基于小波变换的三维表面纹理超分辨率及评价
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摘要
近年来,超分辨率已成为图像处理领域中的研究热点。所谓图像超分辨率处理就是从一序列降质的低分辨率图像中获取高分辨率的图像。超分辨率技术已经广泛应用在卫星遥感、军事侦察、医学诊断等很多领域。目前三维表面纹理超分辨率技术的研究是图像处理领域的前沿方向,有着广阔的研究前景和现实意义。
小波分析是一种时频分析工具,它在图像处理领域有重要应用。小波分析用于超分辨率是当前研究的一个热点。小波变换具有多分辨率特性和时频局部化特性,但它只有有限的方向表示,不能很好地表征图像中的方向信息。Contourlet变换是在小波变换的基础上发展而来的,它继承了小波变换的优点,而且具有多方向性和各向异性,能够更有效地捕捉图像中的边缘信息。Contourlet变换在图像超分辨率处理方面具有广阔的应用前景。
本论文在前期三维表面纹理超分辨率方法的基础上,结合小波变换和Contourlet变换,提出了小波超分辨率法和Contourlet超分辨率法。在小波超分辨率法中,针对具有相似纹理结构的多类三维表面纹理,应用小波变换,并结合textons替代方法,对三维表面纹理进行超分辨处理。文章用多组实验数据证明了该方法的一定有效性。
为克服小波变换的不足,本文进而使用Contourlet系数表示图像特征,采用Contourlet变换域系数关系,构建统一的训练集,并结合支持向量机方法,发展了更具通用性和有效性的Contourlet超分辨率算法。最后,本文对不同超分辨方法的结果图像进行了评价研究。实验表明,Contourlet超分辨率法的结果图像具有较高的信息熵和清晰度,更好地保留了图像的纹理信息,且有较好的视觉效果。
Image super-resolution has become an active research area. It refers to as image processing algorithms to reconstruct high-resolution(HR) images. Super-resolution technology has been widely used in remote sensing, military surveillance and medical diagnosis, etc. Currently, super resolution technique of 3D surface textures is an less studied research field.
As a time-frequency analysis tool, wavelet analysis is important in image processing applications. Wavelet analysis for super-resolution is an active research topic in the field. Wavelet transform has good properties such as multi-resolution and time frequency localization, but it has only limited direction that cannot capture the direction information of the image properly. Contourlet transform is developed based on wavelet transform, and it not only inherits the advantages of wavelet transform, but has multi directional and anisotropy. Therefore it is promising to image super-resolution processing since it can capture significant information from a natural image.
Based on the previous super resolution methods for 3D surface textures, this paper proposes two new super resolution methods. One is based on wavelet transform and developed for many types of 3D surface textures with similar texture, by using wavelet transform and textons substitution. Finally a set of experiments and results are produced to show the evidences of the efficiency of the algorithm.
To overcome the weaknesses of wavelet transform, the other algorithm is proposed based on contourlet transform. It has a general training set by studying the correlation of contourlet coefficients and combining the SVM theories. This algorithm makes the super resolution of 3D surface textures easier and more practical. Finally, the super-resolution results produced by different methods are evaluated. This new algorithm can effectively maintain the edges information and outperforms other methods in terms of entropy, definition and visual qualities.
小波分析是一种时频分析工具,它在图像处理领域有重要应用。小波分析用于超分辨率是当前研究的一个热点。小波变换具有多分辨率特性和时频局部化特性,但它只有有限的方向表示,不能很好地表征图像中的方向信息。Contourlet变换是在小波变换的基础上发展而来的,它继承了小波变换的优点,而且具有多方向性和各向异性,能够更有效地捕捉图像中的边缘信息。Contourlet变换在图像超分辨率处理方面具有广阔的应用前景。
本论文在前期三维表面纹理超分辨率方法的基础上,结合小波变换和Contourlet变换,提出了小波超分辨率法和Contourlet超分辨率法。在小波超分辨率法中,针对具有相似纹理结构的多类三维表面纹理,应用小波变换,并结合textons替代方法,对三维表面纹理进行超分辨处理。文章用多组实验数据证明了该方法的一定有效性。
为克服小波变换的不足,本文进而使用Contourlet系数表示图像特征,采用Contourlet变换域系数关系,构建统一的训练集,并结合支持向量机方法,发展了更具通用性和有效性的Contourlet超分辨率算法。最后,本文对不同超分辨方法的结果图像进行了评价研究。实验表明,Contourlet超分辨率法的结果图像具有较高的信息熵和清晰度,更好地保留了图像的纹理信息,且有较好的视觉效果。
Image super-resolution has become an active research area. It refers to as image processing algorithms to reconstruct high-resolution(HR) images. Super-resolution technology has been widely used in remote sensing, military surveillance and medical diagnosis, etc. Currently, super resolution technique of 3D surface textures is an less studied research field.
As a time-frequency analysis tool, wavelet analysis is important in image processing applications. Wavelet analysis for super-resolution is an active research topic in the field. Wavelet transform has good properties such as multi-resolution and time frequency localization, but it has only limited direction that cannot capture the direction information of the image properly. Contourlet transform is developed based on wavelet transform, and it not only inherits the advantages of wavelet transform, but has multi directional and anisotropy. Therefore it is promising to image super-resolution processing since it can capture significant information from a natural image.
Based on the previous super resolution methods for 3D surface textures, this paper proposes two new super resolution methods. One is based on wavelet transform and developed for many types of 3D surface textures with similar texture, by using wavelet transform and textons substitution. Finally a set of experiments and results are produced to show the evidences of the efficiency of the algorithm.
To overcome the weaknesses of wavelet transform, the other algorithm is proposed based on contourlet transform. It has a general training set by studying the correlation of contourlet coefficients and combining the SVM theories. This algorithm makes the super resolution of 3D surface textures easier and more practical. Finally, the super-resolution results produced by different methods are evaluated. This new algorithm can effectively maintain the edges information and outperforms other methods in terms of entropy, definition and visual qualities.
引文
[1]K. J. Dana, B. van Ginneken, S. K. Nayar and J. J.Koenderink. Reflectance and texture of real-world surfaces.IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 1997.151-157.
[2]K. J. Dana, B. van Ginneken, S. K. Nayar and J. J. Koenderink. Reflectance and texture of real-world surfaces. ACM Transactions on Graphics,1999,18 (1):1-34.
[3]J. J. Koenderink and A. J. van Doom. Illuminance texture due to surface mesostructure, J. Optical Soc. Am. A,1996,13:452-463.
[4]Dong J. Three-dimensional surface texture synthesis:[PhD thesis].UK:Department of Computer Science, Heriot-Watt University, Edinburgh, UK.
[5]Junyu Dong and Mike Chantler. Capture and synthesis of 3D surface texture. International Journal of Computer Vision (IJCV),2005,62 (1-2):177-194.
[6]Junyu Dong, Zuozhen Liu, Jing Ren. Super Resolution of 3D Surface Texture.2007 International Conference on Computational Intelligence and Securiy Workshops (CISW 2007), cisw,2007.279-282.
[7]Junyu Dong, Zuozhen Liu, Zhenwei Yu. Super Resolution of 3D Surface Texture Based on Eigen Images.2008 International Congress on Image and Signal Processing (CISP 2008), 2008, in press.
[8]Leung, T. and Malik, J. Representing and recognizing the visual appearance of materials using three-dimensional textons. International Journal of Computer Vision,2001,43 (1): 7-27.
[9]Nicodemus, F. E., Richmond, J.C. and Hsai, J. J. Geometrical considerations and nomenclature for reflectance. U.S. Dept. of Commerce, Natioal Bureau of Standards Monograph 160,1977, October.
[10]Tong, X., Zhang, J., Liu L., Wang X., Guo B. and Shum, H.Y. Synthesis of bidirectional texture functions on arbitrary surfaces. ACM Transactions on Graphics (TOG). Proc. of the 29th annual conference on Computer graphics and interactive techniques,2002,21(3): 665-672.
[11]R.Tsai and T.Huang. Multiframe image restoration and registration. In Adavance in Computer Vision and Image Processing, Volume 1, Greenwich, CT,1984.JAI.
[12]S.Kim,N.Bose and H.Valenzuela. Recursive reconstruction of high resolution image from noisy undersampled multiframes. IEEE Trans.on Acoustics, Speech, and Signal Processing, 38(6), Jun.1990.
[13]T. Komatsu, T. Igarashi, K. Aizawa and T. Saito. Very High Resolution Imaging Scheme with Multiple Different Aperture Cameras. Signal Processing Image Communication, 1993,5:511-526.
[14]M.Irani and S.Peleg. Improving resoltion by image registration. CVGIP:Graph. Models Image Processing.53:324-335,Dec.1993.
[15]K.Sauer and J.Allebach. Iterative reconstruction of band-limited images from non-uniformly spaced samples.IEEE Trans. Circuits Syst., CAS-34:1497-1505,1987.
[16]R.R.Schutz, R.L.Stevenson. Extraction of High-resolution Frames from Video Sequences. IEEE Trans IP,1996,5(6):996-1011.
[17]M.Elad, A.Feuer. Restoration of a single super-resolution image from several blurred, noisy and undersampled measured images. IEEE Transactions on Image Processing,1997, 6(12):1646-1658.
[18]M.Elad, Y.Hel-Or. A fast super-resolution reconstruction algorithm for pure translational motion and common space invariant blur. IEEE Trans Image Process 2001, 10:1187-1193.
[19]N.K.Bose and S.Lertrattanapanich. Advance in wavelet superresolution. SAMPTA 2001: Proceeding of International Conference on Sampling Theory and Application. pp.5-12. May 13 17.2001.
[20]刘良云,李英才,相里斌.超分辨率图像重构技术的仿真实验研究[J].中国图像图形学报,2001,6A(6)：629-635.
[21]孟庆武.与估计混叠度的MAP超分辨率处理算法[J].软件学报,2004,15(2)：207-214.
[22]田岩,田金文,柳健等.超分辨率技术的实现-一种改善的小波插值方法[J].中国图像图形学报,2003,9A(12)：1422-1426.
[23]J. O. Stromberg. A modified Franklin system and higher order spline systems on Rn as unconditional bases for Hardy spaces, International Conference on Harmonic Analysis in Honor of Antoni Zygmund, Univ. of Chicago Press,1982, Vol.2, pp:475-494.
[24]A. Grossman, J. Morlet, Decompositon of hardy functions into square integrable wavelets of constant shape. SIAM Journal on March. Anal., Vol.15, NO.4, pp:723-736,1984.
[25]Y. Meyer. Wavelets and Operators. Cambridge University Press, U.K.,1992.
[26]S. Mallat. Multiresolution approximations and wavelet orthonormal bases of L2 (R). Trans. Amer. Math. Soc., Vol.31, No.5, pp:69-87,1989.
[27]Daubechies I. Orthonormal bases of compactly supported wavelets[J]. Communications on Pure and Applied mathematics,1988,41(11):909-996.
[28]C.K. Chui, J.Z. Wang. A cardiral spline approach to wavelets. Proc. Amer. Math. Coc, Vol.113, No.3,pp:785-793,1991.
[29]王建忠.多尺度B样条小波边缘检测算子,中国科学(A辑),Vol.25,No.4,pp：426-437,1995.
[30]M. V. Wickerhauser. Adapted wavelet analysis from theory to software. IEEE Press, New York,1994.
[31]Calderbank A R, Daubechies I, Sweldens W, Yeo B. wavelet transforms that map integers to integers. Technical report, Department of Mathmatics, Princeton University,1996.
[32]Song C. Zhu, Cheng-en Guo, Ying N. Wu, and Yizhou Wang. "What are Textons?". Department of Statistics Papers (2002010123). Department of Statistics, UCLA.2002.
[33]Vladimir N Vaapnik著.张学工译.统计学习理论的本质.北京：清华大学出版社,2000.
[34]王晓云.Journal of Western Chongqing University (Nature Science Edition):SVM算法分析与研究.2005,4(3)：15-18.
[35]Spence, A. D. and Chantler, M. J. Optimal illumination for three-image photometric stereo acquisition of surface texture. The 3rd International workshop on texture analysis and synthesis,2003, Nice, France,89-94.
[36]Shashua, A. Geometry and Photometry in 3D visual Recognition:[Ph.D. thesis]. USA:MIT, 1992.
[37]Woodham, R. Analysing images of curved surfaces. Artificial Intelligence,1981,17: 117-140.
[38]Malzbender, T., Gelb D. and Wolters, H. Polynomial Texture Maps. Proceedings of the 28th annual conference on Computer graphics and interactive techniques, Los Angeles, California, 2001.519-528.
[39]Dana, K.J. and Nayar, S. K.3D Textured Surface Modelling. Proceedings Workshop on the Integration of Appearance and Geometric Methods in Object Recognition,1999.46-56.
[40]Epstein, R., Hallinan, P.W., Yuille, A.L.5±2 eigenimages suffice:an empirical investigation of low-dimensional lighting models. Proceedings of the Workshop on Physics-Based Modeling in Computer Vision, pp.1995.108-116.
[41]VIO R, NAGY J, TENORIO L, at al. A simple but efficient algorithm for multiple-image deblurring [J]. Astronomy & Astrophysics,2003,416:403-410.
[42]C.Ford and D.M.Etter. Wavelet basis reconstruction of nonuniformly sampled data. IEEE Transactions Circuits and System Ⅱ:Analog and Digital Signal Processing, vol.45, No.8.pp:1165-1168,August 1998.
[43]N.Nguyen and P.Milanfar, A wavelet-based interpolation-restoration method for superresolution(wavelet superresolution). Circuits Systems Signal Process. Vol.19,No.4, pp:321-338,2000.
[44]Kamimura. K, Tsumura. N, etc. Substituting crossed textons for super resolution of video. SIGGRAPH poster.2006.
[45]Vladimir N Vaapnik著.张学工译.统计学习理论的本质.北京：清华大学出版社,2000.
[46]Candes E J. Ridgelets:Theory and Application. Ph.D. Thesis. Department of Statistics, Stanford University,1998
[47]Candes E. J. and Dohono D L. Curvelet-A surprisingly effective nonadaptive representation for objects with edges. Technical Report, Department of Statistics, Stanford University,1999.
[48]Do M. N., Vetterli M. Contourlet:A Directional Multiresolution Image Representation. Proc of IEEE International Conference on Image Processing, Rochester, NY,2002:357-360
[49]R.H.Bamberger. The directional filter bank:a multirate filter bank for the directional decomposition of images. PhD. Dissertation, Georgia Institute of Technology,1990.
[50]R. H. Bamberger and M. J. T. Smith. A filter band for directional decomposition of images:theory and design. IEE Trans. Signal Proc, April 1992,40(4):882-893
[51]M. N. Do. Drectional Multiresolution image reprentations PhD.dissertation, Swiss Federal Institute of Technology, Lausanne, Switzerland, December 2001.
[52]M. N. Do and M.Vetterli. The contourlet transform:an efficient drectional multiresolution image reprentation. IEEE Trans, on Image Processing,2005,14(12):2091-2106.
[53]D. D.-Y. Po and M. N. Do. Directional muliscale modeling of images using contourlet transtorm. IEEE Transactions on Image Processing,2006,15(6):1610-1620.
[54]J. M. Shapiro, "Embedded image coding using zerotrees of wavelet coefficients," IEEE Transactions on Signal Processing, vol.41, no.12, pp.3445-3462,1993.
[55]C. V. Jiji and Subhasis Chaudhuri. Single-Frame Image Super-resolution through Contourlet Learning. EURASIP Journal on Applied Signal Processing. Volume 2006, Article ID 73767, Pages 1-11.
[56]刘作珍.基于通用训练集的三维表面纹理超分辨率研究：[硕士毕业论文].青岛：中国海洋大学,2008.42-50.
[2]K. J. Dana, B. van Ginneken, S. K. Nayar and J. J. Koenderink. Reflectance and texture of real-world surfaces. ACM Transactions on Graphics,1999,18 (1):1-34.
[3]J. J. Koenderink and A. J. van Doom. Illuminance texture due to surface mesostructure, J. Optical Soc. Am. A,1996,13:452-463.
[4]Dong J. Three-dimensional surface texture synthesis:[PhD thesis].UK:Department of Computer Science, Heriot-Watt University, Edinburgh, UK.
[5]Junyu Dong and Mike Chantler. Capture and synthesis of 3D surface texture. International Journal of Computer Vision (IJCV),2005,62 (1-2):177-194.
[6]Junyu Dong, Zuozhen Liu, Jing Ren. Super Resolution of 3D Surface Texture.2007 International Conference on Computational Intelligence and Securiy Workshops (CISW 2007), cisw,2007.279-282.
[7]Junyu Dong, Zuozhen Liu, Zhenwei Yu. Super Resolution of 3D Surface Texture Based on Eigen Images.2008 International Congress on Image and Signal Processing (CISP 2008), 2008, in press.
[8]Leung, T. and Malik, J. Representing and recognizing the visual appearance of materials using three-dimensional textons. International Journal of Computer Vision,2001,43 (1): 7-27.
[9]Nicodemus, F. E., Richmond, J.C. and Hsai, J. J. Geometrical considerations and nomenclature for reflectance. U.S. Dept. of Commerce, Natioal Bureau of Standards Monograph 160,1977, October.
[10]Tong, X., Zhang, J., Liu L., Wang X., Guo B. and Shum, H.Y. Synthesis of bidirectional texture functions on arbitrary surfaces. ACM Transactions on Graphics (TOG). Proc. of the 29th annual conference on Computer graphics and interactive techniques,2002,21(3): 665-672.
[11]R.Tsai and T.Huang. Multiframe image restoration and registration. In Adavance in Computer Vision and Image Processing, Volume 1, Greenwich, CT,1984.JAI.
[12]S.Kim,N.Bose and H.Valenzuela. Recursive reconstruction of high resolution image from noisy undersampled multiframes. IEEE Trans.on Acoustics, Speech, and Signal Processing, 38(6), Jun.1990.
[13]T. Komatsu, T. Igarashi, K. Aizawa and T. Saito. Very High Resolution Imaging Scheme with Multiple Different Aperture Cameras. Signal Processing Image Communication, 1993,5:511-526.
[14]M.Irani and S.Peleg. Improving resoltion by image registration. CVGIP:Graph. Models Image Processing.53:324-335,Dec.1993.
[15]K.Sauer and J.Allebach. Iterative reconstruction of band-limited images from non-uniformly spaced samples.IEEE Trans. Circuits Syst., CAS-34:1497-1505,1987.
[16]R.R.Schutz, R.L.Stevenson. Extraction of High-resolution Frames from Video Sequences. IEEE Trans IP,1996,5(6):996-1011.
[17]M.Elad, A.Feuer. Restoration of a single super-resolution image from several blurred, noisy and undersampled measured images. IEEE Transactions on Image Processing,1997, 6(12):1646-1658.
[18]M.Elad, Y.Hel-Or. A fast super-resolution reconstruction algorithm for pure translational motion and common space invariant blur. IEEE Trans Image Process 2001, 10:1187-1193.
[19]N.K.Bose and S.Lertrattanapanich. Advance in wavelet superresolution. SAMPTA 2001: Proceeding of International Conference on Sampling Theory and Application. pp.5-12. May 13 17.2001.
[20]刘良云,李英才,相里斌.超分辨率图像重构技术的仿真实验研究[J].中国图像图形学报,2001,6A(6)：629-635.
[21]孟庆武.与估计混叠度的MAP超分辨率处理算法[J].软件学报,2004,15(2)：207-214.
[22]田岩,田金文,柳健等.超分辨率技术的实现-一种改善的小波插值方法[J].中国图像图形学报,2003,9A(12)：1422-1426.
[23]J. O. Stromberg. A modified Franklin system and higher order spline systems on Rn as unconditional bases for Hardy spaces, International Conference on Harmonic Analysis in Honor of Antoni Zygmund, Univ. of Chicago Press,1982, Vol.2, pp:475-494.
[24]A. Grossman, J. Morlet, Decompositon of hardy functions into square integrable wavelets of constant shape. SIAM Journal on March. Anal., Vol.15, NO.4, pp:723-736,1984.
[25]Y. Meyer. Wavelets and Operators. Cambridge University Press, U.K.,1992.
[26]S. Mallat. Multiresolution approximations and wavelet orthonormal bases of L2 (R). Trans. Amer. Math. Soc., Vol.31, No.5, pp:69-87,1989.
[27]Daubechies I. Orthonormal bases of compactly supported wavelets[J]. Communications on Pure and Applied mathematics,1988,41(11):909-996.
[28]C.K. Chui, J.Z. Wang. A cardiral spline approach to wavelets. Proc. Amer. Math. Coc, Vol.113, No.3,pp:785-793,1991.
[29]王建忠.多尺度B样条小波边缘检测算子,中国科学(A辑),Vol.25,No.4,pp：426-437,1995.
[30]M. V. Wickerhauser. Adapted wavelet analysis from theory to software. IEEE Press, New York,1994.
[31]Calderbank A R, Daubechies I, Sweldens W, Yeo B. wavelet transforms that map integers to integers. Technical report, Department of Mathmatics, Princeton University,1996.
[32]Song C. Zhu, Cheng-en Guo, Ying N. Wu, and Yizhou Wang. "What are Textons?". Department of Statistics Papers (2002010123). Department of Statistics, UCLA.2002.
[33]Vladimir N Vaapnik著.张学工译.统计学习理论的本质.北京：清华大学出版社,2000.
[34]王晓云.Journal of Western Chongqing University (Nature Science Edition):SVM算法分析与研究.2005,4(3)：15-18.
[35]Spence, A. D. and Chantler, M. J. Optimal illumination for three-image photometric stereo acquisition of surface texture. The 3rd International workshop on texture analysis and synthesis,2003, Nice, France,89-94.
[36]Shashua, A. Geometry and Photometry in 3D visual Recognition:[Ph.D. thesis]. USA:MIT, 1992.
[37]Woodham, R. Analysing images of curved surfaces. Artificial Intelligence,1981,17: 117-140.
[38]Malzbender, T., Gelb D. and Wolters, H. Polynomial Texture Maps. Proceedings of the 28th annual conference on Computer graphics and interactive techniques, Los Angeles, California, 2001.519-528.
[39]Dana, K.J. and Nayar, S. K.3D Textured Surface Modelling. Proceedings Workshop on the Integration of Appearance and Geometric Methods in Object Recognition,1999.46-56.
[40]Epstein, R., Hallinan, P.W., Yuille, A.L.5±2 eigenimages suffice:an empirical investigation of low-dimensional lighting models. Proceedings of the Workshop on Physics-Based Modeling in Computer Vision, pp.1995.108-116.
[41]VIO R, NAGY J, TENORIO L, at al. A simple but efficient algorithm for multiple-image deblurring [J]. Astronomy & Astrophysics,2003,416:403-410.
[42]C.Ford and D.M.Etter. Wavelet basis reconstruction of nonuniformly sampled data. IEEE Transactions Circuits and System Ⅱ:Analog and Digital Signal Processing, vol.45, No.8.pp:1165-1168,August 1998.
[43]N.Nguyen and P.Milanfar, A wavelet-based interpolation-restoration method for superresolution(wavelet superresolution). Circuits Systems Signal Process. Vol.19,No.4, pp:321-338,2000.
[44]Kamimura. K, Tsumura. N, etc. Substituting crossed textons for super resolution of video. SIGGRAPH poster.2006.
[45]Vladimir N Vaapnik著.张学工译.统计学习理论的本质.北京：清华大学出版社,2000.
[46]Candes E J. Ridgelets:Theory and Application. Ph.D. Thesis. Department of Statistics, Stanford University,1998
[47]Candes E. J. and Dohono D L. Curvelet-A surprisingly effective nonadaptive representation for objects with edges. Technical Report, Department of Statistics, Stanford University,1999.
[48]Do M. N., Vetterli M. Contourlet:A Directional Multiresolution Image Representation. Proc of IEEE International Conference on Image Processing, Rochester, NY,2002:357-360
[49]R.H.Bamberger. The directional filter bank:a multirate filter bank for the directional decomposition of images. PhD. Dissertation, Georgia Institute of Technology,1990.
[50]R. H. Bamberger and M. J. T. Smith. A filter band for directional decomposition of images:theory and design. IEE Trans. Signal Proc, April 1992,40(4):882-893
[51]M. N. Do. Drectional Multiresolution image reprentations PhD.dissertation, Swiss Federal Institute of Technology, Lausanne, Switzerland, December 2001.
[52]M. N. Do and M.Vetterli. The contourlet transform:an efficient drectional multiresolution image reprentation. IEEE Trans, on Image Processing,2005,14(12):2091-2106.
[53]D. D.-Y. Po and M. N. Do. Directional muliscale modeling of images using contourlet transtorm. IEEE Transactions on Image Processing,2006,15(6):1610-1620.
[54]J. M. Shapiro, "Embedded image coding using zerotrees of wavelet coefficients," IEEE Transactions on Signal Processing, vol.41, no.12, pp.3445-3462,1993.
[55]C. V. Jiji and Subhasis Chaudhuri. Single-Frame Image Super-resolution through Contourlet Learning. EURASIP Journal on Applied Signal Processing. Volume 2006, Article ID 73767, Pages 1-11.
[56]刘作珍.基于通用训练集的三维表面纹理超分辨率研究：[硕士毕业论文].青岛：中国海洋大学,2008.42-50.