基于图像的单视点场景建模
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摘要
基于图像的建模和绘制是近年来兴起的一种倍受关注的场景建模和绘制方法,也是目前虚拟场景漫游领域中的主流技术。其最大的特点就是克服了传统的基于几何建模和绘制的不足,可以实现真实感很强的场景漫游效果。在此背景下,本文对于基于图像的单视点场景建模问题进行了探讨和研究。首先,本文综述了IBM、IBR以及单视点场景建模的各自技术特点和最新研究进展,对所涉及的射影几何、仿射几何及线性摄像机模型等相关理论和概念进行了介绍。在此基础上,本文研究了特定场景的单视点建模以及纹理平滑等问题。本文的主要内容包括:
1.针对某些特定的场景,尤其是含有人造建筑物的场景,结合具体实例提出了一种基于图像的单视点场景建模方法。该方法以射影几何、仿射几何为理论基础,以线性摄像机为模型,在满足两个前提假设条件下,利用图像中灭点、灭线等几何信息并通过灭线变换、对应点变换等数学方法,求得图像点与实际空间点之间的变换关系,进而恢复出场景的几何信息并为场景建模。与一般的IBM技术相比,该方法最大的特点就是计算量小、建模过程简单。
2. 由于受到图像分辨率的影响,从场景图像中获取的纹理图像往往分辨率不够,因此漫游过程中纹理易出现马赛克现象,从而影响了场景的真实感。针对这一点,本文提出了一种基于Catmull-Clark细分曲面的纹理平滑算法。该算法具有计算简单、易实现、图像放大时相邻像素的RGB值能够平滑过渡等特点。平滑后的纹理图像在场景漫游中仍保持了良好的连续性和光滑性,取得了比较理想的效果。
虽然从目前来看,基于图像的单视点场景建模无论在理论框架的建构,还是在具体问题的解决能力上还处于比较初步的阶段,还有大量的工作需要做,但是从长远来看,基于图像的单视点场景建模仍然是人们所关注的领域,而且只要多借鉴其它领域的技术(如纹理映射技术、绘制技术等)来完善自身的不足,它仍将是一个充满巨大发展潜力的领域。同时,这也是本文对今后工作的一个展望。
Image Based Modeling & Rendering is a fascinating technique of modeling and rendering in recent years, and now it is also the essential technique in the field of virtual navigating. Furthermore, it’s most remarkable that IBMR can overcome the deficiency of traditional Geometry Based Modeling and Rendering technique, so IBMR can realize photorealistic modeling and navigating. Under this background, the thesis researches and discusses the problem of scene modeling based on single view image, which people have focused on for many years. Firstly, we summarize the speciality and new development of IBM、IBR and single view modeling respectively. At the same time, we introduce some key concepts and theories which are touched in this thesis. In this framework, the thesis presents a modeling method of special scene based on single view and an algorithm of texture smoothing, and so on. The main content of this thesis is as follows:
1. With a given in-door scene, this thesis puts forward a modeling method of special scene based on single view image, which applies to some scenes, especially scenes containing man-made buildings. This method is based on Projective Geometry and Affine Geometry with the linear camera model. Satisfying two premise conditions, this method can obtain the transforming matrix between image pixels and points in the world by some geometry information, such as vanishing point、vanishing line, and some math transformations, such as Vanishing-Line-Transformation、Correspondences-Transformation. So the 3D scene can be modeled according to these geometry information of the scene. This method has the property of smaller computations and simpler modeling procedure.
2. This thesis presents an algorithm of texture smoothing based on Catmull-Clark subdivision surfaces. Influenced by image resolution, the scene texture image always has lower resolution so that mosaic appears easily in course of virtual navigating. Because Catmull-Clark subdivision surfaces have better property of continuity and smoothness, the algorithm of texture smoothing based on Catmull-Clark subdivision surfaces has such characters as simpler computations, better continuity and smoothness in color space, and so on. Finally this algorithm get a more perfect result.
1.针对某些特定的场景,尤其是含有人造建筑物的场景,结合具体实例提出了一种基于图像的单视点场景建模方法。该方法以射影几何、仿射几何为理论基础,以线性摄像机为模型,在满足两个前提假设条件下,利用图像中灭点、灭线等几何信息并通过灭线变换、对应点变换等数学方法,求得图像点与实际空间点之间的变换关系,进而恢复出场景的几何信息并为场景建模。与一般的IBM技术相比,该方法最大的特点就是计算量小、建模过程简单。
2. 由于受到图像分辨率的影响,从场景图像中获取的纹理图像往往分辨率不够,因此漫游过程中纹理易出现马赛克现象,从而影响了场景的真实感。针对这一点,本文提出了一种基于Catmull-Clark细分曲面的纹理平滑算法。该算法具有计算简单、易实现、图像放大时相邻像素的RGB值能够平滑过渡等特点。平滑后的纹理图像在场景漫游中仍保持了良好的连续性和光滑性,取得了比较理想的效果。
虽然从目前来看,基于图像的单视点场景建模无论在理论框架的建构,还是在具体问题的解决能力上还处于比较初步的阶段,还有大量的工作需要做,但是从长远来看,基于图像的单视点场景建模仍然是人们所关注的领域,而且只要多借鉴其它领域的技术(如纹理映射技术、绘制技术等)来完善自身的不足,它仍将是一个充满巨大发展潜力的领域。同时,这也是本文对今后工作的一个展望。
Image Based Modeling & Rendering is a fascinating technique of modeling and rendering in recent years, and now it is also the essential technique in the field of virtual navigating. Furthermore, it’s most remarkable that IBMR can overcome the deficiency of traditional Geometry Based Modeling and Rendering technique, so IBMR can realize photorealistic modeling and navigating. Under this background, the thesis researches and discusses the problem of scene modeling based on single view image, which people have focused on for many years. Firstly, we summarize the speciality and new development of IBM、IBR and single view modeling respectively. At the same time, we introduce some key concepts and theories which are touched in this thesis. In this framework, the thesis presents a modeling method of special scene based on single view and an algorithm of texture smoothing, and so on. The main content of this thesis is as follows:
1. With a given in-door scene, this thesis puts forward a modeling method of special scene based on single view image, which applies to some scenes, especially scenes containing man-made buildings. This method is based on Projective Geometry and Affine Geometry with the linear camera model. Satisfying two premise conditions, this method can obtain the transforming matrix between image pixels and points in the world by some geometry information, such as vanishing point、vanishing line, and some math transformations, such as Vanishing-Line-Transformation、Correspondences-Transformation. So the 3D scene can be modeled according to these geometry information of the scene. This method has the property of smaller computations and simpler modeling procedure.
2. This thesis presents an algorithm of texture smoothing based on Catmull-Clark subdivision surfaces. Influenced by image resolution, the scene texture image always has lower resolution so that mosaic appears easily in course of virtual navigating. Because Catmull-Clark subdivision surfaces have better property of continuity and smoothness, the algorithm of texture smoothing based on Catmull-Clark subdivision surfaces has such characters as simpler computations, better continuity and smoothness in color space, and so on. Finally this algorithm get a more perfect result.
引文
[Adelson 1991] E.H.Adelson and J.Bergen. The Plenoptic Function and the Elements of Early Vision. In Computer Models of Visual Processing, MIT Press, Cambridge, pages 3~20, MA 1991.
[Beardsley 1996] P.Beardsley, P.Torr and A.Zisserman. 3D Model Acquisition from Extended Image Sequences. In European Conference of Computer Vision, volume 2, Springer-Verlag, pages 683~695, Cambridge, England, April 1996.
[Beardsley 1997] P.Beardsley, A.Zisserman and D.Murray. Sequential Updating of Projective and Affine Structure from Motion. International Journal of Computer Vision (23), No. 3, Jun-Jul 1997, pp. 235~259, 1997.
[BLINN 1998] J.F.Blinn. Ten More Unsolved Problems in Computer Graphics. IEEE Computer Graphics & Applications, September/October 1998, pp.86~89.
[Bougnoux 1998] S.Bougnoux. From Projective to Euclidean Space under Any Practical Situation: a Criticism of Self-calibration. Proc. International Conference on Computer Vision, Narosa Publishing House, London, pp. 790-796, 1998.
[Catmull 1974] E.Catmull. A Subdivision Algorithm for Computer Display of Curved Surfaces. Ph.D. thesis, Report UTEC-CSc-74-133, Computer Science Department, University of Utah, Salt Lake City, UT, December 1974.
[Catmull 1978] E.Catmull, J.Clark. Recursively Generated B-spline Surfaces on Arbitrary Topological Meshes. Computer Aided Design, 1978, 10(6): 350~355.
[Caprile 1990] B.Caprile and V.Torre. Using Vanishing Points for Camera Calibration. International Journal of Computer Vision, pages: 127~140, 1990.
[Chen 1993] S.E.Chen and L.Willams. View Interpolation for Image Synthesis. Computer Graphics (SIGGRAPH’93), pages 279~288, August 1993.
[Chen 1995] S.E.Chen. QuickTime VR— an Image-Based Approach to Virtual Environment Navigation. Computer Graphics (SIGGRAPH’95), pages 29~38, August 1995.
[Chen 1999] S.E.Chen and Li-Wei He. Rendering with Concentric Mosaics. Proc. SIGGRAPH’99 (Los Angels), August 1999, ACM SIGGRAPH.
[Collins 1990] R.T.Collins and R.S.Weiss. Vanishing point calculation as a statistical inference on the unit sphere. In Proc. 3rd International Conference on Computer Vision, Osaka, pp.400~403, December 1990.
[Criminisi 1999] A.Criminisi, I.Reid and A.Zisserman. Single View Metrology. In Proc. 7th International Conference on Computer Vision, Kerkeyra, Greece, pages: 434~442, Sep, 1999.
[Criminisi 1999a] Antonio Criminisi. Accurate Visual Metrology from Single and Multiple Uncalibrated Images. Ph.D. thesis, pages: 42~46, 1999.
[Cruz-Neira 1993] Carolina Cruz-Neira, Virtual Reality Overview: Overview of Virtual Reality Systems, Course Notes 23 of SIGGRAPH’93.
[Doo 1978] Doo D, Sabin M. Behaviour of Recursive Division Surfaces near Extraordinary Points. Computer Aided Design, 1978, 10(6): 350~360.
[Dyn 1990] N.Dyn, D.Levin and J.A.Gregory. A Butterfly Subdivision Scheme for Surface Interpolation with Tension Control. ACM Transactions on Graphics, 1990(2): 160~169.
[Faugeras 1993] O.D.Faugeras. Three-Dimensional Computer Vision: a Geometric Viewpoint. MIT Press, 1993.
[Gortler 1996] S.J.Gortler, R.Grzeszczuk, R.Szeliski and M.F.Cohen. The Lumigraph. In Computer Graphics Proceedings, Annual Conference Series, pages 43-54, Proc. SIGGRAPH’96 (New Orleans), August 1996, ACM SIGGRAPH.
[Hartley 1994] R.Hartley. Euclidean Reconstruction from Uncalibrated Views: Applications of Invariance in Computer Vision. Lecture Notes in Computer Science, Vol. 825, Springer-Verlag, pp. 237~256, 1994.
[Herman] I.Herman. The Use of Projective Geometry in Computer Graphics. Lecture Notes in Computer Science, Vol. 564, Springer-Verlag, pp. 8~41, 1991.
[Horry 1997] Youichi Horry, Ken-ichi Anjyo and Kiyoshi Arai. Tour Into the Picture: Using a Spidery Mesh Interface to Make Animation from a Single Image. In Computer Graphics Proceedings, Annual Conference Series, pages 225~232, Proc. SIGGRAPH’97 (Los Angeles), August 1997, ACM SIGGRAPH.
[Joseph 1992] L.Joseph and Andrew Zisserman. Geometric Invariance in Computer Science. The MIT Press, Cambridge, Massachusetts, London, England, 1992.
[Kobbelt 1996] L.Kobbelt. A Variational Approach Subdivision. Computer Aided Geometric Design, 1996, 13(8): 743~761.
[Koch 1998] R.Koch, M.Pollefeys and L.Van Gool. Multi Viewpoint Stereo from Uncalibrated Video Sequences. Proc. European Conference on Computer Vision, 1998.
[Lancaster 1985] P.Lancaster and M.Tismenetaky. The Theory of Matrices (Second Edition). Academic Press, 1985.
[Levoy 1996] M.Levoy and P.Hanrahan. Light Field Rendering. In Computer Graphics Proceedings, Annual Conference Series, pages 31~42, Proc. SIGGRAPH’96 (New Orleans), August 1996, ACM SIGGRAPH.
[Liebowitz 1998] D.Liebowitz and A.Zisserman. Metric Rectification for Perspective Images of Planes. In Proceedings of the Conference on Computer Vision and Pattern Recognition, pages 482~488, June 1998.
[Liebowitz 1999] David Liebowitz and Andrew Zisserman. Combining Scene and Auto-calibration Constraints. In Proc. 7th International Conference on Computer Vision, Kerkyra, Greece, Sep, 1999.
[Loop 1987] C.Loop. Smooth Subdivision Surfaces Based on Triangles. MS thesis, Utah: Department of Mathematics, University of Utah, 1987.
[McMillan 1995] L.McMillan and G.Bishop. Plenoptic Modeling: An Image-based Rendering System. Computer Graphics (SIGGRAPH’95), pages 39~46, August 1995.
[Mohr 1998] R.Mohr, B.Boufama, and P.Brand. Understanding Positioning from Multiple Images. Artificial Intelligence, vol.78, pp. 213~238, 1995.
[Pollefeys 1998] Marc Pollefeys. Self-calibration and Metric 3D Reconstruction from Uncalibrated Image Sequences. Ph.D. thesis, Katholieke Universiteit Leuven, 1998.
[Rothwell 1993] C.A.Rothwell, D.A.Forsyth, A.Zisserman and J.L.Munday. Extracting Projective Structure from Single Perspective Views of 3D Points Sets. In Proc. International Conference on Computer Vision, pages: 573~582, 1993
[Semple 1952] J.G.Semple and G.T.Kneebone. Algebraic Projective Geometry. Oxford University Press, Amen House, London E.C.4, 1952.
[Sturm 1999] Peter F Sturm and Stephen J Maybank. A Method for Interactive 3D Reconstruction of Piecewise Planar Objects from Single Image. BMVC, Nottingham, England, pp. 265-274, September 1999.
[Tomasi 1992] C.Tomasi and T.Kanade. Shape and Motion from Image Streams under Orthography: A Factorization Approach. International Journal of Computer Vision, 9(2): 137-154, 1992.
[Vieville 1994] T.Vieville, C.Zeller and L.Robert. Using Collineations to Compute Motion and Structure in an Uncalibrated Image Sequence. I.J.C.V, 1994.
[Willson 1994] R.G.Willson. Modeling and Calibration of Automated Zoom Lenses. Ph.D. thesis, Carnegie Mellon University, 1994.
[Zhang 1996] Z.Zhang, Q.T.Luong and O.Faugeras. Motion of an Uncalibrated Stereo Rig: Self-calibration and Metric Reconstruction. IEEE Transaction on Robotics and Automation, vol.12, pp.103~113, Feb 1996.
[Zhang 1998] Z.Zhang. Image_Based Geometrically-Correct Photorealistic Scene/Object Modeling (IBPhM): A Review. In the Proceedings of the Asian Conference on Computer Vision (ACCV’98), Hong Kong, Jan 8~11, 1998.
[李象霖 1998] 李象霖 著. 数字图像处理. 中国科学技术大学研究生院, 1998, pp.193-199.
[鲍虎军 1998] 鲍虎军, 彭群生. 基于图像的图形绘制技术. 计算机世界专题报道, 1998 年 9 月 21 日.
[李桂清 2001] 李桂清, 李现民, 李华. 二次样条曲面顶点及法向插值. 计算机辅助设计与图形学学报, 2001, 13(5): 461~466.
[李桂清 2001a] 李桂清. 细分曲面造型及应用. 博士论文, 中国科学院计算技术研究所, 北京, 中国, 2001.
[马颂德 1996] 马颂德, 张正友 著: 计算机视觉— 计算理论与算法基础. 科学出版社, 1996.
[切特维鲁新 1955] 切特维鲁新 著, 东北师范大学几何教研室 译. 射影几何学. 高等教育出版社, 1955.
[彭群生 1999] 彭群生, 鲍虎军, 金小刚 著: 计算机真实感图形的算法基础. 科学出版社, 1999.
[石睿 2001] 石睿, 李华. 一种基于透视投影的纹理校正方法. 计算机应用, 2001, Vol.21, No.4, pp.30~32.
[孙家广 1995] 孙家广, 杨长贵 著: 计算机图形学(新版). 清华出版社, 1995.
[Beardsley 1996] P.Beardsley, P.Torr and A.Zisserman. 3D Model Acquisition from Extended Image Sequences. In European Conference of Computer Vision, volume 2, Springer-Verlag, pages 683~695, Cambridge, England, April 1996.
[Beardsley 1997] P.Beardsley, A.Zisserman and D.Murray. Sequential Updating of Projective and Affine Structure from Motion. International Journal of Computer Vision (23), No. 3, Jun-Jul 1997, pp. 235~259, 1997.
[BLINN 1998] J.F.Blinn. Ten More Unsolved Problems in Computer Graphics. IEEE Computer Graphics & Applications, September/October 1998, pp.86~89.
[Bougnoux 1998] S.Bougnoux. From Projective to Euclidean Space under Any Practical Situation: a Criticism of Self-calibration. Proc. International Conference on Computer Vision, Narosa Publishing House, London, pp. 790-796, 1998.
[Catmull 1974] E.Catmull. A Subdivision Algorithm for Computer Display of Curved Surfaces. Ph.D. thesis, Report UTEC-CSc-74-133, Computer Science Department, University of Utah, Salt Lake City, UT, December 1974.
[Catmull 1978] E.Catmull, J.Clark. Recursively Generated B-spline Surfaces on Arbitrary Topological Meshes. Computer Aided Design, 1978, 10(6): 350~355.
[Caprile 1990] B.Caprile and V.Torre. Using Vanishing Points for Camera Calibration. International Journal of Computer Vision, pages: 127~140, 1990.
[Chen 1993] S.E.Chen and L.Willams. View Interpolation for Image Synthesis. Computer Graphics (SIGGRAPH’93), pages 279~288, August 1993.
[Chen 1995] S.E.Chen. QuickTime VR— an Image-Based Approach to Virtual Environment Navigation. Computer Graphics (SIGGRAPH’95), pages 29~38, August 1995.
[Chen 1999] S.E.Chen and Li-Wei He. Rendering with Concentric Mosaics. Proc. SIGGRAPH’99 (Los Angels), August 1999, ACM SIGGRAPH.
[Collins 1990] R.T.Collins and R.S.Weiss. Vanishing point calculation as a statistical inference on the unit sphere. In Proc. 3rd International Conference on Computer Vision, Osaka, pp.400~403, December 1990.
[Criminisi 1999] A.Criminisi, I.Reid and A.Zisserman. Single View Metrology. In Proc. 7th International Conference on Computer Vision, Kerkeyra, Greece, pages: 434~442, Sep, 1999.
[Criminisi 1999a] Antonio Criminisi. Accurate Visual Metrology from Single and Multiple Uncalibrated Images. Ph.D. thesis, pages: 42~46, 1999.
[Cruz-Neira 1993] Carolina Cruz-Neira, Virtual Reality Overview: Overview of Virtual Reality Systems, Course Notes 23 of SIGGRAPH’93.
[Doo 1978] Doo D, Sabin M. Behaviour of Recursive Division Surfaces near Extraordinary Points. Computer Aided Design, 1978, 10(6): 350~360.
[Dyn 1990] N.Dyn, D.Levin and J.A.Gregory. A Butterfly Subdivision Scheme for Surface Interpolation with Tension Control. ACM Transactions on Graphics, 1990(2): 160~169.
[Faugeras 1993] O.D.Faugeras. Three-Dimensional Computer Vision: a Geometric Viewpoint. MIT Press, 1993.
[Gortler 1996] S.J.Gortler, R.Grzeszczuk, R.Szeliski and M.F.Cohen. The Lumigraph. In Computer Graphics Proceedings, Annual Conference Series, pages 43-54, Proc. SIGGRAPH’96 (New Orleans), August 1996, ACM SIGGRAPH.
[Hartley 1994] R.Hartley. Euclidean Reconstruction from Uncalibrated Views: Applications of Invariance in Computer Vision. Lecture Notes in Computer Science, Vol. 825, Springer-Verlag, pp. 237~256, 1994.
[Herman] I.Herman. The Use of Projective Geometry in Computer Graphics. Lecture Notes in Computer Science, Vol. 564, Springer-Verlag, pp. 8~41, 1991.
[Horry 1997] Youichi Horry, Ken-ichi Anjyo and Kiyoshi Arai. Tour Into the Picture: Using a Spidery Mesh Interface to Make Animation from a Single Image. In Computer Graphics Proceedings, Annual Conference Series, pages 225~232, Proc. SIGGRAPH’97 (Los Angeles), August 1997, ACM SIGGRAPH.
[Joseph 1992] L.Joseph and Andrew Zisserman. Geometric Invariance in Computer Science. The MIT Press, Cambridge, Massachusetts, London, England, 1992.
[Kobbelt 1996] L.Kobbelt. A Variational Approach Subdivision. Computer Aided Geometric Design, 1996, 13(8): 743~761.
[Koch 1998] R.Koch, M.Pollefeys and L.Van Gool. Multi Viewpoint Stereo from Uncalibrated Video Sequences. Proc. European Conference on Computer Vision, 1998.
[Lancaster 1985] P.Lancaster and M.Tismenetaky. The Theory of Matrices (Second Edition). Academic Press, 1985.
[Levoy 1996] M.Levoy and P.Hanrahan. Light Field Rendering. In Computer Graphics Proceedings, Annual Conference Series, pages 31~42, Proc. SIGGRAPH’96 (New Orleans), August 1996, ACM SIGGRAPH.
[Liebowitz 1998] D.Liebowitz and A.Zisserman. Metric Rectification for Perspective Images of Planes. In Proceedings of the Conference on Computer Vision and Pattern Recognition, pages 482~488, June 1998.
[Liebowitz 1999] David Liebowitz and Andrew Zisserman. Combining Scene and Auto-calibration Constraints. In Proc. 7th International Conference on Computer Vision, Kerkyra, Greece, Sep, 1999.
[Loop 1987] C.Loop. Smooth Subdivision Surfaces Based on Triangles. MS thesis, Utah: Department of Mathematics, University of Utah, 1987.
[McMillan 1995] L.McMillan and G.Bishop. Plenoptic Modeling: An Image-based Rendering System. Computer Graphics (SIGGRAPH’95), pages 39~46, August 1995.
[Mohr 1998] R.Mohr, B.Boufama, and P.Brand. Understanding Positioning from Multiple Images. Artificial Intelligence, vol.78, pp. 213~238, 1995.
[Pollefeys 1998] Marc Pollefeys. Self-calibration and Metric 3D Reconstruction from Uncalibrated Image Sequences. Ph.D. thesis, Katholieke Universiteit Leuven, 1998.
[Rothwell 1993] C.A.Rothwell, D.A.Forsyth, A.Zisserman and J.L.Munday. Extracting Projective Structure from Single Perspective Views of 3D Points Sets. In Proc. International Conference on Computer Vision, pages: 573~582, 1993
[Semple 1952] J.G.Semple and G.T.Kneebone. Algebraic Projective Geometry. Oxford University Press, Amen House, London E.C.4, 1952.
[Sturm 1999] Peter F Sturm and Stephen J Maybank. A Method for Interactive 3D Reconstruction of Piecewise Planar Objects from Single Image. BMVC, Nottingham, England, pp. 265-274, September 1999.
[Tomasi 1992] C.Tomasi and T.Kanade. Shape and Motion from Image Streams under Orthography: A Factorization Approach. International Journal of Computer Vision, 9(2): 137-154, 1992.
[Vieville 1994] T.Vieville, C.Zeller and L.Robert. Using Collineations to Compute Motion and Structure in an Uncalibrated Image Sequence. I.J.C.V, 1994.
[Willson 1994] R.G.Willson. Modeling and Calibration of Automated Zoom Lenses. Ph.D. thesis, Carnegie Mellon University, 1994.
[Zhang 1996] Z.Zhang, Q.T.Luong and O.Faugeras. Motion of an Uncalibrated Stereo Rig: Self-calibration and Metric Reconstruction. IEEE Transaction on Robotics and Automation, vol.12, pp.103~113, Feb 1996.
[Zhang 1998] Z.Zhang. Image_Based Geometrically-Correct Photorealistic Scene/Object Modeling (IBPhM): A Review. In the Proceedings of the Asian Conference on Computer Vision (ACCV’98), Hong Kong, Jan 8~11, 1998.
[李象霖 1998] 李象霖 著. 数字图像处理. 中国科学技术大学研究生院, 1998, pp.193-199.
[鲍虎军 1998] 鲍虎军, 彭群生. 基于图像的图形绘制技术. 计算机世界专题报道, 1998 年 9 月 21 日.
[李桂清 2001] 李桂清, 李现民, 李华. 二次样条曲面顶点及法向插值. 计算机辅助设计与图形学学报, 2001, 13(5): 461~466.
[李桂清 2001a] 李桂清. 细分曲面造型及应用. 博士论文, 中国科学院计算技术研究所, 北京, 中国, 2001.
[马颂德 1996] 马颂德, 张正友 著: 计算机视觉— 计算理论与算法基础. 科学出版社, 1996.
[切特维鲁新 1955] 切特维鲁新 著, 东北师范大学几何教研室 译. 射影几何学. 高等教育出版社, 1955.
[彭群生 1999] 彭群生, 鲍虎军, 金小刚 著: 计算机真实感图形的算法基础. 科学出版社, 1999.
[石睿 2001] 石睿, 李华. 一种基于透视投影的纹理校正方法. 计算机应用, 2001, Vol.21, No.4, pp.30~32.
[孙家广 1995] 孙家广, 杨长贵 著: 计算机图形学(新版). 清华出版社, 1995.