Research Proposal: Computational Geometry with Applications on Medical Images

Transcription

1 Research Proposal: Computational Geometry with Applications on Medical Images MEI-HENG YUEH National Chiao Tung University 1 Introduction My research mainly focuses on the issues of computational geometry and their applications. In particular, the surface parameterization problem is one of the widely studied issues. A surface parameterization is a bijective mapping that maps a surface to a simply shaped domain. Large deformations can be easily handled by the surface parameterization while preserving much geometric information. The surface parameterization has been widely applied to tasks of digital image and geometry processing, such as surface registration, surface resampling, surface remeshing and texture mapping. Most commonly used parameterizations for surfaces are based on minimizing either the angle or area distortions. An angle-preserving mapping is also called a conformal mapping while an area-preserving mapping is also called an equiareal mapping. Figure 1 shows the mesh models of Human Brain, and its conformal and equiareal parameterizations. Practical applications of 3D animation and medical image analysis can be smoothly carried out via the one-to-one correspondence between the surface and the domain of simple shape. However, when it comes to real-time applications, the efficiency still needs to be dramatically improved. To overcome the bottleneck of the computational efficiency, we need to study the structure of the linear systems based on numerical analysis and matrix computations. (a) (b) (c) Figure 1: (a) The mesh model of Human Brain. (b) The conformal parameterization of the human brain surface. (c) The equiareal parameterization of the human brain surface. 1

2 1.1 Goals and Overview The goal of my research is to develop efficient and reliable algorithms of surface parameterizations for the real-time applications in medical image analysis. Based on the recent works that we have done, the task is to generalize our proposed parameterization algorithms to deal with surfaces of higher genus, and the 3-manifolds as well. This proposal is organized as follows. In Section 2, we summarize some previous works for computing conformal and equiareal parameterizations of surfaces. In Section 3, we introduce the research topics of the proposal. 2 Previous Works In the recent decade, several numerical methods for the computation of surface parameterizations have been developed by different research groups. Table 1 summarizes some previous works on the surface parameterizations of a given shape of the domain. Table 1: Previous works on manifold parameterizations. Method D Properties Finite Element Method [1, 2] S 2 conformal, efficient Holomorphic Differentials [3] C conformal Heat Flow [3, 4] S 2 conformal Fast Stretch Minimization [5] D equiareal Quasi-Implicit Euler Method [6] S 2 conformal FLASH [7] S 2 conformal Fast Disk Map [8] D conformal Linear Disk Map [9] D conformal Conformal Energy Minimization [10] D conformal, efficient Stretching Energy Minimization [11] D equiareal, efficient Optimal Mass Transportation [12] B volume-preserving In 1999, Angenent et al. [1] proposed an algorithm for computing conformal parameterizations of genus zero closed surfaces by solving a Laplace-Beltrami equation with three-point constraint using the finite element method (FEM), which is efficient but not accurate. The angular distortion might be especially large around the constrained vertices. Two years later, Gu and Yau [3] proposed a systematic algorithm for the conformal parameterizations of closed surfaces of genus larger than or equals to 1 using the holomorphic differentials. For genus zero closed surfaces, Gu and Yau [3] proposed an algorithm based on the nonlinear heat flow. The mapping is more accurate that has less angular distortion but it is more 2

3 time-consuming. The conformal parameterizations of genus zero closed surfaces have been applied to the registration problem between surfaces of human brains [4]. A decade later, Huang et al. [6] further improved the efficiency for the convergence of the heat flow by applying the quasi-implicit Euler method (QIEM). However, the mapping produced by the heat flow is usually not bijective, especially when the geometry is very far from convex. A year later, Choi et al. [7] proposed a fast landmark aligned spherical harmonic (FLASH) algorithm for the computation of conformal parameterizations of genus zero closed surfaces, which usually produces a bijective mapping in general. For the simply connected open surfaces, Choi and Lui [8] proposed an algorithm for computing conformal parameterizations based on the composition of Cayley transforms and quasiconformal mappings. To further improve the efficiency of computations, Choi and Lui [9] developed a linear algorithm for computing conformal parameterizations based on the composition of the spherical conformal mapping [7] of the double covered surface and a quasiconformal mapping. 3 Research Topics In this research, we aim to develop efficient numerical algorithms for the following tasks. 1. Conformal/equiareal parameterizations of arbitrary surfaces. 2. Volume-preserving parameterizations of 3-manifolds. 3. Applications to the problems arising from medical images. In this section, we introduce each topic in detail in Sections 3.1, 3.2, and 3.3, respectively. 3.1 Parameterizations of Arbitrary Surfaces In 2017, we have proposed an efficient conformal energy minimization (CEM) algorithm [13, 10] for computing conformal parameterizations of genus zero surfaces with any number of boundaries, as illustrated in Figure 2. The algorithm is based on minimizing the conformal energy functional defined by E C ( f ) = f 2 dv M A( f ) M in which A( f ) denotes the area of the image f (M). It is known that E C ( f ) = 0 if and only if f is conformal. Our numerical results indicate that a conformal parameterization can be computed in less than one second for a mesh model of more than 100,000 triangular elements by the proposed 3

4 (a) (b) Figure 2: (a) The model of Ho s Face with Holes. (b) The conformal parameterization obtained by the generalized CEM algorithm. v j v l α i,j α j,i v k v i Figure 3: An illustration for the cotangent weights. CEM algorithm. The existence of a nontrivial accumulation point of CEM algorithm is guaranteed. On the other hand, we have proposed a novel stretch energy minimization (SEM) algorithm [13, 11] for the computation of equiareal parameterizations of simply connected open surfaces as well. The algorithm is based on minimizing the discrete stretching energy defined as E S (f) = 1 2 w i,j (f) f i f j 2, [v i,v j ] E(M) where w i,j (f) is the stretched cotangent weight given by ( ω i,j (f) = 1 ( cot αi,j (f) ) 2 σ f ([v i, v j, v k ]) + cot ( αj,i (f) ) ) σ f ([v j, v i, v l ]) in which α i,j (f) and α j,i (f) are two angles opposite to the edge [f i, f j ] connecting vertices f i and f j on C, as illustrated in Figure 3, σ f is the local stretch factor with respect to f defined by σ f ([v i, v j, v k ]) = [v i, v j, v k ] [f i, f j, f k ]. 4

5 Figure 4: The chess horse model and its fundamental domain. The existence of nontrivial limit points of the SEM algorithm is guaranteed under some mild assumptions of the mesh quality. Numerical experiments indicate that the efficiency, accuracy, and robustness of the proposed SEM algorithm outperform other state-of-the-art algorithms. On this topic, we aim to generalize the efficient energy minimization algorithms of the conformal and equiareal parameterizations to deal with genus-one and higher genus surfaces, respectively. In particular, for the genus-one surfaces, note that its universal covering is conformally equivalent to the complex plane C. Intuitively, the conformal mapping between the surface and its fundamental domain can be approached by minimizing the conformal energy on C with an appropriate boundary constraint, as illustrated in Figure 4. If the idea works well, a similar approach might be applied to the parameterizations of surfaces of higher genus. 3.2 Parameterizations of 3-manifolds In the medical image, it is even more important to develop efficient parameterization and registration algorithms for volume mesh due to the fact that the images obtained from magnetic resonance imaging (MRI) are solid. Although several algorithms for volume mesh parameterization based on optimal mass transportation have been proposed by some research groups. It would take several hours for solving a volume mesh parameterization problem by using the currently existing algorithms, which is impractical in real-time applications. In other words, the efficiency of the algorithms still needs to be dramatically improved. On this topic, we aim to develop an efficient volume-preserving parameterization algorithm for 3-manifolds based on the stretching energy minimization. First, the boundary vertices are mapped to the unit sphere by the spherical harmonic mapping. Then the mapping for the interior 5

6 vertices are obtained by solving a Laplace-Beltrami equation in R 3. To reach a volume-preserving mapping, an iterative process is required to minimize the stretching factor for the volume metric. 3.3 Applications to Medical Images Surface parameterizations have been applied to deal with the issues from the medical image analysis [14, 15]. However, the efficiency of the computations is not satisfactory, especially when the data size is huge. Due to this reason, the technology of parameterization cannot be widely used in clinical medicine at present. To overcome this issue, we aim to apply the proposed algorithm to solve the problem efficiently. Some of the problems arising from medical images are listed as follows. 1. Brain Mappings and Deformations The deformation of human brain is subtle and not easy to be observed. In order to efficiently detect the deformation, the one-to-one correspondence between two brain surfaces is necessary. With the efficient parameterization algorithms for genus zero closed surfaces, the registration problem between two brains is reduced into the registration problem on the unit sphere, which is much easier since the shape of domain are identical. Then the deformation from one brain to another can be visualized via the homotopy between the identity mapping and the registration mapping between two brains. Figure 5 shows the surfaces of healthy brain (top) and the slightly depauperated brain (bottom), respectively. A demo video of the brain deformations can be found at 2. Area-Preserving Intestine Flattening Polyps are the predecessor of intestinal cancers. However, polyp detection is usually not an easy task, especially when the polyps are hidden in the folds. In order to efficiently detect the polyps on the intestinal surface, an area-preserving mapping can be applied to flatten the surface to a domain of rectangle, so that the polyps can be easily found in the rectangle domain. Then the exact locations of polyps can be obtained by the one-to-one correspondence between the intestinal surface and the rectangle domain. Figure 6 shows a surface of intestine model (left) and its area-preserving parameterization (right). A demo video of the area-preserving intestine flattening can be found at https: //mhyueh.github.io/intestine.html. 3. Hippocampus Deformation It is known that Alzheimer s disease can be detected by the deformation of the hippocampus. The deformation can be measured via 6

7 Figure 5: The healthy brain (top) and the depauperated brain (bottom), and their spherical parameterizations. Figure 6: The surface of intestine model (left) and its area-preserving parameterization (right). 7

8 the one-to-one correspondence between two surfaces of hippocampi. Similar as the brain mappings, the mapping between two hippocampi can be obtained via the registration mapping on the spherical domain of parameterization mappings. Figure 7 shows the original surface (left) and the deformed surface (right) of a hippocampus. The color represents the quantity of the displacement. A demo video of the hippocampus deformation can be found at io/hippocampus.html. Figure 7: The original surface (left) and the deformed surface (right) of a hippocampus. The color represents the quantity of the displacement. References 1. S. Angenent, S. Haker, A. Tannenbaum, and R. Kikinis. On the Laplace- Beltrami operator and brain surface flattening. IEEE Trans. Med. Imaging, 18(8): , S. Haker, S. Angenent, A. Tannenbaum, R. Kikinis, G. Sapiro, and M. Halle. Conformal surface parameterization for texture mapping. IEEE Trans. Vis. Comput. Graph., (2): , X Gu and S.-T. Yau. Computing conformal structures of surfaces. Communications in Information and Systems, 2(2): , X. Gu, Y. Wang, T. F. Chan, P. M. Thompson, and S.-T. Yau. Genus zero surface conformal mapping and its application to brain surface mapping. IEEE Trans. Med. Imaging, 8: , S. Yoshizawa, A. Belyaev, and H. P. Seidel. A fast and simple stretchminimizing mesh parameterization. In Proceedings Shape Modeling Applications, 2004., pages , June

9 6. W.-Q. Huang, X. D. Gu, T.-M. Huang, S.-S. Lin, W.-W. Lin, and S.-T. Yau. High performance computing for spherical conformal and Riemann mappings. Geom. Imag. Comput., 1(2): , P. T. Choi, K. C. Lam, and L. M. Lui. FLASH: Fast landmark aligned spherical harmonic parameterization for genus-0 closed brain surfaces. SIAM J. Imaging Sci., 8(1):67 94, P. T. Choi and L. M. Lui. Fast disk conformal parameterization of simplyconnected open surfaces. J. Sci. Comput., 65(3): , G. P.-T. Choi and L. M. Lui. A linear formulation for disk conformal parameterization of simply-connected open surfaces. Advances in Computational Mathematics, pages 1 28, M.-H. Yueh, W.-W. Lin, C.-T. Wu, and S.-T. Yau. An efficient energy minimization for conformal parameterizations. J. Sci. Comput., 73(1): , M.-H. Yueh, W.-W. Lin, C.-T. Wu, and S.-T. Yau. A novel stretch energy minimization algorithm for equiareal parameterizations arxiv: [cs.gr]. 12. K. Su, W. Chen, N. Lei, J. Zhang, K. Qian, and X. Gu. Volume preserving mesh parameterization based on optimal mass transportation. Computer- Aided Design, 82:42 56, Isogeometric Design and Analysis. 13. M.-H. Yueh. Surface Conformal and Equiareal Parameterizations with Applications. PhD thesis, National Chiao Tung University, R. Jiang, H. Zhu, W. Zeng, X. Yu, Y. Fan, X. Gu, and Z. Liang. Bladder wall flattening with conformal mapping for MR cystography, H. L. Chan, H. Li, and L. M. Lui. Quasi-conformal statistical shape analysis of hippocampal surfaces for Alzheimer s disease analysis. Neurocomputing, 175(Part A): ,