醫學影像的曲面模型(Surface model)重建對於醫學工程領域來說，是非常重要的工具，尤其是在骨科手術中，常會以患者骨組織的幾何模型來協助術前的診斷與規劃，因此骨組織模型的重建，為首要的步驟，然而骨模型的重建流程含括了醫學影像與幾何模型技術，重建步驟複雜且需耗費許多時間。本研究目標為整合「影像組織分離技術」以及「曲面模型重建技術」，發展電腦斷層影像的骨組織模型重建程序，以提高骨組織模型的重建效率。「影像組織分離技術」方面，以疊代式區域成長法為基礎，發展半自動化的骨組織分離程序，本程序能夠於不同的骨頭組織上自動分配種子區域，正確的完成骨組織分離，另外適用性高以及分離效率佳也是本程序最大的優勢之一。「曲面模型重建技術」方面結合了模型簡化、網格四邊化、網格後處理以及特徵邊搜尋等技術，使能夠於網格化後的骨網格模型上自動規劃出曲面的四邊架構曲線，並經由此架構曲線重建曲面模型。最後整合以上程序，以多組實際的人體骨組織影像為例，重建其曲面模型，以證明其本研究之可行性。

Surface model reconstruction is an extremely important technique in biomedical engineering as it can assist in implant design and preoperative planning of orthopaedics. However, bone model reconstruction is composed of medical image process and computational geometry algorithms, which is complex and difficult to be integrated. Therefore, the objective of this study is to integrate above approaches into a procedure to simplify the operation course and improve overall efficiency of bone model reconstruction. The procedure can be divided into two steps: (1) CT images segmentation: the proposed process is based on iterative regions growing. In which, the seed regions generation algorithm is proposed to automatically generate an initial region on each bone of interest. Finally, the individual bone region will be extracted by seed regions expanded iteratively. (2) Surface model reconstruction: a method for building quadrilateral network of curves automatically from triangular mesh is proposed in this study, which mainly includes mesh simplification, quadrangulation and curve net generation. When curve net is produced, it can be served as the framework of automatic surface reconstruction. Finally, several sets of bone images have been presented to demonstrate the feasibility of this integrated procedure.

[1] J. M. Geusebroek, A. W. M. Smeulders and J. V. D Weijer, “Fast Anisotropic Gauss Filtering” , ECCV 2002, LNCS 2350, pp. 99-112, 2002.
[2] A. Huertas and G. Medioni, “Detection of Intensity Changes with Subpixel Accuracy Using Laplacian-Gaussian Masks”, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. PAMI-8, No. 5, pp. 651-664, 1986.
[3] J. Shen and S. Castan, “An Optimal Linear Operator for Step Edge Detection”, CVGIP: Graphical Models And Image Processing, Vol. 54, No. 2, pp. 112-133, 1992.
[4] P. Bao, L. Zhang and X. Wu, “Canny Edge Detection Enhancement by Scale Multiplication”, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 27, No. 9, pp. 1485-1490, 2005.
[5] R. Adams and L. Bischof, “Seeded Region Growing”, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 16, No. 6, pp. 641-647, 1994.
[6] R. Malladi, J. A. Sethian and B.C. Vemuri, “Shape Modeling with Front Propagation: A Level Set Approach”, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 17, No. 2, pp.158-175, 1995.
[7] S. Geman and D. Geman, “Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images”, Journal of Applied Statistics, Vol. 20, Issue 5, pp. 25-62, 1993.
 
[8] Z. Peter, V. Bousson, C. Bergot and F. Peyrin, “A Constrained Region Growing Approach Based on Watershed for The Segmentation of Low Contrast Structure in Bone Micro-CT Images”, Pattern Recognition 41, pp. 2358-2361, 2008.
[9] K. Nagamune, “Automated Extraction Method of Bone Tunnel after the Anterior Cruciate Ligament Reconstruction from Knee MDCT Image by Fuzzy Inference”, FUZZ-IEEE 2009, pp. 933-938, 2009.
[10] S. M. Carlos, “Self-assessed Contrast-Maximizing Adaptive Region Growing”, ACIVS 2009, pp. 652-663, 2009.
[11] J. L. Rose, “Shape Prior Criterion Based on Tchebichef Moment in Variational Region Growing”, Image Processing (ICIP), pp. 1081-1084, 2009.
[12] R. Yanagisawa and S. Omachi, “Extraction of 3D Shape of a Tooth from Dental CT Images with Region Growing Method”, Lecture Notes in Computer Science, Vol. 6540, pp. 68-77, 2011.
[13] M. Alazab, M. Islam and S. Venkatraman, “Towards Automatic Image Segmentation Using Optimised Region Growing Technique”, Australasian Conference on Artificial Intelligence 2009, Vol. 5866, pp. 131-139, 2009.
[14] D. T. Lin, C. C. Lei and S. W. Hung, “Computer-aided kidney segmentation on abdominal CT images”, IEEE Transaction on information technology in biomedicine, Vol. 10, No. 1, pp. 59-65, 2006.
[15] A. Fabijańska, “Two-pass region growing algorithm for segmenting airway tree from MDCT chest scans”, Computerized Medical Imaging and Graphics, Vol. 33, pp. 537–546, 2009.
 
[16] M. A. Palomera-Pérez, M. E. Martinez-Perez, H. Benítez-Pérez, J. L. Ortega-Arjona, “Parallel Multiscale Feature Extraction and Region Growing: Application in Retinal Blood Vessel Detection”, IEEE Transaction on information technology in biomedicine, Vol. 14, No. 2, 2010.
[17] M. Eck, T. DeRose, H. Duchamp, H. Hoppe, M. Lounsbery and W. Stuetzle, “Multiresolution Analysis of Arbitrary Meshes”, In Proceedings of SIGGRAPH’1995, pp. 173-182, 1995.
[18] I. Guskov, K. Vidimče, W. Sweldens and P. Schröder, “Normal Meshes”, In Proceedings of SIGGRAPH’2000, pp. 95-120, 2000.
[19] H. Ying and Q. Hong, “Surface Reconstruction with Triangular B-splines”, Journal of Computer Science and Technology, Vol. 21, No.2, pp. 232-237, 2006.
[20] W. F. Lee, W. Sweldens, P. Schröder, L. Cowsar, and D. Dobkin, “MAPS: Multiresolution Adaptive Parameterization of Surfaces”, In Proceedings of SIGGRAPH’1998, pp. 95-104, 1998.
[21] Y. Alex, H. Stefanie and B. Georges-Pierre, “Smooth Adaptive Fitting of 3D Models Using Hierarchical Triangular Splines”, International conference on shape modeling and applications, pp. 13-22, 2005.
[22] M. Eck and H. Hoppe, “Automatic Reconstruction of B-Spline Surfaces of Arbitrary Topological Type”, In Proceedings of SIGGRAPH’1996, pp. 209-216, 1996.
[23] E. Catmull, and J. Clark, “Recursively Generated B-spline Surfaces on Arbitrary Topological Meshes”, Computer Aided Design, No. 10, Issue 6, pp. 350-355, 1978.
[24] M. Ioana, H. Rushmeier and J. Jin, “Parameterization of Triangle Meshes over Quadrilateral Domains”, Symposium on Geometry Processing, pp. 197-208, 2004.
[25] L. Kobbelt, J. Vorsatz, U. Labsik and H. Seidel, “A Shrink Wrapping Approach to Remeshing Polygonal Surfaces”, Computer Graphics Forum (Proc. EUROGRAPHICS ''99), pp. 119-130, 1999.
[26] W. Ma and N. Zhao, “Catmull-Clark Surface Fitting for Reverse Engineering Applications”, Geometric Modeling and Processing, pp. 274-284, 2000.
[27] S. Takeuchi, T. Kanai, H. Suzuki, K. Shimada and F. Kimura, “Subdivision Surface Fitting with QEM-Based Mesh Simplification and Reconstruction of Approximated B-spline Surfaces”, The 8th Pacific Conference on Computer Graphics and Applications, pp. 202-212, 2000.
[28] W. Schroeder, J. Zarge and W. Lorensen, “Decimation of Triangle Meshes”, In Proceedings of SIGGRAPH’1992, Vol.26, pp. 65-70, 1992.
[29] Cohen, J., Varshney, V., Manocha, D., Turk, G. and Weber, H., “Simplification Envelop”, In Proceedings of SIGGRAPH’1996, pp. 119-128, 1996.
[30] H. Hoppe, “Progressive Meshes”, In Proceedings of SIGGRAPH’1996, pp. 99-108, 1996.
[31] M. Garland and P. Heckbert, “Surface Simplification Using Quadric Error Metrics”, In Proceedings of SIGGRAPH’1997, pp. 209-216, 1997.
[32] M. Garland and Y. Zhou, “Quadric-Based Simplification in any Dimension”, ACM Transactions on Graphics (TOG), Vol.24, No.2, pp. 209-239, 2005.
 
[33] Y. Chen and T. Nishita, “An Efficient Mesh Simplification Method with Feature Detection for Unstructured Meshes and Web Graphics”, Proceedings of IEEE Computer Graphics International, pp. 34-41, 2003.
[34] T. S. Lau, S. H. Lo and C. K. Lee, “Generation of Quadrilateral Mesh Over Analytical Curved Surface”, Finite Elements in Analysis and Design , Vol. 27, Issue 3, pp. 251-272, 1997.
[35] H. Borouchaki and P. Frey, “Adaptive Triangular-Quadrilateral Mesh Generation”, International Journal for Numerical Methods in Engineering. Vol.41, No. 15, pp. 915-934, 1998.
[36] S. Maza, F. Noel and J. C. Leon, "Generation of Quadrilateral Meshes on Free-Form Surfaces", Computers and Structures, Vol. 71, Issue 5, pp. 505-524, 1999.
[37] K. Y. Lee, I. I. Kim, D. Y. Cho and T. W. Kim, “An Algorithm For Automatic 2D Quadrilateral Mesh Generation with Line Constraints”, Computer-Aided Design, Vol. 35, Issue 12, pp. 1055-1068, 2003.
[38] P. Alliez, D. Cohen-Steiner, D. Devillers, B. L′evy and M. Desbrun, “Anisotropic Polygonal Remeshing”, In Proceedings of SIGGRAPH’2003, pp. 485–493, 2003.
[39] M. Marinov and L. Kobbelt, “Direct Anisotropic Quad-Dominant Remeshing”, Computer Graphics and Applications, 2004. PG 2004. Proceedings. 12th Pacific Conference on, pp. 207-216, 2004.
[40] S. H. Lo, “Generating Quadrilateral Element on Plane and Over Curved Surface”, Computers and Structures, Vol. 31, No. 3, pp. 421-426, 1989.
[41] R. Garimella and M. Shashkov, “Optimization of Surface Mesh Quality Using Local Parameterization”, Proceedings of the 11th International Meshing Roundtable, pp. 41-52, 2002.
[42] Semenova, V. Savchenko and I. Hagiwara, “Two Techniques to Improve Mesh Quality and Preserve Surface Characteristics”, Proceedings 13th International Meshing Rountable, pp. 277-288, 2004.
[43] Semenova, V. Savchenko and I. Hagiwara, “Improvement of Triangular and Quadrileteral Surface Meshes”, Proceedings 14th International Conference on Computer Graphics and Vision, pp. 79-87, 2004.
[44] H. Hoppe, T. DeRose, Y. Duchamp, J. McDonald and W. Stuetzle, “Mesh Optimization”, In Proceedings of SIGGRAPH’1993, pp. 19-26, 1993.
[45] K. Hormann, U. Labsik and G. Greiner, “Remeshing Triangulated Surfaces with Optimal Parameterizations”, Computer-Aided Design, Vol. 33, No. 11, pp.779-788, 2001.
[46] P. J. Frey and H. Boroucraki, “Surface Mesh Quality Evaluation”, International Journal for Numerical Methods in Engineering, Vol. 45, Issue 1, pp. 101-118, 1999.
[47] G. Taubin, “Estimating the Tensor of Curvature of a Surface from a Polyhedral Approximation”, Proceedings of the 5th International Conference on Computer Vision, pp. 902-907, 1995.
[48] P. Krsek, G. Lukacs and R. R. Martin, “Algorithms for Computing Curvatures from Range Data”, Mathematics of Surfaces VIII, pp. 1-16, 1998.
[49] T. Surazhsky, E. Magid, O. Soldea, G. Elber and E. Rivlin, “A Comparison of Gaussian and Mean Curvatures Estimation Methods on Triangular Meshes”, Proceedings of The IEEE International Conference on Robotics and Automation, Vol. 1, pp. 1021-1026, 2003.
[50] F. Koschan, D. L. Page and M. A. Abidi, “Perception-based 3D Triangle Mesh Segmentation Using Fast Marching Watersheds”, 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Vol. 2, pp. II-27-II-32, 2003.
[51] 蔡耀震, 賴景義, 鍾永彬, “網格資料特徵分離”, 第十三屆自動化科技研討會論文集, 1998.
[52] J. Huang and C. H. Menq, “Automatic Data Segmentation for Geometric Feature Extraction from Unorganized 3-D Coordinate Points”, IEEE Transactions on Robotics and Automation, Vol. 17, No. 3, pp. 268-279, 2001.
[53] G. Lavoue, F. Dupont, and A. Baskurt, “Curvature Tensor Based Triangle Mesh Segmentation with Boundary Rectification”, Proceedings of International Conference on Computer Graphics, pp. 10-17, 2004.
[54] Razdan and M. S. Bae, “A Hybrid Approach to Feature Segmentation of Triangle Meshes”, Computer-Aided Design, Vol. 35, Issue 9, pp. 783-789, 2003.
[55] H.C. Kim, S.M. Hur and S.H. Lee, “Segmentation of the Measured Point Data in Reverse Engineering”, The International Journal of Advanced Manufacturing Technology, Vol. 20, pp. 571-580, 2002.
[56] M. Meyer, M. Desbrun, P. Schroder and A. H. Barr, “Discrete Differential Geometry Operators for Triangulated 2-Manifolds”, Visualization and Mathematics III, pp. 35-57, 2003.
[57] N. Gelfand and L. J. Guibas, “Shape Segmentation Using Local Slippage Analysis”, Eurographics Symposium on Geometry Processing, pp. 214-223, 2004.
 
[58] M. Vieira and K. Shimada, “Surface Mesh Segmentation and Smooth Surface Extraction Through Region Growing”, Computer Aided Geometric Design, Vol. 22, No. 8, pp. 771-792, 2005.
[59] 張義宏，「光學掃描量測資料之二次曲面特徵分離」，國立中央大學，碩士論文, 民國95年。
[60] P. Benko, R. R. Martin and T. Varady, “Algorithms for Reverse Engineering Boundary Representation Models”, Computer-Aided Design, Vol. 33, No. 11, pp. 839-851, 2001.
[61] M. Vanco, “A Direct Approach for the Segmentation of Unorganized Points and Recognition of Simple Algebraic Surfaces”, Ph.D. Dissertation, University of Technology Chemnitz, 2002.
[62] S. Lloyd, “Least square quantization in PCM”, IEEE Transactions on Information Theory, Vol. 28, Issue 2, pp. 975-987, 1982.
[63] R. O. Duda, P. E. Hart, D.G. Stork, “Pattern classification (2nd ed.)”, Wiley Interscience, ACM, New York, NY October 2000.
[64] S. Shlafman, A. Tal and S. Katz, “Metamorphosis of Polyhedral Surfaces Using Decomposition”, Computer Graphics Forum, Vol. 21, Issue 3, pp. 219-228, 2002.
[65] K. Fukunaga and L. Hostetler, “The Estimation of the Gradient of a Density Function, with Applications in Pattern Recognition”, IEEE Transactions on Information Theory, Vol. 21, No. 1, pp. 32-40, 1975.
[66] Y. Cheng, “Mean Shift, Mode Seeking, and Clustering”, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 17, Issue 8, pp. 790-799, 1995.
[67] H. Yamauchi, S. Lee, Y. J. Lee and Y. Ohtake, “Feature Sensitive Mesh Segmentation with Mean Shift”, In Proceedings of Shape Modeling International, pp. 236-243, 2005.
 
[68] X. Zhang, G. Li, Y. H. Xiong and F. H. He, “3D Mesh Segmentation Using Mean-Shifted Curvature”, Advances in Geometric Modeling and Processing, pp. 465-474, 2008.
[69] G. Hu, Q. Peng and A. Robin Forrest, “Mean Shift Denoising of Point-Sampled Surfaces”, The Visual Computer, Vol. 22, No. 3, pp. 147-157, 2006.
[70] R. Wang and J. Li, “Feature-Preserving Smoothing of Point-Sampled Geometry”, International Journal of Advanced Science and Technology, Vol. 4, pp.39-45, 2009.
[71] Y. Miao, R. Pajarola and J. Feng, “Curvature-Aware Adaptive Re-sampling for Point-sampled Geometry”, Computer-Aided Design, Vol. 41, Issue 6, pp. 395-403, 2009.
[72] L. James and C. D. Twigg, “Skinning Mesh Animations”, ACM Trans. Graphics (TOG), Vol.24, No.3, pp.399-407, 2005.
[73] W. E. Lorenson and H. E. Cline, “Marching cubes: a high resolution 3D surface construction algorithm”, SIGGRAPH ''87 Proceedings, pp. 163-169, 1987.
[74] 羅賴鈞，「醫學影像之三維顯示與骨組織三角網格重建技術探討」，國立中央大學，碩士論文, 民國97年。
[75] L. Wang and J. Bai, “Threshold selection by clustering gray levels of boundary”, Pattern Recognition Letters, Vol. 24, Issue 12, pp. 1983-1999, 2003.