces.
If the distortion is greater we have a quasiconfor-mal mapping, so we can use other methods to find it. It is known that the growth is conformal, but disease is quasiconformal. We can use Beltrami coefficient method or Yamabe flow or other methods to test if there is a problem. 6. Conclusions
The algorithm finds a unique conformal mapping between genus zero manifolds. So, we can detect abnormalities in early stages in spleen surface. This method depends only on the surface geometry, not on the mesh structure and resolution. This algorithm is very fast and stable.
References:
1. Ahlfors L. V. Lectures on quasiconformal mappings, University Lecture Series 38(2006), Amer. Math. Soc., Providence, RI.
2. Gu X., Wang Y., Chan T. F., Thompson P. M., and Yau Sh. Genus Zero Surface Conformal Mapping and Its Application to Brain Surface Mapping. Ieee Transactions On Medical Imaging,- Vol. 23.- No. 8.-August, 2004.
3. Gu D., Sh-T. Yau. Computational Conformal Geometry, ALM3, ISBN978-1-57146-171-1.
4. Periodic conformal maps. Sechelmann S., Rorigh T.
5. Springborn B., Schroder P., and Pinkall U. 2008. Conformal equivalence of triangle meshes. ACM Trans. Graph. 27, 3 (Aug.), 77: 1-77: 11.
6. Levy B. S. Petitjean N. Ray J. Mallyot. Least Squares Conformal Maps for Automatic Texture Atlas Generation. SIGGRAPH 02
7. Jin M., Wang Y., Gu D., Yau Sh-T. Optimal Global Conformal Surface Parameterization.
tional to the square of the distance between the mass center and the origin.
With some constraints, u maps infinity to infinity, u = az + b, piq are landmarks [7] on the surfaces. So the functional of u is:
« 2
E(u) =Yg(z>)lazi +b-r4 ,
i=1
where z{ is the stereographic projection of ^, T is the projection of ^ ang g is the conformal factor from the plane to the sphere,
4
g (z) =-3, so it is converted in a Least Square
1 + zz Problem.
5. Application in Human spleen
Human spleen is a genus zero surface, we can use the algorithm for genus zero to find a conformal spherical map.
Figure 1. 3D human spleen and spherized spleen
This algorithm uses covariant differentiation to solve a geometric nonlinear partial differential equation. The complexity of the algorithm is O(m, n), where m is the number of the vertices of the spleen
