臺灣博碩士論文加值系統

摘 要
在醫學影像分析中，圖像配準（image registration）是根據給定的兩張影像選取特定的特徵點集合（landmark points），其中P在第一張影像I0上，Q在第二張影像I1上，透過這些點集合可以造出兩張影像一對一且映成的對應關係。LDDMM是目前廣泛使用的非剛性圖像配準（non-rigid image registration）方法之一。LDDMM的計算可以當作是一個最佳化問題。因此，選擇一個合適的起始值（initial）是非常重要的。此篇論文主要的目標是利用thin-plate spline 和möbius transformation的圖像配準方法造出一個計算LDDMM合適的起始值，我們使用的方法不同於以往Marsland and Twining 造出的起始值。並且利用我們計算出的起始值去計算LDDMM造出一個diffeomorphic map。我使用以下的步驟製造起始值：首先我們用möbius transformation 表先出兩組點的仿射變換（affine transformation）。接下來使用thin-plate spline的方法造出一條線性的路徑，這就是我們造出合適的起始值。在論文的最後，我們根據特徵點個數測試不同資料造出的diffeomorphic map和以往Marsland and Twining 造出的起始值計算的LDDMM去比較。並且，我們使用一些線性組合的技巧讓LDDMM可以合理的應用在大腦影像上。

Image registration in medical images analysis is to find a corresponding map via landmar- ks, p and q which is prescribed in two given images, respectively. LDDMM is one of the most commonly used methods for non-rigid medical image registration. Computation of LDDMM is an optimization problem. It is important to find a suitable initial for LDDM- M computation. The goal of this thesis is to find the suitable initial. In this thesis, the initial of computing LDDMM is obtained from the thin-plate spline and möbius transformation, instead of original initial path constructed by Marsland and Twining. We use following steps to construct the initial. First, we find p ̂ by applying möbius transformation on p in order to perform the affine registration. Next, we use thin-plate spline method to find a lin- ear path from p ̂ to q. Finally, a diffeomorphic map is constructed by LDDMM based on geodesic spline interpolation. The proceess of computing the initial is also demonstrated. To examine the initial path, the deformation fields obtained by computing the LDDMM with different initial are listed for comparison. At the end of the thesis, we apply the LDD- MM on brain image registration.

Contents iv
List of Tables v
List of Figures vi
1 Introduction 1
2 Image Registration 4
2.1 Rigid Registration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Non-Rigid Registration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Affine Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Mobius Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Thin-plate Splines(TPS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Large Deformation Diffeomorphic Metric Mapping 16
3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Large Deformation Diffeomorphic Metric Mapping (LDDMM) . . . . . . 17
3.2.1 Large Deformation Diffeomorphic Metric Mapping . . . . . . . . . . . . 17
3.2.2 Computing Large Deformation Diffeomorphic Metric Mapping . . . . 18
3.3 Optimizing the LDDMM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Implementation initial for LDDMM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 Numerical Results 29
5 Conclusion 41

[1]. Vincent Camion, Laurent Younes. Geodesic interpolation splines. In M.A.T. Figueiredo, J. Zerubia, A K. Jain ed., volume 2134 of Energy Minimization Methods in Computer Vision and Pattern Recognition, Lecture notes in Computer Science, pages 513-527. Springer-Verlag, 2001.
[2]. A. M. Mills. Optimizing the paths for geodesic interpolating splines. Master's thesis, University of Manchester, 2003.
[3]. John B. Conway. Functions of one complex variable. Springer-Verlag, 1978.
[4]. S. Marsland and C. Twining. Measuring geodesic distances on the space of bounded diffeomorphisms. British Machine Vision Conference (BMVC), 2002.
[5]. Fred L. Bookstein. Principal Warps: Thin-Plate Splines and the Decomposition of Deformation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(6):567-585, 1989.
[6]. Gianluca Donato, Serge Belogie. Approximate Thin Plate Spline Mappings. A. Heyden et al. ECCV 2002, LNCS 2352, pp. 21-32, 2002.
[7]. N. Chumchob, K. Chen. A robust affine image registration method. International Journal of Numerical Analysis and Modeling 6 (2)pp. 311-334, 2009.
[8]. M.F. Beg, M.I. Miller, A. Trouve, L. Younes. Computing large deformation metric mappings via geodesic flows of diffeomorphisms. Int. J. Comput. Vis., 61 (2) (2005), pp. 139-157 (February).
[9]. J. Modersitzki. Numerical Methods for Image Registration. Oxford University Press, 2004.
[10].Dianne P. O'Leary. Scientific Computing with Case Studies. Chapter 12 Achieving a Common Viewpoint: Yaw, Pitch, and Roll. SIAM Press, 2009.
[11].T. J. Willmore. A survey on Willmore immersions. In Geometry and Topology of Submanifolds. IV, pp. 11-16. World Sci. Pub., 1992.
[12]. Y.-S. Liu, H.-B. Yan, and R. R. Martin. As-rigid-as-possible surface morphing. Journal of Computer Science and Technology, 26:548-557, 2011.
[13]. D. DeCarlo and J. Gallier. Topological evolution of surfaces. In WA Davis,
editor, Proc Graphics Interface, pp. 194–203. Canadian. Information Processing Society, Toronto, Ont., Canada, 1996.
[14]. M.P. do Carmo. Differential Geometry of Curves and Surfaces. Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1976.
[15]. S. Marsland and C. Twining. Splines and diffeomorphisms. May 2003.
[16]. R. Courant and F. John. Introduction to Calculus and Analysis, volume II.
John Wiley and Sons, 1974.
[17]. L. Fonseca and M. Costa. Automatic registration of satellite images. Brazilian Symposium on Graphic Computation and Image Processing, volume 10, pp. 219-226, Los Amitos, IEEE Computer Society, 1997.
[18]. T. Sabisch, A. Ferguson, and H. Bolouri. Automatic registration of complex images using a self organizing neural system. In Proc. IEEE International Joint Conference on Neural Networks, 1998.
[19]. B. Zitova, J. Flusser. Image registration methods: a survey. Image Vis. Comput., vol. 21, pp.977 -1000, 2003.
[20]. T. Boggio. Sulle funzioni di green d'ordine m. Circolo Matematico diPalermo, 20:97-135, February 1905.
[21]. Duchon, J. Splines minimizing rotation-invariant seminorms in Sobolev spaces, in Constructive Theory of Functions of Several Variables, Lecture Notes in Mathematics 571, W. Schempp and K. Zeller, eds., Springer-Verlag (Berlin), pp. 85-100, 1997.
[22]. Harald E. Krogstad. Least Squares Optimization. Harald E. Krogstad, rev. 2010.
[23]. Y. X. Yuan, A review of trust region algorithms for optimization. In: J.M. Ball and J.C.R. Hunt, eds., ICM99: Proceedings of the Fourth International Congress on Industrial and Applied Mathematics, Oxford University Press, pp. 271-282, 2000.
[24]. J. Nocedal and S. J. Wright. Numerical Optimization. Springer series in operations research, 1999.\newline
[25]. N. I. M. Gould and S. Leyffer. An introduction to algorithms for nonlinear optimization. Computational Science and Engineering Department, Atlas Centr, Rutherford Appleton Laboratory, Oxfordshire, December 2002.