核磁共振影像(MRI)或功能性核磁共振影像(functional MRI)之平面化(flattening)演算法，就是要將隱含在高度皺摺的腦皮層結構性或功能性資訊以近似無扭曲(distortion less)的方式展開到二度空間的平面上，方便醫師進一步分析及診斷。本論文提出的方法首先在腦皮層表層定義對應於平面化(flatten)影像的原點以及二維參考軸，然後將腦皮層表面輪廓(profile)編成鏈碼(chain code) 。將三度空間物體編成鏈碼的好處是串成鏈碼的字串(string)同時保存三度空間物體全域(global)及區域(local)的拓撲(topology)關係。本文的做法是將三度空間物體先分解成相互垂直的剖面，然後將物體輪廓轉成鏈碼，因此三度空間物體的結構可以很簡潔地由兩組鏈碼字串表示。從腦皮層不同剖面定義物體輪廓的起始點切開，將此封閉曲線拉開成直線，就相當於鏈碼字串的序列式排序，經過適當的調整後，腦皮層3D座標與平面化影像2D座標的對應關係以及鄰域關係可以由鏈碼(chain code)決定，並且可以由此定義出最佳化(optimal)的誤差能量函數(error energy function)。因此最佳化的步驟，就是憑藉物體表層空間結構的關係並且經由deterministic annealing的最佳化技術，在三度空間物體的兩組鏈碼字串分別對應的平面化影像之間尋找最佳平衡點，然後將二張影像混合(blending)成一張近似無扭曲的完整影像。根據此張平面化的腦皮層影像，醫師能夠有效的觀察皺摺的腦溝部位，當發現異狀時，也能由平面影像資料快速的對應至三維的體積資料上，作更進一步的診斷分析。本論文除將此技術應用於醫學影像的平面化外，也進一步擴展至常用的三維貼圖及三維物體變形的技術。因為不同於傳統表面三角化的處理方式，因此得以保留內部的三維影像資訊，而能獲得更好的醫學影像顯示與量測效果，幫助醫生診斷病情。

Magnetic resonance image (MRI) or functional MRI is the most widely used sensory devices for analyzing the functions of human brain. But, the visualization of human brain function based on cortical activity is extremely difficult, because most of the cortical surface areas are buried inside folds. Flattening is a useful tool to map a complex 3D surface to an undistorted flatten map. In this dissertation, we have proposed a voxel-based flattening method that can flatten the cortical surface directly from the 3D voxel-based volume data. In the proposed method, we first defined a surface-based coordinate system including system origin and reference axes on the 3D surface. Then, the cortex surface is encoded into the chain code format before flattening. The global and local topological relationships of surface voxeles are embedded in the chain codes. Based on chain codes, we can define the energy function for minimizing the distortion error in the flattening process and improve the computation efficiency. In the implementation, we decomposed the 3D surface into two sets of cross-sections along the vertical and horizontal directions respectively. Each cross section in the two sets is then encoded into chain code format. Thus, the two sets of cross sections generate two sets of chain code strings, one represents the horizontal codes and the other represents the vertical ones. The chain code strings can be used to express the cross section as a stretched curve line after cutting it at a given point. All stretched curve lines are used to convert the 3D cortical surface directly to a 2D map. A flattened map is then obtained from warping and blending the 2D maps to the position of minimum distortions by using an optimized algorithm (deterministic annealing). As the data are processed in voxel format, the inherent information of the medical data will be preserved at the same time; physicians can make further clinical analysis. The proposed voxel-based ideas are also extended to the algorithm for texture mapping and 3D morphing. By using the properties of flattening (voxel-based approach), the developed methods in texture mapping and 3D morphing have been proved very efficient in the applications of medical diagnostics. Using these tools, it is possible to develop useful visual systems to assist physicians in several clinical diagnoses.

[1] E.L. Schwartz, A. Shaw, and E. Wolfson, A Numerical Solution to the Generalized Mapmaker’s Problem: Flattening Nonconvex Polyheadral Surfaces, IEEE Transaction on Pattern Analysis and Machine Intelligence, Vol.11, No.9, pp.1005-1008, 1989.
[2] E. Wolfson, and E. L. Shwrtz, Computing Minimal Distances on Polyhedral Surface, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol.11, No.9, pp.1001-1005, 1989.
[3] H. A. Drury, D. C. VanEssen, C. H. Anderson, C. W. Lee, T. A. Coogan, and J. W. Lewis, Computerized Mappings of the Cerebral Cortex: A Multi-resolution Flattening Method and a Surface-Based Coordinate System, Journal of Cognitive Neuroscience , Vol.8, No.1, pp.1-28, 1996.
[4] B. Fischl, M. I. Sereno, and M. D. Anders, Cortical Surface-Based Analysis II: Inflation, Flattening, and a Surface-Based Coordinate System, NeuroImage , Vol.9, No.2, pp.195-207, 1999.
[5] D. C. VanEssen, H. A. Drury, S. Joshi ,and M. I. Miller, Functional and Structural Mapping of Human Cerebral Cortex: Solution are in the Surface, Proc. Natl. Acad. Sci. USA , Vol.95, pp.788-795, 1998.
[6] E.L. Schwartz, E.L. Merker, B. Wolfson, and E. Shaw, A Applications of computer graphics and image processing to 2D and 3D modeling of the functional architecture of visual cortex, IEEE Computer Graphics and Applications , Vol.8, No.4, pp.13-23, 1988.
[7] P.C. Teo, G. Sapiro, and B.A. Wandell, Creating connected representations of cortical gray matter for functional MRI visualization, IEEE Transactions on Medical Imaging ,Vol.16, No.6, pp.852-863, 1997.
[8] J. Assa, D. Cohen-Or, and T. Milo, Displaying data in multidimensional relevance space with 2D visualization maps, Visualization '97 Proceedings , Vol.534, pp127-134, 1997.
[9] Ge Wang, G. McFarland, B.P. Brown, and M.W. Vannier, GI tract unraveling with curved cross sections, IEEE Transactions on Medical Imaging , Vol.17, No.2, pp.318-322, 1998.
[10] S. Angenent, S. Haker, A. Tannenbaum, and R. Kikinis, On the Laplace-Beltrami operator and brain surface flattening, IEEE Transactions on Medical Imaging , Vol.18, No.8, pp.700-711, 1999.
[11] S. Halier, S. Angenent, A. Tannenbaurn, and R. Kikinis, Nondistorting flattening maps and the 3-D visualization of colon CT images, IEEE Transactions on Medical Imaging , Vol.19, No.7, pp.665-670, 2000.
[12] P.G. Lacroute, Fast Volume Rendering Using A Shear-Warp Factorization of the Viewing Transformation, Technical Report: CSL-TR-95-678, Departments of Electrical Engineering and Computer Science, Stanford University, Sept. 1995
[13] J.K. Udupa, and D. Odhner, Fast Visualization, Manipulation, and Analysis of Binary Volumetric Objects, IEEE Computer Graphics and Applications , Vol. 11, No.6, pp.53-62, 1991.
[14] M. Levoy, Display of surfaces from volume data, IEEE Computer Graphics and Applications , Vol.8, No.3, pp.29-37, 1998.
[15] L. Westover, Interactive volume rendering, Workshop on Volume Visualization, Chapel Hill, NC, pp.9-16, 1989.
[16] T.L. Weng, T.Y. Lee, and Y.N. Sun, New rendering method: scan-line based semi-boundary algorithm, Proceedings of SPIE , Vol.3335, pp.20-27, 1997.
[17] Stan Z. Li, Markov Random Field Modeling in Image Analysis, Springer-Verlag Tokyo Berlin Heidelberg New York.
[18] W. E. Lorensen, and H. E. Cline, Marching Cubes: A High Resolution 3D Surface Construction Algorithm, Computer Graphics , Vol.21, No.4, pp.163-169, 1987.
[19] Y. Zimmer, R. Tepper, and S. Akselrod, An Improved Method to Compute the Convex Hull of A Shape A Binary Image, Pattern Recognition , Vol.30, No.3, pp.397-402, 1997.
[20] S.S. Rao, Engineering Optimization Theory and Practice, Third Edition.
[21] J. Malliot, H. Yahia, and A. Verroust, Interactive Texture Mapping, Computer Graphics Proceedings, Annual Conference Series, pp.27-34, 1993.
[22] E.A. Bier and K.R. Sloan, Two-Part Texture Mappings, IEEE Computer Graphics and Applications , Vol.6, No.9, pp.40-53, 1986.
[23] C. Bennis, J.M. Vézien, and G.Iglésias, Piecewise Surface Flattening for Non-Distorted Texture Mapping, SIGGRAPH ’91 , Vo1.25, No.4, pp.237-246, 1991.
[24] R. Grossmann, N. Kiryati, and R. Kimmel, Computational surface flattening: a voxel-based approach, IEEE Trans. On PAMI, Vol.24, No.4, pp.433-441, 2002.
[25] H.H.S. Ip and L. Yin, Constructing a 3D individualized head model from two orthogonal views, The Visual Computer, Vol.12, No.6, pp.254-266, 1996.
[26] U. Akdemir, B. Özgüç, U. Güdükbay, and A. Selçuk, Right-triangular subdivision for texture mapping ray-traced objects, The Visual Computer , Vol.14, pp.445-454, 1998.
[27] B. Lévy and J.L. Mallet, Non-Distorted Texture Mapping For Sheared Triangulated Meshes, SIGGRAPH ’98 , pp.343-352, 1998.
[28] T. Beier and S. Neely, Feature-Based Image Metamorphosis, SIGGRAPH ’92 , Vol.26, No.2, pp.35-42, 1992.
[29] P.J. Burt and E.H. Adelson, A Multiresolution Spline With Application to Image Mosaics, ACM Trans. on Graphics , Vol.2, No.4, pp217-236, 1983.
[30] P.J. Burt and E.H. Adelson, The Laplacian Pyramid as a Compact Image Code, IEEE Trans. On Communications , Vol.31, No.4, pp532-540, 1983.
[31] X. Chenyang, L.P. Dzung, E.R. Maryam, N.Y. Daphne, and L.P. Jerry, Reconstruction of the Human Cerebral Cortex from Magnetic Resonance Images, IEEE Trans. on Medical Imaging, Vol.18, No.6, pp467-480, 1999.
[32] Z. Xiaolan, H.S. Lawrence, T.S. Robert, and S.D. James, Segmentation and Measurement of the Cortex from 3-D MR Images Using Coupled-Surfaces Propagaction, IEEE Trans. on Medical Imaging, Vol.18, No.10, pp.927-937, 1999.
[33] C. Bennis, J.M. Vezien, and G. Iglesias, Piecewise flattening for non-distorted texture mapping, SIGGRAPH 91, Vol.25, No.4, pp237-246, 1991.
[34] R. Grossmann, N. Kiryati, and R. Kimmel, Computational Surface Flattening: A Voxel-based Approach, IEEE Trans. On PAMI. , Vol.24, No.4, pp.433-441, 2002.
[35] T. Kanai, H. Suzuki, and F. Kimuraf, Three-dimensional geometric meta- morphosis based on harmonic maps, The Visual Computer , Vol.4, No.4, pp.166-176, 1998.
[36] B. LÉvy, and J.L. Mallet, Non-distorted texture mapping for sheared triangulated meshes, SIGGRAPH , pp.343-352, 1998.
[37] P. Dan, and B. George, Seamless texture mapping of subdivision surfaces by model pelting and texture blending, SIGGRAPH , pp.471-478, 2000.
[38] Bruno Levy, Sylvain Petitjean Nicolas Ray and Jerome Maillor, Least squares conformal maps for automatic texture atlas generation, SIGGRAPH , pp362-371, 2000.
[39] G. Zigelman, R. Kimmel, and N. Kiryati, Texture mapping using surface flattening via multidimensional scaling, IEEE Trans. On Visualization and Computer Graphics, Vol.8, No.2, pp198-207, 2002.
[40] M.W. Trosset, Applications of Multidimensional Scaling to Molecular Conformation, Computing Science and Statistics , Vol.29, pp.148-152, 1998.
[41] D. J. Williams, and M. Shah, A fast algorithm for active contours and curvature estimation, CVGIP: Image Understanding , Vol.55, No.1, pp14-26, 1992.
[42] T. Hofmann, and J. M. Buhmann, Pairwise data clustering by deterministic annealing, IEEE Trans. On PAMI , Vol.19, No.1, pp1-13, 1997.
[43] S.J. Lee, Accelerated deterministic annealing algorithms for transmission CT reconstruction using ordered subsets, IEEE Trans. Nuclear Science , Vol.49, No. 5, pp.2373-2380, 2002.
[44] J. Puzicha, T. Hofmann, and J.M. Buhmann, Deterministic annealing: fast physical heuristics for real-time optimization of large systems, Proceedings of the 15th IMACS World Conference on Scientific Computation, Modelling and Applied Mathematics, Berlin, 1997.
[45] Chandler, D. Introduction to modern statistical mechanics. Oxford University Press.
[46] U. Akdemir, B. Özgüç, U. Güdükbay and A. Selçuk, Right-triangular subdivision for texture mapping ray-traced objects, The Visual Computer , Vol.14, No.10, pp.33-57, 1998.
[47] R.O. Duda, and P.E. Hart, Pattern Classification and Scene Analysis, Wiley, New York, 1973
[48] W.S. Lee, and N.M. Thalmann, From Real Faces To Virtual Faces: Problems and Solutions, Proc. 3IA'98, pp.5-19, 1998.
[49] A. Watt, and M. Watt, Advanced Animation and Rendering Techniques: Theory and Practice, Addison-Wesley Publishing Company, 1992
[50] T. Kanai, H. Suzuki and F. Kimura, 3D geometric metamorphosis based on harmonic map, The Visual Computer , Vol.14, No.4, pp.166-176, 1998.
[51] W.S. Lee, and N.M. Thalmann, Head Modeling from Pictures and Morphing in 3D with Image Metamorphosis based on triangulation, Proc. Captech 98 , pp.254-267, 1998.
[52] F. Lazarus and A. Verroust, Three-dimensional metamorphosis: a survey, The Visual Computer , Vol.14, pp.373-389, 1998.
[53] H.H.S. Ip and L. Yin, Constructing a 3D individualized head model from two orthogonal views, The Visual Computer , Vol.12, pp254-266, 1996.
[54] T.L. Weng, C.Y. Yu, and Y.N. Sun, Visualization of human Cerebral Cortex by Voxel-based Flattening Algorithm, BME'99 conference , pp369-370, 1999.
[55] T. Hofmann, and J.M. Buchmann, Pairwise data clustering by deterministic annealing, IEEE Trans. On PAMI, Vol.19, No.1, pp.1-14, 1997.