從影像中重建物體的三維模型，一直以來，在電腦視覺與圖學領域裡是個很熱門的研究題目。在重建的過程中，利用從影像中所獲取的資訊來校正相機以求得相機參數。而這個相機校正的步驟，扮演了很重要的角色。在傳統以特徵點為基礎的三維重建演算法中，例如從運動求得結構(Structure from Motion)，是利用影像與影像間的特徵點對應來校正相機。然而，從影像間求得對應點是一項具挑戰性的問題，尤其是對於寬基線(wide baseline)的影像或是物體本身缺少材質(texture-less)的情況而言。因此，在本論文中，我們提出了利用物體剪影資訊來校正相機的演算法。不用事先尋求影像間的對應點，也不需要特需的樣型輔助。
對於一個任意移動的相機而言，如果其相機資訊沒有一個很接近的初始猜測，僅利用物體剪影資訊是不足以校正相機。因此，我們必須限制相機的移動方式來使其變為可能。基於不同的場景設定下，我們提出了兩種三維重建的演算法，分別為從鏡面反射重建物體模型(Shape from Mirror Reflection)與從環狀運動重建物體模型(Shape from Circular Motion)。在第一個演算法中，物體被放置於平面鏡前，利用相機拍攝其影像。相機的位置因為鏡面反射的特性而有限制。在第二個演算法中，物體被放置於一個轉盤上，利用相機拍攝其影像。相機的位置因為環狀運動的特性而有限制。針對不同的特性，我們推導出其對應的限制(constraints)，僅利用物體剪影資訊來校正相機與重建其物體三維模型。實驗結果分別呈現於多組模擬與真實的影像資料，用以驗證我們演算法的強固性與其表現。

3D modeling from images is a popular research topic in computer vision and computer graphics. During the 3D reconstruction process, the camera calibration step plays an important role that estimates camera parameters from the information extracted from images. Traditional feature-based reconstruction algorithm, i. e. the Structure from Motion (SfM) algorithm, exploits the feature correspondences across views for camera calibration. However, to robustly estimate correspondences is a challenging problem, especially for wide-baseline images or texture-less objects. Instead of making use of feature correspondences, in this thesis, we propose to exploit the object silhouettes for camera calibration without identifying any feature points in advance or using any specific calibration pattern.
From an arbitrary moving camera, the captured image silhouettes of an object can not provide enough constraints for estimating camera parameters without a good initial guess; therefore, we need to restrict the camera motion that makes camera calibration from silhouettes possible. We hereby propose two 3D modeling algorithms based on different scene settings; namely the Shape from Mirror Reflection (SfMR) and the Shape from Circular Motion (SfCM) algorithms. In the SfMR algorithm, an object is put in front of planar mirrors, and the camera positions are constrained by mirror reflection property. In the SfCM algorithm, an object is put on a turn-table, and the camera positions are constrained by the circular motion property. From different properties, we derive constraints to calibrate the camera and reconstruct the 3D shape of a model using only the object silhouettes. Experimental results on both synthetic and real data sets demonstrate the robustness of the proposed algorithms and their performance.

[A. Laurentini, 1994] A. Laurentini, “The Visual Hull Concept for Silhouette Based Image Understanding,” IEEE Trans. Pattern Analysis and Machine Intelligence (TPAMI’94), vol. 16, no. 2, pp. 150-162, 1994.
[A. R. J. Francois, 2002] A. R. J. Francois, G. G. Medioni, and R. Waupotitsch, “Reconstructing Mirror Symmetric Scenes From a Single View Using 2-View Stereo Geometry,” Proc. IEEE International Conference on Pattern Recognition (ICPR’02), vol. 4, pp. 12-16, 2002.
[A. W. Fitzgibbon, 1998] A. W. Fitzgibbon, G. Cross, and A. Zisserman, “Automatic 3D Model Construction for Turn-Table Sequences,” Proc. 3D Structure from Multiple Images of Large-Scale Environments (SMILE’98), pp. 155-170, 1998.
[B. Hu, 2009] B. Hu, “It's All Done with Mirrors: Calibration-and-Correspondence- Free 3D Reconstruction,” Proc. Canadian Conference on Robot Vision (CRV’09), pp. 148-154, 2009.
[C. A. Rothwell, 1993] C. A. Rothwell, D. A. Forsyth, A. Zisserman, and J. L. Mundy, “Extracting Projective Structure from Single Perspective Views of 3D Point Sets,” Proc. IEEE International Conference on Computer Vision (ICCV’93), pp. 573-582, 1993.
[C. Hern□ndez, 2004] C. Hern□ndez and F. Schmitt, “Silhouette and Stereo Fusion for 3D Object Modeling,” Computer Vision and Image Understanding (CVIU’04), vol. 96, no. 3, pp. 367–392, 2004.
[C. Hern□ndez, 2007] C. Hern□ndez, F. Schmitt, and R. Cipolla, “Silhouette Coherence for Camera Calibration under Circular Motion,” IEEE Trans. Pattern Analysis and Machine Intelligence (TPAMI’07), vol. 29, no. 2, pp. 343-349, 2007.
[D. Q. Huynh, 1999] D. Q. Huynh, “Affine Reconstruction from Monocular Vision in the Presence of a Symmetry Plane,” Proc. IEEE International Conference on Computer Vision (ICCV’99), pp. 476-482, 1999.
[D. G. Lowe, 2004] D. G. Lowe, “Distinctive Image Features from Scale-Invariant Keypoints,” International Journal of Computer Vision (IJCV’04), vol. 60, no. 2, pp. 91-110, Nov. 2004.
[F. Blais, 2004] F. Blais, “Review of 20 Years of Range Sensor Development,” J. Electronic Imaging (EI’04), vol. 13, no. 1, pp. 231-240, Jan. 2004.
[E. Boyer, 2006] E. Boyer, “On Using Silhouettes for Camera Calibration,” Proc. Asian Conference on Computer Vision (ACCV’06), pp. 1-10, 2006.
[G. Jiang, 2002] G. Jiang, H. T. Tsui, L. Quan, and A. Zisserman, “Single Axis Geometry by Fitting Conics,” IEEE Trans. Pattern Analysis and Machine Intelligence (TPAMI’02), vol. 25, no. 10, pp. 1343-1348, 2002.
[G. Jiang, 2004] G. Jiang, L. Quan, and H. T. Tsui, “Circular Motion Geometry Using Minimal Data,” IEEE Trans. Pattern Analysis and Machine Intelligence (TPAMI’04), vol.26, no. 6, pp. 721-731, 2004.
[G. K. M. Cheung, 2005] G. K. M. Cheung, S. Baker, and T. Kanade, “Shape-From- Silhouette across Time Part I: Theory and Algorithms,” International Journal of Computer Vision (IJCV’05), vol. 62, No. 3, pp. 221-247, May 2005.
[G. G. Slabaugh, 2001] G. G. Slabaugh, W. B. Culbertson, T. Malzbender, and R. W. Schafer, “A Survey of Methods for Volumetric Scene Reconstruction from Photographs,” Proc. International Workshop on Volume Graphics, pp. 81-100, 2001.
[G. Zhang, 2006] G. Zhang, H. Zhang, and K.-Y. K. Wong, “1D Camera Geometry and Its Application to Circular Motion Estimation,” Proc. British Machine Vision Conference (BMVC’06), vol. 1, pp. 67-76, 2006.
[H. Mitsumoto, 1992] H. Mitsumoto, S. Tamura, K. Okazaki, N. Kajimi, and Y. Fukui, “3-D Reconstruction Using Mirror Images Based on a Plane Symmetry Recovering Method,” IEEE Trans. Pattern Analysis and Machine Intelligence (TPAMI’92), vol. 14, no. 9, pp. 941-946, 1992.
[H. Zhang, 2005] H. Zhang, G. Zhang, and K.-Y. K. Wong, “Auto-Calibration and Motion Recovery from Silhouettes for Turntable Sequences,” Proc. British Machine Vision Conference (BMVC’05), vol. 1, pp. 79-88, 2005.
[H. Zhong, 2006] H. Zhong, W. F. Sze, and Y. S. Hung, “Reconstruction from Plane Mirror Reflection,” Proc. IEEE International Conference on Pattern Recognition (ICPR’06), vol. 1, pp. 715-718, Aug. 2006.
[H. Zhong, 2007] H. Zhong and Y. S. Hung, “Multi-Stage 3D Reconstruction Under Circular Motion,” Image and Vision Computing (IVC’07), vol. 25, no. 11, pp. 1814-1823, 2007.
[J. Gluckman, 2001] J. Gluckman and S. K. Nayar, “Catadioptric Stereo Using Planar Mirrors,” International Journal of Computer Vision (IJCV’01), vol. 44, no. 1, pp. 65-79, Aug. 2001.
[J. Gluckman, 2002] J. Gluckman and S. K. Nayar, “Rectified Catadioptric Stereo Sensors,” IEEE Trans. Pattern Analysis and Machine Intelligence (TPAMI’02), vol. 24, no. 2, pp. 224–236, Feb. 2002.
[J. Mundy, 1993] J. Mundy and A. Zisserman, “Repeated Structure: Image Correspondence Constraints and 3D Structure Recovery,” Proc. The 2nd Joint European - US Workshop on Applications of Invariance in Computer Vision, J. L. Mundy, A. Zisserman, and D. Forsyth, eds., vol. 825, pp.89-106, Oct. 1993.
[K. Forbes, 2004] K. Forbes, A. Voigt, and N. Bodika, “Visual Hulls from Single Uncalibrated Snapshots Using Two Planar Mirrors,” Proc. 15th Annual Symposium of the Pattern Recognition Association of South Africa (PRASA’04), Nov. 2004.
[K. Forbes, 2006] K. Forbes, F. Nicolls, G. de Jager and A. Voigt, “Shape-from- Silhouette with Two Mirrors and an Uncalibrated Camera,” Proc. 9th European Conference on Computer Vision (ECCV’06), vol. 2, pp. 165-178, 2006.
[K. H. Jang, 2004] K. H. Jang, D. H. Lee, and S. K. Jung, “A Moving Planar Mirror Based Approach for Cultural Reconstruction,” Journal of Visualization and Computer Animation (JVCA’04), vol. 15, no. 3-4, pp. 415-423, 2004.
[K.-Y. K. Wong, 2002] K.-Y. K. Wong, P. R. S. Mendonca, and R. Cipolla, “Structure and Motion Estimation from Apparent Contours under Circular Motion,” Image and Vision Computing (IVC’02), vol. 20, no. 5, pp. 441-448, 2002.
[K.-Y. K. Wong, 2004] K.-Y. K. Wong and R. Cipolla, “Reconstruction of Sculpture from Its Profiles with Unknown Camera Positions,” IEEE Trans. Image Processing (TIP’04), 13(3):381-389, 2004.
[M. Pollefeys, 2004] M. Pollefeys, L. Van Gool, M. Vergauwen, F. Verbiest, K. Corneils, J. Tops, and R. Koch, “Visual Modeling with a Hand-Held Camera,” International Journal of Computer Vision (IJCV’04), vol. 59, no. 3, pp. 207-232, 2004.
[M. Kazhdan, 2006] M. Kazhdan, M. Bolitho, and H. Hoppe, “Poisson Surface Reconstruction,” Proc. Fourth Eurographics Symposium on Geometry Processing, pp. 61-70, 2006.
[M. Naganawa, 2005] M. Naganawa, S. Sawa, M. Sakata, Y. Yasumuro, M. Imura, Y. Manabe, and K. Chihara, “3D Imaging System for Visualizing and Monitoring Patients,” Proc. IEEE Engineering in Medicine and Biology Society (EMBS’05), vol. 7, no. 4, pp. 3-10, Sep. 2005.
[M. Potmesil, 1987] M. Potmesil, “Generating Octree Models of 3D Objects from their Silhouettes in a Sequence of Images,” Computer Vision, Graphics, and Image Processing (CVGIP’87), vol. 40, pp. 1-29, 1987.
[N. Snavely, 2008] N. Snavely, S.M. Seitz, and R. Szeliski, “Modeling the World from Internet Photo Collections,” International Journal of Computer Vision (IJCV’08), vol. 80, no. 2, pp. 189-210, 2008.
[P. A. Redert, 1999] P. A. Redert and E. A. Hendriks, “Acquisition of 3-D Scenes with a Single Hand Held Camera,” Proc. International Workshop on Synthetic-Natural Hybrid Coding and 3D Imaging, pp. 87-90, 1999.
[P. Favaro, 2008] P. Favaro, S. Soatto, M. Burger, and S. J. Osher, “Shape from Defocus via Diffusion,” IEEE Trans. Pattern Analysis and Machine Intelligence (TPAMI’08), vol. 30, no. 3, pp. 518-531, Mar. 2008.
[P.-H. Huang, 2006] P.-H. Huang and S.-H. Lai, “Contour-Based Structure from Reflection,” Proc. IEEE Computer Vision and Pattern Recognition (CVPR’06), vol. 1, pp. 379-386, 2006.
[P.-H. Huang, 2007] P.-H. Huang and S.-H. Lai, “Camera Calibration from Silhouette under Incomplete Circular Motion with a Constant Interval Angle,” Proc. Asian Conference on Computer Vision (ACCV’07), vol. 1, pp. 106-115, 2007.
[P.-H. Huang, 2008] P.-H. Huang and S.-H. Lai, “Silhouette-Based Camera Calibration from Sparse Views under Circular Motion,” Proc. IEEE Computer Vision and Pattern Recognition (CVPR’08), pp. 1-8, 2008.
[P. R. S. Mendonca, 1999] P. R. S. Mendonca and R. Cipolla, “Estimation of Epipolar Geometry from Apparent Contours: Affine and Circular Motion Cases,” Proc. IEEE Computer Vision and Pattern Recognition (CVPR’99), vol.1, pp. 9-14, 1999.
[P. R. S. Mendonca, 2001] P. R. S. Mendonca, K.-Y. K. Wong, and R. Cipolla, “Epipolar Geometry from Profiles under Circular Motion,” IEEE Trans. Pattern Analysis and Machine Intelligence (TPAMI’01), vol. 23, no. 6, pp. 604-616, 2001.
[R. Hartely, 2004] R. Hartely and A. Zisserman, Multiple View Geometry in Computer Vision Second Edition. Cambridge University Press, 2004.
[R. K. Kumar, 2008] R. K. Kumar, A. Ilie, J.-M. Frahm, and M. Pollefeys, “Simple Calibration of Non-Overlapping Cameras with a Mirror,” Proc. IEEE Computer Vision and Pattern Recognition (CVPR’08), 2008.
[R. Szeliski, 1993] R. Szeliski, “Rapid Octree Construction from Image Sequences,” Computer Vision, Graphics, and Image Processing (CVGIP): Image Understanding, vol. 58, no. 1, pp. 23-32, Jul. 1993.
[R. Zhang, 1999] R. Zhang, P.-S. Tsai, J. E. Cryer, and M. Shah, “Shape from Shading: A Survey,” IEEE Trans. Pattern Analysis and Machine Intelligence (TPAMI’99), vol. 21, no. 8, pp. 690-706, Aug. 1999.
[S. Kuthirummal, 2006] S. Kuthirummal and S. K. Nayar, “Multiview Radial Catadioptric Imaging for Scene Capture,” ACM Trans. on Graphics (ToG’06), vol. 25, no. 3, pp. 916-923, Jul. 2006.
[S. N. Sinha, 2004] S. N. Sinha, M. Pollefeys, and L. McMillian, “Camera Network Calibration from Dynamic Silhouettes,” Proc. IEEE Computer Vision and Pattern Recognition (CVPR’04), vol. 1, pp. 195-202, 2004.
[S. N. Sinha, 2005] S. N. Sinha and M. Pollefeys, “Multi-View Reconstruction Using Photo-Consistency and Exact Silhouette Constraints: A Maximum-Flow Formulation,” Proc. IEEE International Conference on Computer Vision (ICCV’05), vol. 1, pp. 349-356, 2005.
[S. Nayar, 1994] S. Nayar and Y. Nakagawa, “Shape from focus,” IEEE Pattern Analysis and Machine Intelligence, vol. 16, no. 8, pp. 824-831, Aug. 1994.
[W. Matusik, 2000] W. Matusik, C. Buehler, R. Raskar, S. J. Gortler, and L. McMillan, “Image-Based Visual Hulls,” Proc. ACM SIGGRAPH 2000, pp. 369-374, Jul. 2000.
[X. Cao, 2006] X. Cao, J. Xiao, H. Foroosh, and M. Shah, “Self-Calibration from Turn-Table Sequences in Presence of Zoom and Focus,” Computer Vision and Image Understanding (CVIU’06), vol. 102, no. 3, pp. 227-237, 2006.
[Y. Uranishi, 2006] Y. Uranishi, M. Naganawa, Y. Yasumuro, M. Imura, Y. Manabe, and K. Chihara, “Whole Shape Measurement System Using a Single Camera and a Cylindrical Mirror,” Proc. IEEE International Conference on Pattern Recognition (ICPR’06), vol. 4, pp. 866-869, 2006.
[Z. Y. Zhang, 1998] Z. Y. Zhang and H. T. Tsui, “3D Reconstruction from a Single View of an Object and Its Image in a Plane Mirror,” Proc. IEEE International Conference on Pattern Recognition (ICPR’98), vol. 2, pp. 1174-1176, 1998.
[Z. Zhang, 1994] Z. Zhang, “Iterative Point Matching for Registration of Free-Form Curves and Surfaces,” International Journal of Computer Vision (IJCV’94), vol. 13, no. 2, pp. 119-152, 1994.
[Z. Zhang, 2000] Z. Zhang, “A Flexible New Technique for Camera Calibration,” IEEE Trans. Pattern Analysis and Machine Intelligence (TPAMI’00), vol. 22, no. 11, pp. 1330-1334, 2000.