基于光度立体的高质量表面重建研究
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摘要
场景表面的三维立体重建是计算机视觉领域的一个重要分支,其在产品质量控制、工业测量、医学诊断、文物的数字化保存、现场取证、数字娱乐以及虚拟现实等多种领域都有着广泛的应用。在一些应用领域,基于摄像机多视角几何的方法往往不能精确重建场景表面的局部细节特征,使得最后输出三维模型的真实感大大降低,不能够满足实际应用要求。而基于光度立体的测量方案恰好能够较好地表现重建物体的局部细节,极大地补充和完善了基于多视角几何的方法,在研究领域得到了广泛的关注。
自从Woodham第一次提出光度立体视觉的概念以及实现方法以来,由于其在三维场景细节重建方面的巨大潜力,三十年来一直保持着持续的关注度,相关学者在这些方面做出了许多有意义的研究与探索。本文在深入分析了近年来国际上光度立体视觉领域相关问题的基础上,重点研究了光度立体视觉中的光源参数标定问题、非朗伯表面法向估计问题以及基于梯度场的表面重建问题,通过对光度立体视觉中这三个关键步骤高精度算法的实现,保证了最后高质量的光度立体三维重建结果。本文的主要工作和创新之处在于：
(1)参考传统摄像机标定算法,利用镜面反射和理想漫反射中的几何光度信息,提出了一种基于平面靶标的光源标定算法。区别于传统基于球面的光源标定算法,算法对标定靶的要求比较低,仅仅利用常见的平面镜,划分出镜面反射与漫反射区域,就可以方便地实现光源标定。本文提出的光源标定算法精度相对于传统标定算法提高了一个数量级,从而能够保证后续光度立体视觉法向估计的精度。
(2)基于线性光源系统,利用光度图像之间的线性偏差,提出了一种非朗伯表面的法向估计算法。算法在线性光源结构下,首先把高光检测问题转化成线性偏差的模式分类问题。利用经过实际数据训练或者模拟数据训练的分类器,可以较好地把高光像素从漫反射像素中进行检测,使得光度图像中剩余的少量高光误差呈现较为稀疏的分布。最后利用基于e1范数逼近的法向估计算法,能够较好地校正稀疏分布的高光误差,使得法向估计精度进一步提高。
(3)借鉴模式识别中的核方法,提出了一种基于核化泊松方程梯度重建算法。算法对传统的基于泊松方程的梯度重建算法进行核化改造,使得重建过程中每个像素的对应曲面高度不仅与该点观测到的梯度相关,并且与多个邻域像素相关。通过对重建过程中边界问题的仔细考虑,保证重建曲面的边界平滑,使得从具有高斯分布噪声的梯度场数据能够较好地重建目标曲面。
(4)参考压缩感知领域稀疏表达的思想,提出一种基于e1线性解码的梯度重建算法。对于呈稀疏分布的梯度噪声,算法利用e1线性解码方法从受到污染的梯度数据到目标曲面的高度数据进行重建,并在解码框架中加入拉普拉斯正则项,极大地增加了算法在稀疏噪声下的解码能力。实验证明,在多达30%以上像素点存在稀疏噪声的情况下,算法依然能够稳定地从梯度数据进行表面重建。
Three-dimensional surface reconstruction is an important branch of Computer Vision. It's widely used in the quality control of products, industrial measurement, medical diagnosis, digital preservation of relic, scene evidence, digital entertainment and virtual reality. In some application areas, local features of measured object can not be well reconstructed based on the geometric measurement methods which makes the details of reconstructed surface quality greatly degraded, and often can not meet the requirements. On the other hand, Photometric Stereo scheme can keep local details of measured objects well. It's quite satisfactory for Computer Vision applications and greatly complement geometric measurement based methods. Thus Photometric Stereo has received wide attention in the research field.
Since Woodham first proposed the concept and realization method of Photometric Stereo, its huge potential in the three-dimensional reconstruction of scene details has been noticed. In the past thirty years, scholars have done many meaningful research and exploration in this area and it is still a hot topic and trend now. After analyzing the recent research approach in Photometric Stereo field, we have focused on the light sources calibration, non-Lambertian surface normal estimation and shape from gradient fields. By keeping highly precision in each step of Photometric Stereo using proposed methods, the final reconstruction outperform state-of-the-art techniques. The major work and contributions lie in several fields as follows:
(1) Using the parameters acquired by camera calibration and the geometric&photometric in-formation from the planar surface, we propose a novel method for light sources calibration. Differ-ing from the traditional sphere based methods, proposed method just uses a plane mirror which is divided into mirror reflection area and diffuse area respectively. The accuracy of proposed method is improved by an order of magnitude compared to the traditional methods. So it can guarantee the precision of normal estimation in Photometric Stereo.
(2) Based on co-linear light source configuration, we propose a non-Lambertian surface nor-mal estimation method. Using deviations in photometric images under the co-linear light sources, the specularity detection problem is converted to pattern classification one. In training step we use real or synthetical data to build a robust specularity classifier. After discarding specularity in testing step, the residual error in photometric images is sparsely distributed. An l1-norm approximation method is designed to further correct the sparse error. The final output is quite satisfactory.
(3) We convert the traditional Poisson equation to its kernel representation form using kernel method, the underlying surface can be recovered from Gaussian noise contaminated gradient data by kernel regression. With further considering an even extension of the original gradient field, we assume a Neumann boundary condition which makes final reconstructed surface smooth and reliable.
(4) Inspired by the idea of sparse expression in compressive sensing fields, we propose andecoding method for surface reconstruction from gradient fields. The l1decoding procedure can exactly recover surface from sparse noise contaminated gradient data. The Laplacian term is additionally employed to increase the information in decoding matrix and suppress noise and/or outliers. Experimental results validate that the proposed method significantly outperforms state-of-the-art techniques, and can produce satisfactory reconstruction even in the very extreme situation of60%outliers.
自从Woodham第一次提出光度立体视觉的概念以及实现方法以来,由于其在三维场景细节重建方面的巨大潜力,三十年来一直保持着持续的关注度,相关学者在这些方面做出了许多有意义的研究与探索。本文在深入分析了近年来国际上光度立体视觉领域相关问题的基础上,重点研究了光度立体视觉中的光源参数标定问题、非朗伯表面法向估计问题以及基于梯度场的表面重建问题,通过对光度立体视觉中这三个关键步骤高精度算法的实现,保证了最后高质量的光度立体三维重建结果。本文的主要工作和创新之处在于：
(1)参考传统摄像机标定算法,利用镜面反射和理想漫反射中的几何光度信息,提出了一种基于平面靶标的光源标定算法。区别于传统基于球面的光源标定算法,算法对标定靶的要求比较低,仅仅利用常见的平面镜,划分出镜面反射与漫反射区域,就可以方便地实现光源标定。本文提出的光源标定算法精度相对于传统标定算法提高了一个数量级,从而能够保证后续光度立体视觉法向估计的精度。
(2)基于线性光源系统,利用光度图像之间的线性偏差,提出了一种非朗伯表面的法向估计算法。算法在线性光源结构下,首先把高光检测问题转化成线性偏差的模式分类问题。利用经过实际数据训练或者模拟数据训练的分类器,可以较好地把高光像素从漫反射像素中进行检测,使得光度图像中剩余的少量高光误差呈现较为稀疏的分布。最后利用基于e1范数逼近的法向估计算法,能够较好地校正稀疏分布的高光误差,使得法向估计精度进一步提高。
(3)借鉴模式识别中的核方法,提出了一种基于核化泊松方程梯度重建算法。算法对传统的基于泊松方程的梯度重建算法进行核化改造,使得重建过程中每个像素的对应曲面高度不仅与该点观测到的梯度相关,并且与多个邻域像素相关。通过对重建过程中边界问题的仔细考虑,保证重建曲面的边界平滑,使得从具有高斯分布噪声的梯度场数据能够较好地重建目标曲面。
(4)参考压缩感知领域稀疏表达的思想,提出一种基于e1线性解码的梯度重建算法。对于呈稀疏分布的梯度噪声,算法利用e1线性解码方法从受到污染的梯度数据到目标曲面的高度数据进行重建,并在解码框架中加入拉普拉斯正则项,极大地增加了算法在稀疏噪声下的解码能力。实验证明,在多达30%以上像素点存在稀疏噪声的情况下,算法依然能够稳定地从梯度数据进行表面重建。
Three-dimensional surface reconstruction is an important branch of Computer Vision. It's widely used in the quality control of products, industrial measurement, medical diagnosis, digital preservation of relic, scene evidence, digital entertainment and virtual reality. In some application areas, local features of measured object can not be well reconstructed based on the geometric measurement methods which makes the details of reconstructed surface quality greatly degraded, and often can not meet the requirements. On the other hand, Photometric Stereo scheme can keep local details of measured objects well. It's quite satisfactory for Computer Vision applications and greatly complement geometric measurement based methods. Thus Photometric Stereo has received wide attention in the research field.
Since Woodham first proposed the concept and realization method of Photometric Stereo, its huge potential in the three-dimensional reconstruction of scene details has been noticed. In the past thirty years, scholars have done many meaningful research and exploration in this area and it is still a hot topic and trend now. After analyzing the recent research approach in Photometric Stereo field, we have focused on the light sources calibration, non-Lambertian surface normal estimation and shape from gradient fields. By keeping highly precision in each step of Photometric Stereo using proposed methods, the final reconstruction outperform state-of-the-art techniques. The major work and contributions lie in several fields as follows:
(1) Using the parameters acquired by camera calibration and the geometric&photometric in-formation from the planar surface, we propose a novel method for light sources calibration. Differ-ing from the traditional sphere based methods, proposed method just uses a plane mirror which is divided into mirror reflection area and diffuse area respectively. The accuracy of proposed method is improved by an order of magnitude compared to the traditional methods. So it can guarantee the precision of normal estimation in Photometric Stereo.
(2) Based on co-linear light source configuration, we propose a non-Lambertian surface nor-mal estimation method. Using deviations in photometric images under the co-linear light sources, the specularity detection problem is converted to pattern classification one. In training step we use real or synthetical data to build a robust specularity classifier. After discarding specularity in testing step, the residual error in photometric images is sparsely distributed. An l1-norm approximation method is designed to further correct the sparse error. The final output is quite satisfactory.
(3) We convert the traditional Poisson equation to its kernel representation form using kernel method, the underlying surface can be recovered from Gaussian noise contaminated gradient data by kernel regression. With further considering an even extension of the original gradient field, we assume a Neumann boundary condition which makes final reconstructed surface smooth and reliable.
(4) Inspired by the idea of sparse expression in compressive sensing fields, we propose andecoding method for surface reconstruction from gradient fields. The l1decoding procedure can exactly recover surface from sparse noise contaminated gradient data. The Laplacian term is additionally employed to increase the information in decoding matrix and suppress noise and/or outliers. Experimental results validate that the proposed method significantly outperforms state-of-the-art techniques, and can produce satisfactory reconstruction even in the very extreme situation of60%outliers.
引文
[1]彭群生,鲍虎军,金小刚.计算机真实感图形的算法基础.科学出版社,2009.
[2]李晓彤,岑兆丰.几何光学像差光学设计(第二版).浙江大学出版社,2007.
[3]张广军.机器视觉.科学出版社,2005.
[4]徐海松.颜色信息工程.浙江大学出版社,2006.
[5]章毓晋.图像理解.清华大学出版社,2007.
[6]A. Agrawal, R. Chellappa, and R. Raskar. An algebraic approach to surface reconstruc-tion from gradient fields. In Proceedings of the 2005 IEEE International Conference on Computer Vision,2005.
[7]A. Agrawal, R. Raskar, and R. Chellappa. What is the range of surface reconstructions from a gradient field? In Proceedings of the 2006 IEEE European Conference on Computer Vision,2006.
[8]N. Alldrin and D. J. Kriegman. Toward reconstructing surfaces with arbitrary isotropic reflectance:A stratified photometric stereo approach. In Proceedings of the 2007 IEEE International Conference on Computer Vision,2007.
[9]T. Aoto, T. Taketomi, T. Sato, Y. Mukaigawa, and N. Yokoya. Position estimation of near point light sources using a clear hollow spheren. In Proceedings of the 2012 IEEE Interna-tional Conference on Pattern Recognition,2012.
[10]R. T. Azuma. A survey of augmented reality. Presence-Teleoperators and Virtual Environ-ments,6(4),1997.
[11]S. Barsky and M. Petrou. The 4-source photometric stereo technique for three-dimensional surfaces in the presence of highlights and shadows. IEEE Transactions on Pattern Analysis and Machine Intelligence,25(10):1239-1252,2003.
[12]P. Beckmann and A. Spizzichino. The Scattering of Electromagnetic Waves from Rough Surfaces. Norwood, MA, Artech House, Inc.,1987.
[13]P. N. Belhumeur, D. J. Kriegman, and A. L. Yuille. The bas-relief ambiguity. International Journal of Computer Vision,35(1):33-44,1999.
[14]C. M. Bishop. Pattern Recognition and Machine Learning. Springer,New York,2006.
[15]P. Bloomfield and W. Steiger. Least absolute deviations:theory, applications, and algo-rithms. Birkhaauser, Boston,1983.
[16]B. E. Boser, I. M. Guyon, and V. N. Vapnik. A training algorithm for optimal margin clas-sifiers. In Proceedings of 1992 ACM annual workshop on Computational learning theory, 1992.
[17]C. S. Bouganis and M. Brookes. Multiple light source detection. IEEE Transactions on Pattern Analysis and Machine Intelligence,26(4):509--514,2004.
[18]J. Bouguet. Camera calibration toolbox for matlab. http://www.vision.caltech. edu/bouguetj/calib_doc/index.html.
[19]S. P. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press,2004.
[20]Y. Boykov,O. Veksler, and R. Zabih. Fast approximate energy minimisation via graph cuts. IEEE Transactions on Pattern Analysis and Machine Intelligence,23(11):1222-1239,2001.
[21]M. J. Brooks and B. K. P. Horn. Shape and source from shading. In International Joint Conference on Artilcial Intelligence,1985.
[22]G. Burdea and P. Coffet. Virtual reality technology. Presence:Teleoperators & Virtual Environments,12(6).663-664,2003.
[23]E. J. Candes and T. Tao. Decoding by linear programming. IEEE Transactions on Informa-tion Theory,51(12):4203-4215,2005.
[24]M. Chandraker, S. Agarwal, and D. Kriegman. Shadowcuts:Photometric stereo with shad-ows. In Proceedings of the 2007 IEEE Conference on Computer Vision and Pattern Recog-nition,2007.
[25]C. C. Chang and C. J. Lin. LIBSVM:a library for support vector machines. ACM Trans-actions on Intelligent Systems and Technology,2(3):27:1-27:7,2011. Software available at http://www.csie.ntu.edu.tw/~cjlin/libsvm.
[26]S. S. Chen, D. L. Donoho, and M. A. Saunders. Atomic decomposition by basis pursuit. SI AM journal on scientific computing,20(1):33-61,1998.
[27]Y. Cheng and H. L. Shen. Robust surface reconstruction from gradient fields. In IEEE Conference on Computer Science and Automation Engineering,2012.
[28]Y. Cheng and H. L. Shen. Shape from gradient fields. Electronics Letters,48(7):375-376, 2012.
[29]W. Chojnacki, M. J. Brooks, and D. Gibbins. Revisiting pentland's estimator of light source direction. Journal of the Optical Society of America A,11(1):118-124,1994.
[30]R. L. Cook and K. E. Torrance. A reflectance model for computer graphics. ACM Transac-tions on Graphics, 1(1):7-24,1982.
[31]C. Cortes and V. Vapnik. Support-vector networks. Machine learning,20(3):273-297,1995.
[32]J. Davis, D. Ramamoorthi, and D. R. Nehab. Spacetime stereo:A unifying framework for depth from triangulation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 27(2):296-302,2005.
[33]A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via the em algorithm. Journal of the Royal Statistical Society. Series B (Methodological), pages 1-38,1977.
[34]O. Drbohlav and M. Chantler. Can two specular pixels calibrate photometric stereo? In Proceedings of the 2005 IEEE International Conference on Computer Vision,2005.
[35]O. Drbohlav and R. Sara. Specularities reduce ambiguity of uncalibrated photometric stereo. In Proceedings of the 2002 IEEE European Conference on Computer Vision,2002.
[36]D. A. Forsyth and J. Ponce. Computer Vision:A Modern Approach,Second Edition. Prentice Hall,2011.
[37]R. T. Frankot and R. Chellappa. A method for enforcing integrability in shape from shading algorithms. IEEE Transactions on Pattern Analysis and Machine Intelligence,10(4):439-451,1987.
[38]A. S. Georghiades. Incorporating the torrance and sparrow model of reflectance in uncal-ibrated photometric stereo. In Proceedings of the 2003 IEEE International Conference on Computer Vision,2003.
[39]A. S. Georghiades, P. N. Belhumeur, and D. J. Kriegman. From few to many:Illumination cone models for face recognition under variable lighting and pose. IEEE Transactions on Pattern Analysis and Machine Intelligence,23(6):643-660,2001.
[40]V. P. Godambe. Estimating functions. Oxford University Press,1991.
[41]D. B. Goldman, B. Curless, A. Hertzmann, and S. Seitz. Shape and spatially-varying brdfs from photometric stereo. In Proceedings of the 2005 IEEE International Conference on Computer Vision,2005.
[42]Google. Google glass. http://en.wikipedia.org/wiki/Google_Glass.
[43]J. E. Greivenkamp. Field guide to geometrical optics. SPIE Press,2004.
[44]P. R. Halmos and P. P. R. Halmos. A Hilbert space problem book. Springer Verlag,1982.
[45]M. Harker and P. O'Leary. Least squares surface reconstruction from measured gradien-t fields. In Proceedings of the 2008 IEEE Conference on Computer Vision and Pattern Recognition,2008.
[46]A. Harrison. Maximum likelihood estimation of depth maps using photometric stereo. IEEE Transactions on Pattern Analysis and Machine Intelligence,34(7):1368-1380,2012.
[47]R. Hartley and A. Zisserman. Multiple view geometry in computer vision. Cambridge,2000.
[48]H. Hayakawa. Photometric stereo under a light source with arbitrary motion. Journal of the Optical Society of America A,11(11):3079-3089,1994.
[49]H. V. Helmholtz. Handbuch der physiologischen Optik. Voss,1867.
[50]S. Herbort and C. Wohler. An introduction to image-based 3d surface reconstruction and a survey of photometric stereo method.3D Research,2(3):1-17,2011.
[51]C. Hernandez, G. Vogiatzis, and R. Cipolla. Multiview photometric stereo. IEEE Transac-tions on Pattern Analysis and Machine Intelligence,30(3):548-554,2008.
[52]C. Hernandez, G. Vogiatzis, and R. Cipolla. Shadows in three-source photometric stereo. In Proceedings of the 2008 IEEE European Conference on Computer Vision,2008.
[53]A. Hertzmann and S. M. Seitz. Shape and materials by example:A photometric stereo approach. In Proceedings of the 2003 IEEE Conference on Computer Vision and Pattern Recognition,2003.
[54]A. Hertzmann and S. M. Seitz. Example-based photometric stereo:Shape reconstruction with general, varying brdfs. IEEE Transactions on Pattern Analysis and Machine Intelli-gence,27(8):1254-1264,2005.
[55]B. Horn and M. Brooks. The variational approach to shape from shading. Computer Vision, Graphics, and Image Processing,33(2):174-208,1986.
[56]B. K. P. Horn. Shape from shading:A method for obtaining the shape of a smooth opaque object from one view. AITR-232,1970.
[57]C. W. Hsu and C. J. Lin. A comparison of methods for multiclass support vector machines. IEEE Transactions on Neural Networks,13(2):415-425,2002.
[58]S. Ikehata, D. Wipf, Y. Matsushita, and K. Aizawa. Robust photometric stereo using sparse regression. In Proceedings of the 2012 IEEE Conference on Computer Vision and Pattern Recognition,2012.
[59]K. Ikeuchi and K. Sato. Determining refectance parameters using range and brightness images. In Proceedings of the 1990 IEEE International Conference on Computer Vision, 1990.
[60]E. C. Jr. and R. Jain. Obtaining 3-dimensional shape of textured and specular surfaces using four-source photometry. Computer Graphics and Image Processing,18(4):309-328,1982.
[61]P. Kovesi. Shapelets correlated with surface normals produce surfaces. In Proceedings of the 2005 IEEE International Conference on Computer Vision,2005.
[62]P. Lagger and P. Fua. Retriving multiple light sources in the presence of specular reflections and texture. IEEE Transactions on Pattern Analysis and Machine Intelligence,111(2):207--218,2008.
[63]J. H. Lambert. Photometria, sive de Mensura et Gradibus Luminis, Colorum et Umbrae. Augsburg,1760.
[64]J. Lawrence, A. Ben-Artzi, C. DeCoro, W. Matusik, H. Pfister, R. Ramamoorthi, and S. Rusinkiewicz. Inverse shade trees for non-parametric material representation and editing. ACM Transactions on Graphics,25(3):735-745,2006.
[65]L. Le Cam. Maximum likelihood:an introduction. International Statistical Review/Revue Internationale de Statistique, pages 153-171,1990.
[66]C. H. Lee and A. Rosenfeld. Improved methods of estimating shape from shading using the light source coordinate system. Artificial Intelligence,26(2):125-143,1985.
[67]S. Z. Li. Markov Random Field Modeling in Image Analysis. Springer,2009.
[68]M. Liao, X. Huang, and R. Yang. Interreflection removal for photometric stereo by using spectrum-dependent albedo. In Proceedings of the 2011 IEEE Conference on Computer Vision and Pattern Recognition,2011.
[69]S. P. Mallick, T. E. Zickler, D. J. Kriegman, and P. N. Belhumeur. Beyond lambert:Re-constructing specular surfaces using color. In Proceedings of the 2005 IEEE Conference on Computer Vision and Pattern Recognition,2005.
[70]D. Marr. Vision:A Computational Investigation into the Human Representation and Pro-cessing of Visual Information. New York, Freeman,1982.
[71]W. Matusik, H. Pfister, M. Brand, and L. McMillan. A datadriven reflectance model. ACM Transactions on Graphics,22(3):759-769,2003.
[72]Microsoft. Kinect. http://www.xbox.com/en-US/kinect/default.htm.
[73]P. Milgram and A. F. Kishino. Taxonomy of mixed reality visual displays. IEICE Transac-tions on Information Systems,77(12):1321-1329,1994.
[74]D. Miyazaki, K. Hara, and K. Ikeuchi. Median photometric stereo as applied to the segonko tumulus and museum objects. International journal of computer vision,86(2):229-242, 2010.
[75]D. Miyazaki and K. Ikeuchi. Photometric stereo using graph cut and m-estimation for a virtual tumulus in the presence of highlights and shadows. In IEEE Conference on Computer Vision and Pattern Recognition Workshops,2010.
[76]Y. Mukaigawa, Y. Ishii, and T. Shakunaga. Classification of photometric factors based on photometric linearization. In Proceedings of the 2006 IEEE Asian Conference on Computer Vision,2006.
[77]Y. Mukaigawa, Y. Ishii, and T. Shakunaga. Analysis of photometric factors based on photo-metric linearization. Journal of the Optical Society of America A,24(10):3326-3334,2007.
[78]S. Nayar, K. Ikeuchi, and T. Kanade. Shape from interreflections. International Journal of Computer Vision,6(3):173-195,1991.
[79]S. K. Nayar, K. Ikeuchi, and T. Kanade. Surface reflection:Physical and geometrical per-spectives. IEEE Transactions on Pattern Analysis and Machine Intelligence,13(7):611-634, 1999.
[80]S. K. Nayar, G. Krishnan, M. D. Grossberg, and R. Raskar. Fast separation of direct and global components of a scene using high frequency illumination. ACM Transactions on Graphics,25(3):935-944,2006.
[81]D. Nehab, S. Rusinkiewicz, J. Davis, and R. Ramamoorthi. Efficiently combining positions and normals for precise 3d geometry. ACM Transactions on Graphics,24(3):536-543,2005.
[82]H. S. Ng, T. P. Wu, and C. K. Tang. Surface-from-gradients without discrete integrabili-ty enforcement:a gaussian kernel approach. IEEE Trans. Pattern Analysis and Machine Intelligence,32(11):2085-2099,2009.
[83]R. M. Norton. The double exponential distribution:Using calculus to find a maximum like-lihood estimator. The American Statistician(American Statistical Association),38(2):135-136,1984.
[84]J. Pearl. Reverend Bayes on inference engines:A distributed hierarchical approach. Cog-nitive Systems Laboratory, School of Engineering and Applied Science, University of Cali-fornia, Los Angeles,1982.
[85]A. Pentland. Linear shape from shading. International journal of computer vision, 4(26):153-162,1990.
[86]A. P. Pentland. Finding the illuminant direction. Journal of the Optical Society of America A,72(l):448-455,1982.
[87]N. Petrovic, T. Cohen, B. Frey, R. Koetter, and T. Huang. Enforcing integrability for surface reconstruction algorithms using belief propagation in graphical models. In Proceedings of the 2001 IEEE Conference on Computer Vision and Pattern Recognition,2001.
[88]B. T. Phong. Illumination for computer generated pictures. Communications of ACM, 18(6):311--317,1975.
[89]M. Powell, S. Sarkar, and D. Goldgof. A simple strategy for calibrating the geometry of light sources. PAMI,23(1):1022-1027,2001.
[90]D. Reddy, A. Agrawal, and R. Chellappa. Enforcing integrability by error correction using (?)l-minimization. In Proceedings of the 2009 IEEE Conference on Computer Vision and Pattern Recognition,2009.
[91]D. F. Rogers and J. A. Adams. Mathematical elements for computer graphics. McGraw-Hill Higher Education,1989.
[92]L. I. Rudin, S. sher, and E. Fatemi. Nonlinear total variation based noise removal algorithms. Physica D:Nonlinear Phenomena,60(1):259-268,1992.
[93]Y. Sato and K. Ikeuchi. Temporal-color space analysis of reflection. Journal of Optical Society of America A, 11(11):2990-3002,1994.
[94]Y. Sato and K. Ikeuchi. Shape and spatially-varying brdfs from photometric stereo. IEEE Transactions on Pattern Analysis and Machine Intelligence,32(6):1060-1071,2010.
[95]K. Schluns. Photometric stereo for non-lambertian surfaces using color information. In Proceedings of Computer Analysis of Images and Patterns,1995.
[96]S. Seitz, Y. Matsushita, and K. Kutulakos. A theory of inverse light transport. In Proceedings of the 2005 IEEE International Conference on Computer Vision,2005.
[97]S. Shafer. Using color to separate reflection components. Color Research and Applications, 10(4):210-218,1985.
[98]H. L. Shen and Y. Cheng. Calibrating light sources by using a planar mirror. Journal of Electronic Imaging,20(1),2011.
[99]B. Shi, Y. Matsushita, Y. Wei, C. Xu, and P. Tan. Self-calibrating photometric stereo. In Proceedings of the 2010 IEEE Conference on Computer Vision and Pattern Recognition, 2010.
[100]T. Simchony, R. Chellappa, and M. Shao. Direct analytical methods for solving poisson e- quations in computer vision problems. IEEE Transactions on Pattern Analysis and Machine Intelligence,12(5):435-446,1990.
[101]M. M. Stark, J. Arvo, and B. Smits. Barycentric parameterizations for isotropic brdfs. IEEE Transactions on Visualization and Computer Graphics,11(2):126-138,2005.
[102]P. Tan, S. P. Mallick, L. Quan, D. J. Kriegman, and T. Zickler. Isotropy, reciprocity and the generalized bas-relief ambiguity. In Proceedings of the 2007 IEEE Conference on Computer Vision and Pattern Recognition,2007.
[103]P. Tan and T. Zickler. A projective framework for radiometric image analysis. In Proceedings of the 2009 IEEE Conference on Computer Vision and Pattern Recognition,2009.
[104]K. L. Tang, C. K. Tang, and T. T. Wong. Dense photometric stereo using tensorial belief propagation. In Proceedings of the 2005 IEEE Conference on Computer Vision and Pattern Recognition, pages 132-139,2005.
[105]J. S. Taylor and Cristianini. Kernel Methods for Pattern Analysis. Cambridge University Press,2004.
[106]E. Torrance and E. M. Sparrow. Theory for offspecular reflection from roughened surfaces. Journal of the Optical Society of America A,57(9):1105-1114,1967.
[107]Y. Wang and D. Samaras. Estimation of multiple directional illuminants from a single image. Image and Visual Computing,26(1):1179-1195,2008.
[108]G. J. Ward. Measuring and modeling anisotropic reflection. ACM Transactions on Graphics, 26(2):265-272,1992.
[109]W. L. Wolfe. Introduction to radiometry. SPIE Press,1998.
[110]R. J. Woodham. Photometric method for determining surface orientation from multiple images. Optical Engineering,19(1):139-144,1980.
[111]L. Wu, A. Ganesh, B. Shi, Y. Matsushita, Y. Wang, and Y. Ma. Robust photometric stere-o via low-rank matrix completion and recovery. In Proceedings of the 2010 IEEE Asian Conference on Computer Vision,2010.
[112]T. P. Wu and C. K. Tang. Dense photometric stereo using a mirror sphere and graph cut. In Proceedings of the 2005 IEEE Conference on Computer Vision and Pattern Recognition, 2005.
[113]T. P. Wu and C. K. Tang. Dense photometric stereo by expectation-maximization. In Pro-ceedings of the 2006 IEEE Conference on Computer Vision and Pattern Recognition,2006.
[114]T. P. Wu and C. K. Tang. Visible surface reconstruction from normals with discontinuity consideration. In Proceedings of the 2006 IEEE Conference on Computer Vision and Pattern Recognition,2006.
[115]T. P. Wu and C. K. Tang. Photometric stereo via expectation maximization. IEEE Transac-tions on Pattern Analysis and Machine Intelligence,32(3):546-560,2010.
[116]T. P. Wu, K. L. Tang, C. K. Tang, and T. T. Wong. Dense photometric stereo:a markov random field approach. IEEE Transactions on Pattern Analysis and Machine Intelligence, 28(11):1830-1846,2006.
[117]S. Xu and A. Wallace. Recovering surface reflectance and multiple light locations and in-tensities from image data. Pattern Recognition Letters,29(11):1639-1647,2008.
[118]Y. Yang and A. Yuille. Source from shading. In Proceedings of the 1991 IEEE Conference on Computer Vision and Pattern Recognition,1991.
[119]C. Yu, Y. Seo, and S. W. Lee. Photometric stereo from maximum feasible lambertian reflec-tions. In Proceedings of the 2010 IEEE European Conference on Computer Vision,2010.
[120]A. Yuille and D. Snow. Shape and albedo from multiple images using integrability. In Proceedings of the 1997 IEEE Conference on Computer Vision and Pattern Recognition, 1997.
[121]A. L. Yuille, J. M. Coughlan, and S. Konishi. The generic viewpoint constraint resolves the generalized bas relief ambiguity. In Conference on Information Science and Systems,2000.
[122]YZhang and Y.Yang. Illumination direction determination for multiple light sources. In Proceedings of the 2000 IEEE Conference on Computer Vision and Pattern Recognition, pages 269-276,2000.
[123]L. Zhang, B. Curless, and S. M. Seitz. Spacetime stereo:Shape recovery for dynamic scenes. In Proceedings of the 2003 IEEE Conference on Computer Vision and Pattern Recognition, 2003.
[124]Q. Zhang, M. Ye, R. G. Yang, Y. Matsushita, B. Wilburn, and H. M. Yu. Edge-preserving photometric stereo via depth fusion. In Proceedings of the 2012 IEEE Conference on Com-puter Vision and Pattern Recognition,2012.
[125]R. Zhang, P. S. Tsai, and J. E. Cryer. Shape-from-shading:a survey. IEEE Transactions on Pattern Analysis and Machine Intelligence,21(8):690-706,1999.
[126]Y. Zhang and Y. Yang. Multiple illuminant direction with application to image synthesis. IEEE Transactions on Pattern Analysis and Machine Intelligence,23(8):915-920,2001.
[127]Q. Zheng and R.Chellappa. Estimation of illumination, albedo, and shape from shad. IEEE Transactions on Pattern Analysis and Machine Intelligence,13(1):680-702,1991.
[128]Zhou and P. Tan. Ring-light photometric stereo. In Proceedings of the 2010 IEEE European Conference on Computer Vision,2010.
[129]W. Zhou and C. Kambhamettu. Estimation of illuminant direction and intensity of multiple light sources. In Proceedings of the 2002 IEEE European Conference on Computer Vision, 2002.
[130]W. Zhou and C. Kambhamettu. A unified framework for scene illuminant estimation. Image and Vision Computing,26(3):415-429,2008.
[131]T. E. Zickler, P. N. Belhumeur, and D. J. Kriegman. Helmholtz stereopsis:Exploiting reci-procity for surface reconstruction. International Journal of Computer Vision,49(2-3):215-227,2002.
[2]李晓彤,岑兆丰.几何光学像差光学设计(第二版).浙江大学出版社,2007.
[3]张广军.机器视觉.科学出版社,2005.
[4]徐海松.颜色信息工程.浙江大学出版社,2006.
[5]章毓晋.图像理解.清华大学出版社,2007.
[6]A. Agrawal, R. Chellappa, and R. Raskar. An algebraic approach to surface reconstruc-tion from gradient fields. In Proceedings of the 2005 IEEE International Conference on Computer Vision,2005.
[7]A. Agrawal, R. Raskar, and R. Chellappa. What is the range of surface reconstructions from a gradient field? In Proceedings of the 2006 IEEE European Conference on Computer Vision,2006.
[8]N. Alldrin and D. J. Kriegman. Toward reconstructing surfaces with arbitrary isotropic reflectance:A stratified photometric stereo approach. In Proceedings of the 2007 IEEE International Conference on Computer Vision,2007.
[9]T. Aoto, T. Taketomi, T. Sato, Y. Mukaigawa, and N. Yokoya. Position estimation of near point light sources using a clear hollow spheren. In Proceedings of the 2012 IEEE Interna-tional Conference on Pattern Recognition,2012.
[10]R. T. Azuma. A survey of augmented reality. Presence-Teleoperators and Virtual Environ-ments,6(4),1997.
[11]S. Barsky and M. Petrou. The 4-source photometric stereo technique for three-dimensional surfaces in the presence of highlights and shadows. IEEE Transactions on Pattern Analysis and Machine Intelligence,25(10):1239-1252,2003.
[12]P. Beckmann and A. Spizzichino. The Scattering of Electromagnetic Waves from Rough Surfaces. Norwood, MA, Artech House, Inc.,1987.
[13]P. N. Belhumeur, D. J. Kriegman, and A. L. Yuille. The bas-relief ambiguity. International Journal of Computer Vision,35(1):33-44,1999.
[14]C. M. Bishop. Pattern Recognition and Machine Learning. Springer,New York,2006.
[15]P. Bloomfield and W. Steiger. Least absolute deviations:theory, applications, and algo-rithms. Birkhaauser, Boston,1983.
[16]B. E. Boser, I. M. Guyon, and V. N. Vapnik. A training algorithm for optimal margin clas-sifiers. In Proceedings of 1992 ACM annual workshop on Computational learning theory, 1992.
[17]C. S. Bouganis and M. Brookes. Multiple light source detection. IEEE Transactions on Pattern Analysis and Machine Intelligence,26(4):509--514,2004.
[18]J. Bouguet. Camera calibration toolbox for matlab. http://www.vision.caltech. edu/bouguetj/calib_doc/index.html.
[19]S. P. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press,2004.
[20]Y. Boykov,O. Veksler, and R. Zabih. Fast approximate energy minimisation via graph cuts. IEEE Transactions on Pattern Analysis and Machine Intelligence,23(11):1222-1239,2001.
[21]M. J. Brooks and B. K. P. Horn. Shape and source from shading. In International Joint Conference on Artilcial Intelligence,1985.
[22]G. Burdea and P. Coffet. Virtual reality technology. Presence:Teleoperators & Virtual Environments,12(6).663-664,2003.
[23]E. J. Candes and T. Tao. Decoding by linear programming. IEEE Transactions on Informa-tion Theory,51(12):4203-4215,2005.
[24]M. Chandraker, S. Agarwal, and D. Kriegman. Shadowcuts:Photometric stereo with shad-ows. In Proceedings of the 2007 IEEE Conference on Computer Vision and Pattern Recog-nition,2007.
[25]C. C. Chang and C. J. Lin. LIBSVM:a library for support vector machines. ACM Trans-actions on Intelligent Systems and Technology,2(3):27:1-27:7,2011. Software available at http://www.csie.ntu.edu.tw/~cjlin/libsvm.
[26]S. S. Chen, D. L. Donoho, and M. A. Saunders. Atomic decomposition by basis pursuit. SI AM journal on scientific computing,20(1):33-61,1998.
[27]Y. Cheng and H. L. Shen. Robust surface reconstruction from gradient fields. In IEEE Conference on Computer Science and Automation Engineering,2012.
[28]Y. Cheng and H. L. Shen. Shape from gradient fields. Electronics Letters,48(7):375-376, 2012.
[29]W. Chojnacki, M. J. Brooks, and D. Gibbins. Revisiting pentland's estimator of light source direction. Journal of the Optical Society of America A,11(1):118-124,1994.
[30]R. L. Cook and K. E. Torrance. A reflectance model for computer graphics. ACM Transac-tions on Graphics, 1(1):7-24,1982.
[31]C. Cortes and V. Vapnik. Support-vector networks. Machine learning,20(3):273-297,1995.
[32]J. Davis, D. Ramamoorthi, and D. R. Nehab. Spacetime stereo:A unifying framework for depth from triangulation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 27(2):296-302,2005.
[33]A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via the em algorithm. Journal of the Royal Statistical Society. Series B (Methodological), pages 1-38,1977.
[34]O. Drbohlav and M. Chantler. Can two specular pixels calibrate photometric stereo? In Proceedings of the 2005 IEEE International Conference on Computer Vision,2005.
[35]O. Drbohlav and R. Sara. Specularities reduce ambiguity of uncalibrated photometric stereo. In Proceedings of the 2002 IEEE European Conference on Computer Vision,2002.
[36]D. A. Forsyth and J. Ponce. Computer Vision:A Modern Approach,Second Edition. Prentice Hall,2011.
[37]R. T. Frankot and R. Chellappa. A method for enforcing integrability in shape from shading algorithms. IEEE Transactions on Pattern Analysis and Machine Intelligence,10(4):439-451,1987.
[38]A. S. Georghiades. Incorporating the torrance and sparrow model of reflectance in uncal-ibrated photometric stereo. In Proceedings of the 2003 IEEE International Conference on Computer Vision,2003.
[39]A. S. Georghiades, P. N. Belhumeur, and D. J. Kriegman. From few to many:Illumination cone models for face recognition under variable lighting and pose. IEEE Transactions on Pattern Analysis and Machine Intelligence,23(6):643-660,2001.
[40]V. P. Godambe. Estimating functions. Oxford University Press,1991.
[41]D. B. Goldman, B. Curless, A. Hertzmann, and S. Seitz. Shape and spatially-varying brdfs from photometric stereo. In Proceedings of the 2005 IEEE International Conference on Computer Vision,2005.
[42]Google. Google glass. http://en.wikipedia.org/wiki/Google_Glass.
[43]J. E. Greivenkamp. Field guide to geometrical optics. SPIE Press,2004.
[44]P. R. Halmos and P. P. R. Halmos. A Hilbert space problem book. Springer Verlag,1982.
[45]M. Harker and P. O'Leary. Least squares surface reconstruction from measured gradien-t fields. In Proceedings of the 2008 IEEE Conference on Computer Vision and Pattern Recognition,2008.
[46]A. Harrison. Maximum likelihood estimation of depth maps using photometric stereo. IEEE Transactions on Pattern Analysis and Machine Intelligence,34(7):1368-1380,2012.
[47]R. Hartley and A. Zisserman. Multiple view geometry in computer vision. Cambridge,2000.
[48]H. Hayakawa. Photometric stereo under a light source with arbitrary motion. Journal of the Optical Society of America A,11(11):3079-3089,1994.
[49]H. V. Helmholtz. Handbuch der physiologischen Optik. Voss,1867.
[50]S. Herbort and C. Wohler. An introduction to image-based 3d surface reconstruction and a survey of photometric stereo method.3D Research,2(3):1-17,2011.
[51]C. Hernandez, G. Vogiatzis, and R. Cipolla. Multiview photometric stereo. IEEE Transac-tions on Pattern Analysis and Machine Intelligence,30(3):548-554,2008.
[52]C. Hernandez, G. Vogiatzis, and R. Cipolla. Shadows in three-source photometric stereo. In Proceedings of the 2008 IEEE European Conference on Computer Vision,2008.
[53]A. Hertzmann and S. M. Seitz. Shape and materials by example:A photometric stereo approach. In Proceedings of the 2003 IEEE Conference on Computer Vision and Pattern Recognition,2003.
[54]A. Hertzmann and S. M. Seitz. Example-based photometric stereo:Shape reconstruction with general, varying brdfs. IEEE Transactions on Pattern Analysis and Machine Intelli-gence,27(8):1254-1264,2005.
[55]B. Horn and M. Brooks. The variational approach to shape from shading. Computer Vision, Graphics, and Image Processing,33(2):174-208,1986.
[56]B. K. P. Horn. Shape from shading:A method for obtaining the shape of a smooth opaque object from one view. AITR-232,1970.
[57]C. W. Hsu and C. J. Lin. A comparison of methods for multiclass support vector machines. IEEE Transactions on Neural Networks,13(2):415-425,2002.
[58]S. Ikehata, D. Wipf, Y. Matsushita, and K. Aizawa. Robust photometric stereo using sparse regression. In Proceedings of the 2012 IEEE Conference on Computer Vision and Pattern Recognition,2012.
[59]K. Ikeuchi and K. Sato. Determining refectance parameters using range and brightness images. In Proceedings of the 1990 IEEE International Conference on Computer Vision, 1990.
[60]E. C. Jr. and R. Jain. Obtaining 3-dimensional shape of textured and specular surfaces using four-source photometry. Computer Graphics and Image Processing,18(4):309-328,1982.
[61]P. Kovesi. Shapelets correlated with surface normals produce surfaces. In Proceedings of the 2005 IEEE International Conference on Computer Vision,2005.
[62]P. Lagger and P. Fua. Retriving multiple light sources in the presence of specular reflections and texture. IEEE Transactions on Pattern Analysis and Machine Intelligence,111(2):207--218,2008.
[63]J. H. Lambert. Photometria, sive de Mensura et Gradibus Luminis, Colorum et Umbrae. Augsburg,1760.
[64]J. Lawrence, A. Ben-Artzi, C. DeCoro, W. Matusik, H. Pfister, R. Ramamoorthi, and S. Rusinkiewicz. Inverse shade trees for non-parametric material representation and editing. ACM Transactions on Graphics,25(3):735-745,2006.
[65]L. Le Cam. Maximum likelihood:an introduction. International Statistical Review/Revue Internationale de Statistique, pages 153-171,1990.
[66]C. H. Lee and A. Rosenfeld. Improved methods of estimating shape from shading using the light source coordinate system. Artificial Intelligence,26(2):125-143,1985.
[67]S. Z. Li. Markov Random Field Modeling in Image Analysis. Springer,2009.
[68]M. Liao, X. Huang, and R. Yang. Interreflection removal for photometric stereo by using spectrum-dependent albedo. In Proceedings of the 2011 IEEE Conference on Computer Vision and Pattern Recognition,2011.
[69]S. P. Mallick, T. E. Zickler, D. J. Kriegman, and P. N. Belhumeur. Beyond lambert:Re-constructing specular surfaces using color. In Proceedings of the 2005 IEEE Conference on Computer Vision and Pattern Recognition,2005.
[70]D. Marr. Vision:A Computational Investigation into the Human Representation and Pro-cessing of Visual Information. New York, Freeman,1982.
[71]W. Matusik, H. Pfister, M. Brand, and L. McMillan. A datadriven reflectance model. ACM Transactions on Graphics,22(3):759-769,2003.
[72]Microsoft. Kinect. http://www.xbox.com/en-US/kinect/default.htm.
[73]P. Milgram and A. F. Kishino. Taxonomy of mixed reality visual displays. IEICE Transac-tions on Information Systems,77(12):1321-1329,1994.
[74]D. Miyazaki, K. Hara, and K. Ikeuchi. Median photometric stereo as applied to the segonko tumulus and museum objects. International journal of computer vision,86(2):229-242, 2010.
[75]D. Miyazaki and K. Ikeuchi. Photometric stereo using graph cut and m-estimation for a virtual tumulus in the presence of highlights and shadows. In IEEE Conference on Computer Vision and Pattern Recognition Workshops,2010.
[76]Y. Mukaigawa, Y. Ishii, and T. Shakunaga. Classification of photometric factors based on photometric linearization. In Proceedings of the 2006 IEEE Asian Conference on Computer Vision,2006.
[77]Y. Mukaigawa, Y. Ishii, and T. Shakunaga. Analysis of photometric factors based on photo-metric linearization. Journal of the Optical Society of America A,24(10):3326-3334,2007.
[78]S. Nayar, K. Ikeuchi, and T. Kanade. Shape from interreflections. International Journal of Computer Vision,6(3):173-195,1991.
[79]S. K. Nayar, K. Ikeuchi, and T. Kanade. Surface reflection:Physical and geometrical per-spectives. IEEE Transactions on Pattern Analysis and Machine Intelligence,13(7):611-634, 1999.
[80]S. K. Nayar, G. Krishnan, M. D. Grossberg, and R. Raskar. Fast separation of direct and global components of a scene using high frequency illumination. ACM Transactions on Graphics,25(3):935-944,2006.
[81]D. Nehab, S. Rusinkiewicz, J. Davis, and R. Ramamoorthi. Efficiently combining positions and normals for precise 3d geometry. ACM Transactions on Graphics,24(3):536-543,2005.
[82]H. S. Ng, T. P. Wu, and C. K. Tang. Surface-from-gradients without discrete integrabili-ty enforcement:a gaussian kernel approach. IEEE Trans. Pattern Analysis and Machine Intelligence,32(11):2085-2099,2009.
[83]R. M. Norton. The double exponential distribution:Using calculus to find a maximum like-lihood estimator. The American Statistician(American Statistical Association),38(2):135-136,1984.
[84]J. Pearl. Reverend Bayes on inference engines:A distributed hierarchical approach. Cog-nitive Systems Laboratory, School of Engineering and Applied Science, University of Cali-fornia, Los Angeles,1982.
[85]A. Pentland. Linear shape from shading. International journal of computer vision, 4(26):153-162,1990.
[86]A. P. Pentland. Finding the illuminant direction. Journal of the Optical Society of America A,72(l):448-455,1982.
[87]N. Petrovic, T. Cohen, B. Frey, R. Koetter, and T. Huang. Enforcing integrability for surface reconstruction algorithms using belief propagation in graphical models. In Proceedings of the 2001 IEEE Conference on Computer Vision and Pattern Recognition,2001.
[88]B. T. Phong. Illumination for computer generated pictures. Communications of ACM, 18(6):311--317,1975.
[89]M. Powell, S. Sarkar, and D. Goldgof. A simple strategy for calibrating the geometry of light sources. PAMI,23(1):1022-1027,2001.
[90]D. Reddy, A. Agrawal, and R. Chellappa. Enforcing integrability by error correction using (?)l-minimization. In Proceedings of the 2009 IEEE Conference on Computer Vision and Pattern Recognition,2009.
[91]D. F. Rogers and J. A. Adams. Mathematical elements for computer graphics. McGraw-Hill Higher Education,1989.
[92]L. I. Rudin, S. sher, and E. Fatemi. Nonlinear total variation based noise removal algorithms. Physica D:Nonlinear Phenomena,60(1):259-268,1992.
[93]Y. Sato and K. Ikeuchi. Temporal-color space analysis of reflection. Journal of Optical Society of America A, 11(11):2990-3002,1994.
[94]Y. Sato and K. Ikeuchi. Shape and spatially-varying brdfs from photometric stereo. IEEE Transactions on Pattern Analysis and Machine Intelligence,32(6):1060-1071,2010.
[95]K. Schluns. Photometric stereo for non-lambertian surfaces using color information. In Proceedings of Computer Analysis of Images and Patterns,1995.
[96]S. Seitz, Y. Matsushita, and K. Kutulakos. A theory of inverse light transport. In Proceedings of the 2005 IEEE International Conference on Computer Vision,2005.
[97]S. Shafer. Using color to separate reflection components. Color Research and Applications, 10(4):210-218,1985.
[98]H. L. Shen and Y. Cheng. Calibrating light sources by using a planar mirror. Journal of Electronic Imaging,20(1),2011.
[99]B. Shi, Y. Matsushita, Y. Wei, C. Xu, and P. Tan. Self-calibrating photometric stereo. In Proceedings of the 2010 IEEE Conference on Computer Vision and Pattern Recognition, 2010.
[100]T. Simchony, R. Chellappa, and M. Shao. Direct analytical methods for solving poisson e- quations in computer vision problems. IEEE Transactions on Pattern Analysis and Machine Intelligence,12(5):435-446,1990.
[101]M. M. Stark, J. Arvo, and B. Smits. Barycentric parameterizations for isotropic brdfs. IEEE Transactions on Visualization and Computer Graphics,11(2):126-138,2005.
[102]P. Tan, S. P. Mallick, L. Quan, D. J. Kriegman, and T. Zickler. Isotropy, reciprocity and the generalized bas-relief ambiguity. In Proceedings of the 2007 IEEE Conference on Computer Vision and Pattern Recognition,2007.
[103]P. Tan and T. Zickler. A projective framework for radiometric image analysis. In Proceedings of the 2009 IEEE Conference on Computer Vision and Pattern Recognition,2009.
[104]K. L. Tang, C. K. Tang, and T. T. Wong. Dense photometric stereo using tensorial belief propagation. In Proceedings of the 2005 IEEE Conference on Computer Vision and Pattern Recognition, pages 132-139,2005.
[105]J. S. Taylor and Cristianini. Kernel Methods for Pattern Analysis. Cambridge University Press,2004.
[106]E. Torrance and E. M. Sparrow. Theory for offspecular reflection from roughened surfaces. Journal of the Optical Society of America A,57(9):1105-1114,1967.
[107]Y. Wang and D. Samaras. Estimation of multiple directional illuminants from a single image. Image and Visual Computing,26(1):1179-1195,2008.
[108]G. J. Ward. Measuring and modeling anisotropic reflection. ACM Transactions on Graphics, 26(2):265-272,1992.
[109]W. L. Wolfe. Introduction to radiometry. SPIE Press,1998.
[110]R. J. Woodham. Photometric method for determining surface orientation from multiple images. Optical Engineering,19(1):139-144,1980.
[111]L. Wu, A. Ganesh, B. Shi, Y. Matsushita, Y. Wang, and Y. Ma. Robust photometric stere-o via low-rank matrix completion and recovery. In Proceedings of the 2010 IEEE Asian Conference on Computer Vision,2010.
[112]T. P. Wu and C. K. Tang. Dense photometric stereo using a mirror sphere and graph cut. In Proceedings of the 2005 IEEE Conference on Computer Vision and Pattern Recognition, 2005.
[113]T. P. Wu and C. K. Tang. Dense photometric stereo by expectation-maximization. In Pro-ceedings of the 2006 IEEE Conference on Computer Vision and Pattern Recognition,2006.
[114]T. P. Wu and C. K. Tang. Visible surface reconstruction from normals with discontinuity consideration. In Proceedings of the 2006 IEEE Conference on Computer Vision and Pattern Recognition,2006.
[115]T. P. Wu and C. K. Tang. Photometric stereo via expectation maximization. IEEE Transac-tions on Pattern Analysis and Machine Intelligence,32(3):546-560,2010.
[116]T. P. Wu, K. L. Tang, C. K. Tang, and T. T. Wong. Dense photometric stereo:a markov random field approach. IEEE Transactions on Pattern Analysis and Machine Intelligence, 28(11):1830-1846,2006.
[117]S. Xu and A. Wallace. Recovering surface reflectance and multiple light locations and in-tensities from image data. Pattern Recognition Letters,29(11):1639-1647,2008.
[118]Y. Yang and A. Yuille. Source from shading. In Proceedings of the 1991 IEEE Conference on Computer Vision and Pattern Recognition,1991.
[119]C. Yu, Y. Seo, and S. W. Lee. Photometric stereo from maximum feasible lambertian reflec-tions. In Proceedings of the 2010 IEEE European Conference on Computer Vision,2010.
[120]A. Yuille and D. Snow. Shape and albedo from multiple images using integrability. In Proceedings of the 1997 IEEE Conference on Computer Vision and Pattern Recognition, 1997.
[121]A. L. Yuille, J. M. Coughlan, and S. Konishi. The generic viewpoint constraint resolves the generalized bas relief ambiguity. In Conference on Information Science and Systems,2000.
[122]YZhang and Y.Yang. Illumination direction determination for multiple light sources. In Proceedings of the 2000 IEEE Conference on Computer Vision and Pattern Recognition, pages 269-276,2000.
[123]L. Zhang, B. Curless, and S. M. Seitz. Spacetime stereo:Shape recovery for dynamic scenes. In Proceedings of the 2003 IEEE Conference on Computer Vision and Pattern Recognition, 2003.
[124]Q. Zhang, M. Ye, R. G. Yang, Y. Matsushita, B. Wilburn, and H. M. Yu. Edge-preserving photometric stereo via depth fusion. In Proceedings of the 2012 IEEE Conference on Com-puter Vision and Pattern Recognition,2012.
[125]R. Zhang, P. S. Tsai, and J. E. Cryer. Shape-from-shading:a survey. IEEE Transactions on Pattern Analysis and Machine Intelligence,21(8):690-706,1999.
[126]Y. Zhang and Y. Yang. Multiple illuminant direction with application to image synthesis. IEEE Transactions on Pattern Analysis and Machine Intelligence,23(8):915-920,2001.
[127]Q. Zheng and R.Chellappa. Estimation of illumination, albedo, and shape from shad. IEEE Transactions on Pattern Analysis and Machine Intelligence,13(1):680-702,1991.
[128]Zhou and P. Tan. Ring-light photometric stereo. In Proceedings of the 2010 IEEE European Conference on Computer Vision,2010.
[129]W. Zhou and C. Kambhamettu. Estimation of illuminant direction and intensity of multiple light sources. In Proceedings of the 2002 IEEE European Conference on Computer Vision, 2002.
[130]W. Zhou and C. Kambhamettu. A unified framework for scene illuminant estimation. Image and Vision Computing,26(3):415-429,2008.
[131]T. E. Zickler, P. N. Belhumeur, and D. J. Kriegman. Helmholtz stereopsis:Exploiting reci-procity for surface reconstruction. International Journal of Computer Vision,49(2-3):215-227,2002.