Detection of Shape Deformities using Yamabe ﬂow and Beltrami coefﬁcients

We address the problem of detecting deformities on elastic surfaces. This is of great importance for shape analysis, with applications such as detecting abnormalities in biological shapes (e.g., brain structures). We propose an effective algorithm to detect abnormal deformations by generating quasi-conformal maps between the original and deformed surfaces. We ﬁrstly ﬂatten the 3D surfaces conformally onto 2D rectangles using the discrete Yamabe ﬂow and use them to compute a quasi-conformal map that matches internal features lying within the surfaces. The deformities on the elastic surface are formulated as non-conformal deformations, whereas normal deformations that preserve local geometry are formulated as conformal deformations. We then detect abnormalities by computing the Beltrami coefﬁcient associated uniquely with the quasi-conformal map. The Beltrami coefﬁcient is a complex-valued function deﬁned on the surface. It describes the deviation of the deformation from conformality at each point. By considering the norm of the Beltrami coefﬁcient, we can effectively segment the regions of abnormal changes, which are invariant under normal (non-rigid) deformations that preserve local geometry. Furthermore, by considering the argument of the Beltrami coefﬁcient, we can capture abnormalities induced by local rotational changes. We tested the algorithm by detecting abnormalities on synthetic surfaces, 3D human face data and MRI-derived brain surfaces. Experimental results show that our algorithm can effectively detect abnormalities and capture local rotational alterations. Our method is also more effective than other existing methods, such as the isometric indicator, for locating abnormalities.


I. INTRODUCTION
Detecting abnormal changes on surfaces is a central problem in shape analysis, especially in medical research. For example, neuroscientists commonly aim to identify abnormal deformations of cortical and subcortical structures in the brain in order to detect systematic patterns of alterations in brain diseases. In cardiac imaging and oncology, physicians commonly need to track changes or abnormalities in biological organs or tumors, to evaluate the effectiveness of different treatments, or monitor disease progression.
Detecting and examining abnormalities with the human eye is inefficient and often inaccurate, especially on complicated surfaces such as the cerebral cortex of the brain. Therefore, it is of great importance to develop automatic methods to detect abnormalities and track abnormal geometric changes over time.
In this paper, we propose a framework for detecting abnormal changes on elastic surfaces using quasiconformal geometry. Two issues arise in solving this problem. First, we want to find good registrations between surfaces with enforced landmark-correspondences. Second, we want to define a robust measure of deformity, which is invariant under normal (non-rigid) deformations that preserve local geometry, so that quantitative analysis can be carried out. These goals are related, since good registrations allow accurate detection of abnormalities, while good measures of deformity can be used to set criteria for desirable registrations.
We propose to model deformities between elastic shapes as non-conformal deformations, whereas normal deformations that preserve the local geometry are formulated as conformal deformations. This is a generalization of existing work by Lord et al. and Unal et al. [9] [19], which modeled deformities between shapes as non-isometric diffeomorphisms. Using the isometric indicator, the authors proposed to detect abnormalities that are invariant under rigid transformations. Our proposed method is an extension of their approaches, in that we are trying to detect deformities on elastic surfaces that are invariant under non-rigid normal deformations. This is a more general and accurate definition in some situations. For example, we may consider the growth processes of the human body, which is a key area of study in medical imaging and face recognition. As a person grows, different parts of the human body grow locally and the growth rate is somewhat uniform locally [17]. The same organs of two healthy individuals tend to have similar shapes, although they may not be exactly isometric to each other. In other words, local geometry is well preserved under the normal growth process. Conformal maps are well-known to preserve local geometry. Therefore, a good registration between two organs of the same or different individuals should not be far from conformal. This motivates us to define good registrations as conformal registrations, and measure deformities as regions of non-conformality.
The key contribution in this paper is the use of Beltrami coefficient to detect shape abnormalities.
Suppose we are given a registration between a normal and a deformed surface of the same or different subjects. We can compute the Beltrami coefficient of the registration map, which is a complex-valued function measuring the non-conformal deformation at each point of the map. In our examples, we show that the Beltrami coefficient is an effective measure of shape deformities on elastic surfaces.
To complete our framework for the automatic detection of shape deformities, we propose the following registration algorithm. We first generate a quasi-conformal map between the original 3D surface and the surface to be compared with it. The surfaces are first conformally flattened onto 2D rectangles using a novel Yamabe flow method, which computes accurate conformal parameterizations of surfaces. Then, a quasi-conformal map is computed to match landmark features between both flattened surfaces. By formulating abnormal changes as non-conformal deformations, we detect abnormalities by computing the Beltrami coefficient, which is uniquely associated with the quasi-conformal map. By considering the norm of the Beltrami coefficient, we detect regions with abnormal changes, which are invariant under conformal deformation. Furthermore, by considering the argument of the Beltrami coefficient, we can capture abnormalities induced by local rotational changes. We propose a quantity called the Beltrami index, which allows us to quantitatively describe the degree of abnormality in each region. Experiments applying our algorithm to synthetic surfaces, 3D human face data and real MRI-derived brain surfaces show that our algorithm can effectively detect abnormalities and capture local rotational alterations. The algorithm is also robust under added noise, and it is successful in detecting altered gyrification patterns on cortical surfaces. Quantitative comparisons of the size of abnormality with the Beltrami index show that the detected sizes of abnormalities correspond positively with the sizes of abnormalities. A comparison of the Beltrami coefficient with other commonly used measures like the isometric indicator and gradient show that these measures are not as effective as Beltrami coefficient and may tend to give misleading results when the compared surfaces undergo normal changes that preserve the local geometry.
Our paper is organized as follows: prior work on related topics is presented in section 2. The basic mathematical theory is discussed in section 3. In Section 4, the details of our proposed model are discussed.
In Section 5, our computer algorithm is summarized. Experimental results are discussed in Section 6, and some conclusions and future work are discussed in section 7.

II. PREVIOUS WORK
Many different approaches have been proposed to detect changes in shapes. Most of them take into account properties that depend on the embedding of the shape in space. Tosun et al. [18] proposed the use of three different shape measures -the shape index, curvedness, and L2 norm of mean curvatureto quantify cortical gyrification and complexity. While the curvature measure is an important geometric measure of surface properties, it is also affected by healthy changes like normal growth, which increases surface size and decreases curvature. Moreover, since the curvature is a second order geometric measure, it is more sensitive to noise in data, especially in triangular meshes. Extra preprocessing is required to guarantee the robustness of the measure. In contrast, the Beltrami coefficient we use in this paper is a first order geometric measure. This leads to less perturbation due to noise. By computing surface Jacobian and applying statistical inference via random field theory on cortical surfaces processed by diffusion smoothing, Chung et al. [2] studied changes in cortical surface area, thickness, and curvature, as well as the change of the total gray matter volume over time. While surface Jacobian can detect area changes caused by abnormalities, it can also be affected by healthy deformations like normal growth, which also causes area changes of surface registrations. By looking at the Beltrami coefficients, we are able to minimize the effects caused by such changes. We are also able to detect rotational distortions caused by abnormalities.
Shi et al. [15] suggested the use of Hamilton-Jacobi skeletons on cortical surfaces to study gyrification patterns in Williams syndrome. In this approach, the degree of gyrification is measured by the number of branches in the Hamilton-Jacobi skeleton. While this method is easily computable and gives a fast measure of complexity of the cortical surface, subtle changes could occur within a branch of the cortical surface, which require a finer measure to detect. To overcome this problem, our approach provides an local and intrinsic measure of surface distortion to detect such changes.
Registration is broadly used in medical imaging and face recognition. It aims to produce a desirable registration for obtaining good correspondences or performing further analysis. Grenander et al. [4] surveyed existing methods of computational anatomy and proposed a framework to compute deformation maps and empirical probability laws for disease testing. Early work such as Gaser et al. [3] compute an initial correspondence field between surfaces to ease the detection of abnormalities. Such applications often involve models of the human face or organs of the human body. Some existing methods model deformities between shapes in terms of nonrigid or non-isometric diffeomorphisms [9] [19]. This may make their algorithms classify normal growth in living organisms as abnormal. This motivates us to model abnormalities as non-conformal diffeomorphisms, and good registrations as ones that are as conformal as possible. We support our view with the following points. First, normal changes such as growth in humans tend to preserve local geometry well, and hence are conformal. Second, by modeling good registrations as those that are conformal, we are still able to get a correct registration when the proper registration is indeed isometric.
In most methods for detecting shape changes, an accurate registration between two shapes has to be computed. The area has been extensively studied. Unal et al. [19] proposed the use of coupled PDEs for joint segmentation and registration. However, technique does not support the matching of user-defined landmarks. Pohl et al. [14] proposed an expectation maximization-based method to solve the problem, which takes into account of image artifacts, anatomical labelmaps, and a structure-dependent hierarchical mapping from the atlas to the image space. This allows some controls over the matched areas, but userdefined landmark constraints may not be matched. Lord et al. [9] [8] proposed to match hippocampal surfaces by finding maps minimizing the deviation from isometry under the constraints of piecewise deformation homogeneity, with local asymmetry quantified. The authors defined asymmetries between paired shapes as non-rigid diffeomorphisms between shapes. However, under normal conditions, like growth, non-rigid shape changes naturally occur. In one of our examples, it is shown that the maps of a correctly matched pair of surfaces can be non-isometric in normal as well as abnormal regions. Such occasions call for a more flexible definition of shape abnormality.
It has become increasingly popular to study shape changes by computing diffeomorphisms with desired properties, as they give one-to-one correspondences of shapes for local comparisons. Using control theory and large-deformation continuum mechanics, variational metrics have been developed on the space of diffeomorphisms [4] [20]. In computing diffeomorphisms, one of the goals is to preserve local geometry as far as possible while consistently aligning important anatomical features lying within a surface. Gu et al. [6] proposed to find an optimal Möbius transformation between two surfaces to minimize a landmark mismatch error. Wang et al. [21][11] defined a new energy functional based on harmonic energy to optimize the conformal parameterization of cortical surfaces, while fixing landmark correspondences. The harmonic energy method was applied by Shi et al. [16] to match implicitly-defined surfaces by solving PDEs on them using level set methods. Using integral flows of smooth vector fields, Lui et al. [10] further proposed to compute a landmark-matching diffeomorphism with shape-based correspondences between landmark curves.
Recently, conformal geometry has been used for 3D non-rigid surface matching and registration.
Harmonic maps have been used to track non-rigid motions of 3D surfaces, which requires tracking of boundary correspondences. This challenge was addressed in [23] [25], where different conformal mappings induced by different holomorphic differentials were combined to achieve accurate and robust matching results. Even so, holomorphic differentials cannot handle surfaces with complicated topology; the Ricci flow method is much more flexible and general. The surface-based Ricci flow was introduced in [7]. The theory of discrete surface Ricci flow was introduced in [1]. Euclidean Ricci flow has been applied to 3D shape analysis in [5], where all the Gaussian curvatures of the surface are concentrated on several singularities. To remove these singularities, the discrete hyperbolic Ricci flow was introduced in [22], which improves the accuracy and robustness of the matching and registration. In practice, the Ricci flow method imposes constraints on the triangulation of the discrete surface, whereas the discrete Yamabe flow can handle much more general triangulations. Furthermore, the Yamabe flow can achieve a higher level of conformality than the Ricci flow. The theory of discrete Yamabe flow is established in [12], where the existence, uniqueness and stability of the solution is rigorously proven.

III. MATHEMATICAL BACKGROUND
In this section, we describe some basic mathematical concepts relevant to describing our algorithm.
A surface S with a conformal structure is called a Riemann surface. Given two Riemann surfaces M and N , a map f : M → N is conformal if it preserves the surface metric, up to a multiplicative factor called the conformal factor. An immediate consequence is that every conformal map preserves angles. With the angle-preserving property, a conformal map effectively preserves the local geometry of the surface structure.
A generalization of conformal map is called the quasi-conformal map which is an orientation-preserving homeomorphism between Riemann surfaces with bounded conformality distortion, in the sense that the first order approximation of the quasi-conformal homeomorphism takes small circles to small ellipses of bounded eccentricity. Thus, a conformal homeomorphism that maps a small circle to a small circle can also be regarded as quasi-conformal. Mathematically, f is quasi-conformal provided that it satisfies the Beltrami's equation: for some complex-valued Lebesgue-measurable µ satisfying |µ| ∞ < 1. In terms of the metric tensor, consider the effect of the pullback under f of the usual Euclidean metric ds 2 E ; the resulting metric is given by: which, relative to the background Euclidean metric dz and dz, has eigenvalues (1+|µ|) 2 ∂f ∂z and (1−|µ|) 2 ∂f ∂z . µ is called the Beltrami coefficient, which is a measure of conformality. In particular, the map f is conformal around a small neighborhood of p when µ(p) = 0. Also, the gradient and the Beltrami coefficient of the map are closely related. Suppose f = (f 1 , f 2 ) under the conformal parameter domains. The Jacobian matrix J of f can be written as (∇f 1 , ∇f 2 ) which is a 2 × 2 matrix. The Jacobian is closely related to |µ f |. Mathematically, Infinitesimally, around a point p, f may be expressed with respect to its local parameter as follows: with shrinking factor 1 − |µ(p)|. The distortion or dilation is given by: Thus, the Beltrami coefficient µ gives us all the information about the conformaity of the map (See Figure   1). This motivates its use as a measure of abnormal deformation, as abnormalities tend to cause significant distortions in the conformality of the deformation map. It is often convenient to conformally map 3D surfaces to planar domains, to simplify subsequent computations. Yamabe flow offers a rigorous and efficient way to do this. Suppose S is a surface embedded in the 3D Euclidean space R 3 , then it has the induced Euclidean metric denoted by g = (g ij ). Let u : S → R be a function on S, thenḡ = e 2u g is another Riemannian metric of S, which is conformal to g. If the Gaussian curvatures induced by g andḡ are K andK, then they are related by the following where ∆ g is the Laplace-Beltrami operator under the original metric g.
It is fundamentally important to design a Riemannian metricḡ with a prescribed curvatureK, which is equivalent to solving the Yamabe equation. The surface Yamabe flow (or equivalently the Ricci flow) is a powerful tool for solving it. Intuitively, the surface Yamabe flow deforms the metric in proportion to the Gaussian curvature, such that the Gaussian curvature evolves according to a heat diffusion process.
It has been proven [12] that ifK ≡ 0 everywhere and the total area is preserved during the flow, the Yamabe flow converges and leads to a metric with constant Gaussian curvature. Note that, for higherdimensional Riemannian manifolds, the Ricci flow and Yamabe flow are different. On discrete surfaces, the discrete Yamabe flow produces better conformality than the discrete Ricci flow.

IV. OUR PROPOSED MODEL
In our method, we propose to use the Beltrami coefficient to detect abnormalities on surfaces. This problem. Second, a quasi-conformal map is obtained between the two surfaces by computing a harmonic map between their conformal parameter domains, which also accommodates constraints to match internal landmarks based on their shapes. This gives a one-to-one correspondence (registration) between the two surfaces. Finally, the Beltrami coefficient associated with the quasi-conformal map is computed as an index of abnormal changes. We describe each step in detail next.

A. Conformal parameterization using the Yamabe Flow
To detect regions with abnormalities, the first step is to find a one-to-one correspondence (registration) between the original and deformed surfaces. With this registration, we can detect the distortion in conformality of the deformation at each point by computing the associated Beltrami coefficient. Computing the surface registration directly on surfaces is difficult, especially on complicated surfaces such as the cortical surface of the brain. Therefore, it is advantageous for us to first parameterize the surfaces conformally onto the 2D parameter domain. This simplifies the process by transforming the surface problem into a 2D problem. Under the conformal parameterizations, determining conformality distortion between surfaces is equivalent to determining it between their conformal parameter domains. Of course, the computation of conformal parameterizations has to be accurate in order to effectively detect changes in conformality. The Yamabe flow (6) provides us with an efficient and accurate way to compute these conformal maps.
In practice, all surfaces are approximated by piecewise linear triangular meshes M (V, E, F ), where V , E and F represent the vertex, edge and face set of the mesh. We use v i to denote the i-th vertex, The discrete metric is the edge length function ℓ : E → R + satisfying triangle inequality. The vertex discrete curvature is defined as angle deficiency, where ℓ 0 ij is the original edge length in R 3 . The discrete Yamabe flow is defined as The discrete Yamabe flow converges, and the final discrete metric induces the prescribed curvature; a detailed proof can be found in [12]. The discrete Yamabe flow is the negative gradient flow of the following Yamabe energy, where u 0 = (0, 0, · · · , 0). This energy is convex and has a unique global minimum, which corresponds to the desired metric. Using Newton's method, the Yamabe energy can be optimized very efficiently. The Hessian matrix H = (h ij ) of the Yamabe energy has an explicit form. If [v i , v j ] is an edge on the mesh, To compute a conformal mapping of a simply connected surface S, we set the target curvature to be zeros for all interior vertices, and the total curvature of boundary vertices to be 2π. For example, in order to map a human face surface to a rectangle, we select four corner vertices, and set their target curvatures to be π 2 . The target curvatures for all other vertices are zeros. Using the Yamabe flow, we can obtain the desired metric. Then we flatten the mesh isometrically face by face using the resulting metric. Figure   2 shows two examples of conformal surface mappings using Yamabe flow method. The conformality is demonstrated using checkerboard texture mapping.

B. Quasi-conformal Map between Surfaces
After parameterizing the surfaces conformally, the next step is to determine a quasi-conformal registration between the conformal parameter domains. There may be important curves on the surfaces representing landmark features [24], so we look for the optimized harmonic diffeomorphism that exactly matches landmark features via a variational approach. However, minimizing the energy functional over the search space of landmark-matching surface diffeomorphisms is difficult. Following [10], we formulate our

∂Φ ∂t
⃗ Y is the projection of the vector field ⃗ X i such that it is tangential to C. This ensures the exact landmark matching property off i . The vector fields ⃗ X i = a i ∂ ∂x + b i ∂ ∂y minimizes the following energy functional: The variational problem is then formulated to be defined over the space of C 1 smooth vector fields on Ω.
The last integral in the energy is the smoothness term for the vector fields ⃗ X i . The first two integrals are   the harmonic terms, which aim to preserve the harmonicity of the parameterization as much as possible.
The second term is a symmetric shape term defined as an arc length integral, where the shape measure is defined according to the curvature. The proposed energy functional can be minimized by modifying the vector field iteratively according to the following Euler-Lagrange equation: where: Df 0,i |∇H(ϕ)|; Ψ i is the orthogonal fundamental matrix for the homogeneous system of Figure 3 shows a schematic diagram of our quasi-conformal map construction. Figure 4 shows how the constraint vector field can be used to obtain the desired registration with exact landmark matching based on the shape information. Figure Figure 5 illustrates the matching results for cortical surfaces with several sulcal landmarks labeled. Figure 5(A) shows brain surface 1 with several landmarks labeled. It is mapped to brain surface 2 under the conformal parameterization as shown in Figure 5(B). The sulcal landmarks on Brain 1 are only mapped approximately to the sulcal regions on Brain 2. Figure 5(C) shows the matching result under the parameterization we proposed. The corresponding landmarks are mapped exactly. Also, the correspondence between the landmark curves follows the shape information (corners to corners; see the black dot). Using harmonic maps for registration is beneficial as they tend to preserve conformality as much as possible. They effectively capture the region of conformal deformation and help to identify regions of non-conformal deformation. By computing the Beltrami coefficient µ of the registration, we can detect abnormalities. In cases where the deformed surface does not undergo any abnormal (non-conformal) deformations, the harmonic map obtained will be close to conformal and µ will approximately be zero.

C. Computing the Beltrami coefficient
Given the conformal parameterizations together with the registration between the conformal parameter domains, we can detect the abnormal region by computing the Beltrami coefficient. The Beltrami coefficient measures the change in conformality. Suppose the conformal parameterizations of S o and S d are respectively. Denote the registration between the parameter domains by give a landmark matching harmonic registration between the original and deformed surface. By computing the Beltrami coefficient µF ofF , we can detect which point on the surface undergoes abnormal (non-conformal) deformation. The Beltrami coefficient is invariant under conformal maps. In other words, |µ f •g | = |µ g | and |µ g•f | = |µ f | if g is conformal. Since ϕ o and ϕ d are both conformal diffeomorphisms, we have: Thus, to compute µF , it suffices to compute µ F . Since F is a map defined on the 2D complex plane, we can compute its Beltrami coefficient easily by: The Beltrami coefficient is a complex-valued function. When the deformation near a point p is conformal, then µF (p) = 0. At the point where abnormal deformation happens around its immediate neighborhood, µF will be non-zero. We can easily detect regions of abnormality by computing the norm of the Beltrami coefficient |µF |.
To measure the degree of abnormality quantitatively, we introduce a measure called Beltrami Index (BI) which measures the average of the Beltrami coefficient over a certain region. Mathematically, it is defined as: The Beltrami Index can effectively measure the degree of abnormalities in different regions on the surface (see Figure 17).
Also, as described in section 3, the argument of µF measures the angle of complex dilation. This can be used to describe any local rotational change of the abnormal shape. By representing µF as a vector field ⃗ V = (Re(µF ), Im(µF )) on the parameter domain, we can easily visualize the rotational change of the abnormal shape (see Figure 9).

V. COMPUTER ALGORITHM
The computer algorithm can be summarized as follows: on the parameter domain.

VI. EXPERIMENTAL RESULTS
We tested our proposed algorithm on synthetic surfaces, 3D human face data and MRI-derived models of human brain cortical surfaces, to detect abnormalities.
In Figure 6, we tested our proposed algorithm using synthetic data. Figure 6(A) shows the original surface. Figure 6(B) shows the deformed surface with both conformal and non-conformal deformations. shows the distribution of µ as a complex number. It represents a vector field on the conformal parameter domain-this can be used to visualize the rotational change of the abnormal shape. Again, it is observed that µ can be used effectively as an indicator to segment the abnormal regions on the surface (see the shaped segmented region on the surface in Figure 7(B)).
In Figure 8, we illustrate our idea on a human face.   Figure 9(A) shows a human face with an abnormally swollen area. In Figure 9(B), the swollen area is rotated. Figure 9(C) plots the complex-valued Beltrami coefficients µ on the parameter domain. In Figure 10, we test our proposed algorithm on noisy data. Figure 10(A) shows the original clean human face. Figure 10(B) shows the deformed human face with noise. Figure  In Figure 12, 13 and 14, we illustrate our proposed algorithm on a human brain cortical surface. Figure   12 shows the original brain surface (Brain 1) and the deformed brain surface (Brain 2) with abnormal gyral thickening. The gyral thickening can be observed inside the circled region. Figure 13 shows a zoom-in of the abnormal region. Gyral thickening is clearly observed. Figure 13(A) shows the plot of |µ| versus the parameter domain. Figure 13(B) shows the distribution of |µ| in color. Again, the Beltrami coefficient µ can effectively detect the gyral thickening region.  Figure 15 shows another example of detecting abnormalities on brain surfaces. Figure 15(A) shows the original brain surface. Figure 15(B) shows the deformed brain surface with gyral thickening in the circled regions. Figure 15(C) shows the distribution of |µ|. The Beltrami coefficient µ again clearly reflects the region of abnormal changes on the human brain cortical surface. Figure 16 illustrates how we can use Beltrami coefficient to capture the gyrification pattern on the human brain surface. Figure 16(A) shows a real human brain cortical surface. Figure 16(B) shows the deformed cortical surface. The brain has undergone different degrees of gyrification at different regions.
Figure 16(C) shows the colormap of the brain determined by the Beltrami coefficient. Red color means high value of Beltrami coefficient whereas blue color means low value of Beltrami coefficient. Figure   16(C) shows the colormap on the conformal parameter domain. Observe that the colormap of the Beltrami coefficient effectively captures the gyrification pattern on the brain surface. Figure 17 shows how we can qualitatively measure the degree of abnormalities using the Beltrami Index (BI). The top shows a series of brain surfaces which are undergoing more and more gyral thickening.
The bottom shows the plot of the Beltrami Index (BI) of each deformed brain surfaces. Observe that the value of the Beltrami Index becomes bigger as the gyral thickening becomes more severe. It shows that the Beltrami Index (BI) can effectively measure the degree of the abnormal changes.

VII. CONCLUSION AND FUTURE WORK
In conclusion, we developed an effective algorithm to detect abnormalities on surfaces using quasiconformal geometry. To do this, we computed a landmark-matching harmonic registration between the      original and deformed 3D surfaces, together with its associated Beltrami coefficient, µ. Experimental results show that the Beltrami coefficient can effectively detect regions with abnormalities, which are invariant under normal (conformal) deformation. By visualizing µ as a vector field defined on the parameter domain, we can capture the rotational change of the abnormal shape. In future, we will apply our algorithm to study human brain diseases such as Williams syndrome, which results from geneticallymediated developmental abnormalities in cortical folding. We will also develop more efficient numerical schemes to speed up the computation.

Numerical Implementation of the Quasi-conformal Registration Algorithm
We describe how the quasi-conformal registration algorithm can be implemented. In practice, all surfaces are represented by meshes which consists of vertices, edges and triangular faces. The functions and their partial derivatives in the iterative scheme are defined on each vertex and linearly interpolated to define the value on each triangular face. They can be computed as follows: • Laplacian of a function F can be computed as: ∆F = ∑ [u,v]∈Nv k uv (F (v) − F (u)), where: N v is a set of triangles around v that forms a neighborhood of v; k uv = (cot α + cot β)/2, where α and β   are the opposite angles of the edge [u, v]. For detail, please see [13].  , t))dt). Suppose the interval [0, 1] is discretized as: s 0 = 0 < s 1 < ... < s n = 1. Ψ i (⃗ x, s) can be computed as: )(s j − s j−1 )).
• The function δ ϵ is defined to be a positive function that is compactly supported in (−ϵ, ϵ) and can be computed mathematically as: δ ϵ (x) = 1 a(ϵ) √ π exp(− x 2 a(ϵ) 2 ) • η ep is a smooth function on Ω such that η ep = 0 at the endpoints of the open curves Γ k ⊂ C, k = 1, 2, ..N . It can be computed mathematically as: a 1 , a 2 , ..., a 2N are the set of end points of the landmark curves.
• The initial map f 0i maps the landmark curves C i to the common curve C standard,i . It can be computed as follows: Given a set of landmark curves C i (t) on the parameter domain and a set of corresponding common curves C standard,i (t). Starting from the initial map f 0 = Id, we can iteratively flow the map to get a diffeomorphism which matches C i (t) to C standard,i (t) as follow. We can define a vector fields on f n (C i (t)) as ⃗ V n (t) = C standard,i (t) − f n (C i (t)) and smoothly extend to the parameter domain by Gaussian convolution G * ⃗ V n (t). The iterative scheme can then be written as: f n+1 = f n + dt ⃗ V n (f n ).