摘要
将非对称逆布局模式表示模型用于图像的表示,研究了一种基于光栅扫描顺序搜索起始点,按面积最大值匹配矩形的不重叠分解方案的矩形NAM图像表示方法,给出了该矩形NAM图像表示方法的编码算法和解码算法并分析了算法的时空复杂度,接着给出了采用直接存储方法存储矩形NAM图像时的存储结构并分析了数据量,最后提出了存储方法的改进思路。
二值矩形NAM图像表示的单值存储方法通过减少需要保存的子模式实例数和不保存子模式实例的v域的方式来减少数据量,全值存储方法通过不保存子模式实例的x、y域来减少数据量,理论证明与直接存储相比单值存储和全值存储方法能提高压缩比2倍左右。
限制矩形大小的存储方法通过限制全值存储时的子模式实例的矩形大小来降低w、h域所需要的存储位数,从而达到提高压缩比的效果。限制矩形大小的存储方法需要根据图像复杂度来选择合适的最大矩形才能得到较好的压缩比。
分类存储方法根据矩形大小对子模式实例进行分类,不同分类的子模式实例其w、h域用不同的存储位来保存,通过减少保存小尺寸矩形子模式实例所需要的数据量来达到提高压缩比的目的。实验数据表明分类存储方法能够有效提高单值存储的压缩比。
将子模式实例作为一个整体进行Huffman编码的存储方法可以利用子模式实例之间的相关性提高压缩比。为了统一和减少编码单元,提出了规范子模式实例的存储方法,任何一个矩形可以由一个较小的基础矩形经过不大于三个基础运算而得到,基础矩形和基础操作代替原矩形作为Huffman编码单元。为了不存储Huffman编码树,提出了预定义码字的存储方法,并给出了一个通过对多幅有代表性的二值图像进行统计分析得出的预定义码字表。实验数据表明,与全值存储相比,子模式实例的Huffman编码存储方法、规范子模式实例的存储方法、预定义码字的存储方法压缩比都有显著的提高。
灰度矩形NAM图像表示可分为直接表示和位平面分解表示两种。直接表示的灰度矩形NAM图像可以通过全值存储、限制矩形大小存储以及Huffman编码存储来提高压缩比。位平面分解表示的灰度矩形NAM图像存储时直接利用二值图像的存储方法。针对二进制码位平面分解的不足,提出了格雷码位平面分解方法,格雷码位平面的图像复杂度要低于二进制码位平面,可以获得更高的压缩比。
实验数据表明,改进的二值矩形NAM图像的压缩比是四元树的5.45到9.70倍、行程编码的2.16至5.09倍、CCITT G3标准的1.42至2.82倍、CCITT G4标准的0.55至0.87倍;改进的灰度矩形NAM图像的压缩比是行程编码的1.01到1.72倍,GIF编码的0.95到1.18倍。这些结果表明经过改进的矩形NAM图像表示的存储方法是一个高效的存储方法,适用于二值图像和灰度图像的无损压缩。
矩形NAM图像表示的格式化方法利用五个队列来重建子模式实例之间的位置关系,使子模式实例之间的连通关系处理变得更容易。在对连通关系进行分类的基础上研究了子模式实例连通关系判断、连通关系搜索和连通关系遍历等基础操作,给出了各算法的时间复杂度分析。连通关系搜索算法的最坏情况复杂度为O(logN),N为图像的边长;连通关系遍历算法的时间复杂度为O(N_q),N_q为图像的子模式实例数。
以子模式实例的连通关系处理算法为基础,分别研究了基于矩形NAM图像的二值图像轮廓提取、欧拉数计算、连通区域标记等图像运算,给出了具体的算法和分析,并将其与流行的基于其它图像表示的同类相关运算进行了比较。实验数据表明,采用矩形NAM表示的二值图像轮廓提取算法的执行时间是矩阵表示的0.31至0.86;矩形NAM表示的欧拉数计算执行时间是模板算法的0.11至0.63;矩形NAM表示的连通区域标记算法执行时间是矩阵表示的0.08至0.37,四元树表示的0.05至0.12。理论分析和试验结果表明,基于格式化存储结构的矩形NAM图像在图像运算方面非常有效。
With the image representation based on Non-symmetry Anti-packing pattern representation Model(NAM), a rectangle NAM Image representation is presented, which searched start point based on raster order, matched rectangle according maximum area and decomposed image non-overlapping. Its coding and decoding algorithms are presented and analyzed, then the storage structure of direct storage method is given and data amount is analyzed, finally improvement idea of direct storage method is presented.
Single value storage method of binary rectangle NAM image representation can reduce data amount by reducing the number of sub-pattern instance and not saving the v field of sub-pattern instance. Full value storage method reduce data amount by not saving the x, y fields of sub-pattern instance. The compression ratios of Single value storage method and full value storage method are both near 2 times of that of direct storage method.
Limited rectangle storage method decreased the bits of the w, h fields by limiting the rectangle size of sub-pattern instance of full value storage method. The limited rectangle storage method need choose appropriate maximum rectangle according the complexity of the image.
Classified storage method classified sub-pattern instances by their size, then saving the w, h fields with different bits when the class is different. Less sub-pattern instance needed fewer bits. With experimental data, the compression ratio of classified storage method is bigger than that of single value storage.
Huffman code of sub-pattern instance can improve compression ratio by reducing the relativity of sub-pattern instance. Its coding unit is the sub-pattern instance. For unifying and decreasing the coding unit, a standard sub-pattern instance storage method is presented. Any rectangle can compounded by a base rectangle and base operations less then three. The base rectangles and operations are new coding unit. For not saving Huffman coding tree, a storage method based on predefined code is presented and a predefined code table is given. With experimental data, the compression ratios of Huffman code of sub-pattern instance, Huffman code of standard sub-pattern instance and predefined code storage method are all much bigger than that of full value storage method.
Gray scale NAM image representation has two representations, direct representation and bit plane decompose representation. The direct represented gray scale NAM image can improve compression ratio by full value storage, limited rectangle storage and Huffman code storage. The bit plane decompose represented gray scale NAM image can use the storage method of binary NAM image directly. Aim at the shortage of bit plane decompose based on binary code, a new bit plane decompose base on Gray code is presented. The complexity of bit plane based on Gray code is less than that based on binary code, so its compression ratio is higher than that based on binary code.
With the experimental data, the compression ratio of improved binary NAM image is 5.45 to 9.70 times of that of quadtree, 2.16 to 5.09 times of that of run length code, 1.42 to 2.82 times of that of CCITT G3, and 0.55 to 0.87 times of that of CCITT G4. The compression ratio of improved gray scale NAM image is 1.01 to 1.72 times of that of run length code, 0.95 to 1.18 times of that of GIF image format. As discussed above, the improved storage method of rectangle NAM image is a high performance storage method. It can be used for lossless compression of binary and gray scale image.
The formatted method of rectangle NAM image representation reconstructed the ubiety of sub-pattern instance with five queues. It made connectivity operation between sub-pattern instances easier. With the class of connectivity between sub-pattern instances, the based operations of connectivity such as verdict, search and ergod are researched. The complexity of connectivity search at the worst situation is O(logN), N is the side length of image. The complexity of connectivity ergod is O(N_q), N_q is the number of sub-pattern instances.
With the connectivity operation of sub-pattern instance, contour extraction, Euler number computing and connected component labeling of binary rectangle NAM image are researched. With the experimental data, operation time of contour extraction of binary image represented by rectangle NAM is 0.31 to 0.86 times of that of image represented by matrix, operation time of Euler number computing of binary rectangle NAM image is 0.11 to 0.63 times of that of the pattern algorithm, operation time of connected component labeling of binary image represented by rectangle NAM is 0.08 to 0.37 times of that of image represented by matrix, 0.05 to 0.12 times of that of image represented by quadtree. Either from theoretical or from experiment, the formatted storage of rectangle NAM image is good at image operation.
引文
[1] G.K. Wallace. Overview of the JPEG (ISO/CCITT) Still Image Compression Standard. in: Robert J. Moorhead Ⅱ, Keith S. Pennington eds. Proceedings of SPIE: Image Processing Algorithms and Techniques. Santa Clara, CA, USA. 1990. Bellingham, WA, USA: SPIE, 1990. 220-233
[2] J. Flusser. Refined Moment Calculation using Image Block Representation. IEEE Transactions on Image Processing, 2000, 9(11): 1977-1978
[3] J. Malo, I. Epifanio, R. Navarro, et al. Nonlinear Image Representation for Efficient Perceptual Coding. IEEE Transactions on Image Processing, 2006, 15(1): 68-80
[4] V.M. Kharinov. Representation of Image Information for Machine Computation. Pattern Recognition and Image Analysis, 2005, 15(1): 212-214
[5] P. Tsai, C.Y. Hu, C.C. Chang. Novel image authentication scheme based on quadtree segmentation, Imaging Science Journal, 2005, 53(3): 149-162
[6] H. Samet, A. Rosenfeld, C.A. Shaffer, et al. Quadtree Region Representation in Cartography Experimental Results. IEEE Transactions on Systems, Man, and Cybernetics, 1983, 13(6): 1148-1154
[7] R.Woodwark. Compressed Quadtrees. The Computer Journal, 1984, 27(3): 225-229
[8] E. Kawaguchi, T. Rndo. On a Method of Binary-Picture Representation and Its Application to Data Compression. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1980, 2(1): 27-35
[9] J. Knipe, X. Li. On the reconstruction of quadtree data. IEEE Transactions on Image Processing, 1998, 7(12): 1653-1660
[10] 陈传波.非对称逆布局模式表示方法研究:[博士学位论文].武汉:华中科技大
[11] Y.P. Zheng, C.B. Chen, M. Sarem. A Novel Algorithm for Triangle Non-symmetry and Anti-packing Pattern Representation Model of Gray Images. in: Marco Loog ed. Proceedings of the 2007 International Conference on Intelligent Computing. Qingdao, P. R. C. 2007. Berlin, Germany: Springer-Verlag, 2007. 832-841
[12] I.R. King. Accuracy of Measurement of Star Images on a Pixel Array. Astronomical Society of The Pacific, 1983,95(2): 163-168
[13] B. Carpentieri, M.J. Weinberger, G. Seroussi. Lossless compression of continuous-ton images. Proceedings of the IEEE, 2000, 88(11): 1797-1809
[14] R.E. Roger, M.C. Cavenor. Lossless compression of AVIRIS images. IEEE Transactions on Image Processing, 1996, 5(5): 713-719
[15] S. Andriani, G. Calvagno. Lossless Compression of Color Sequences Using Optimal Linear Prediction Theory. IEEE Transactions on Image Processing, 2008,17(11): 2102-2111
[16] A.E. Laemmel. Efficiency of codes as a function of the message length. in: E.C. Cherry ed. Proceedings of Symposium on Applications of Communication Theory. London, England. 1952. London, England: Butterworths Scientific Publications,1953. 84-87
[17] M.A. Treuhaft. Description of a System for Transmission of Line Drawings with Bandwidth-Time Compression. Proceedings of the IRE, 1958,46(10): 1394-1402
[18] W.S. Michel. Statistical Encoding for Text and Picture Communication. Communication and Electronics, 1958, 35(1): 33-36
[19] H. Meyr, H. Rosdolsky, T. Huang. Optimum Run Length Codes. IEEE Transactions on Communications, 1974, 22(6): 826-835
[20] H. Tanaka, A. Leon-Garcia. Efficient Run-Length Encodings. IEEE Transactions on Information Theory, 1982, 28(6): 880-890
[21] H. Freeman. On the Encoding of arbitrary Geometric Configurations. IRE Transactions on Electronic Computers, 1961, 10(2): 260-268
[22] F.Y. Shih, W.T. Wong. A One-pass Algorithm for Local Symmetry of Contours from Chain Code. Pattern Recognition, 1999, 32(7): 1203-1210
[23] J. Koplowitz, S. Plants. Corner Detection of Chain Code Curves. Pattern Recognition, 1995,28(6): 843-852
[24] G Kim, V. Govindaraju. Efficient Chain Code Based Image Manipulation for Handwritten Word Recognition.in: Patrick Vannoorenberghe ed. Proceedings of SPIE Symposium on Electronic Imaging. San Jose, CA, USA. 1996. Bellingham, WA, USA: SPIE Press. 1996. 262-272
[25] Hui Li, B.S. Manjunath, S.K. Mitra. A Contour-based Approach to Multisensor Image Registration. IEEE Transactions on Image processing, 1995, 4(3): 320-334
[26] C.R. Dyer, A. Rosenfeld, H. Samet. Region Representation: Boundary Codes from Quadtrees. Communications of the ACM, 1980, 23(3): 171-179
[27] J. Koplowitz, J. DeLeone. Hierarchical representation of chain-encoded binary image contours. Computer Vision and Image Understanding, 1996, 63(2): 344-352
[28] S.A. Potapov, S.O. Gamayunova. Information criterion for constructing the hierarchical structural representations of images. Proceedings of SPIE, Automatic Target Recognition XV, 2005, 5807: 443-454
[29] J.I. Wundrich, C.Von der Malsburg, P.R.Wurtz. Image representation by complex cell responses. Neural Computation, 2004, 16(12): 2563-2575
[30] O. Jinsung. Structuring element representation of an image and its applications. International Journal of Control, Automation, and Systems, 2004, 2(4): 509-515
[31] R. Szeliski, Y.H. Shum. Motion estimation with quadtree splines. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1996, 18(12): 1199-1210
[32] A.B. Banister, R.T. Fischer. Quadtree classification and TCQ image coding. IEEE Transactions on Circuits and Systems for Video Technology, 2001, 11(1): 3-8
[33] A. Klinger. Data Structure and Pattern Recognition. Advances in Information Systems Science, 1978, 7(2): 273-310
[34] A. Klinger, M.L. Phodes. Organization and Access of Image Data by Areas. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1979, 1(1): 50-60
[35] G.M. Hunter, K. Steiglitz. Operations on Image Using Quadtree. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1979, 1(2): 145-153
[36] H. Samet. Region Representation: Quadtrees from Binary Arrays. Computer Graphics and Image Processing, 1980, 13(1): 88-98
[37] H. Samet. Region Representation: Quadtrees From Boundary Codes. Communications of the ACM, 1980,23(3): 163-170
[38] H. Samet. Reconstruction of Quadtree From Quadtree Medial Axis Transforms. Computer Graphics and Image Processing, 1985, 29(3): 311-328
[39] C.H. Chien, J.K. Aggarwal. A Normalized Quadtree Representation. Computer Graphics and Image Processing, 1984, 26(3): 331-346
[40] L.P. Jones, S.S. Iyengar. Space and Time Efficient Virtual Quadtrees. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1984, 6(2): 244-247
[41] F. Fabbrini, C. Montani. Autumnai Quadtrees. The Computer Journal, 1986, 29(5): 472-474
[42] I. Gargantini. An Effective Way to Represent Quadtrees. Communications of the ACM, 1982, 25(12): 905-910
[43] C.R. Dyer. The Space Efficiency of Quadtrees. Computer Graphics and Image Processing, 1982, 19(4): 335-348
[44] J.P. Lauzon. Two-Dimensional Run-Encoding for Quadtree Representation. Computer Graphics and Image Processing, 1985, 30(1): 56-69
[45] S. Ranade. Use of Quadtrees for Edge Enhancement. IEEE Transactions on Systems, Man, and Cybernetics, 1981, 11(5): 373-376
[46] S. Rarade, M. Shneier. Using Quadtrees to Smooth Images. IEEE Transactions on System, Man, and Cybernetics, 1981, 11(5): 373-376
[47] A. Unnikrishnan, Y.V. Venkatesh, P. Shankar. Connected Component Labeling Using Quadtrees-A Bottom-up Approach. The Computer Journal, 1987, 30(2):176-182
[48] H. Samet. Algorithms for the conversion of Quadtrees to Raster. Computer Graphics and Image Processing, 1984, 26(1): 1-16
[49] C.E. Shannon. A Mathematical Theory of Communication. Bell System Technical Journal, 1948, 27: 379-423, 623-656
[50] A.K. Jain. Image data compression: a review. Proceedings of the IEEE, 1981, 69(3): 349-389
[51] D.A. Huffman. A Method for the Construction of Minimum-Redundancy Codes. Proceedings of the IRE. 1951,40(9): 1098-1101
[52] D. Hirschberg, D. Lelewer. Efficient Decoding of Prefix Codes. Communications of the ACM. 1990, 33(4): 449-459
[53] A. Sieminski. Fast Decoding of the Huffman Codes. Information Processing Letters 26(5): 237-241
[54] D.E. Knuth. Dynamic Huffman Coding. Journal of Algorithms 1985, 6(1): 163-180
[55] J.S. Vitter. Design and Analysis of Dynamic Huffman Codes. Journal of the ACM. 1987, 34(4): 825-845
[56] J.B. Connell. A Huffman-Shannon-Fano code. Proceedings of the IEEE. 1973,61(7): 1046-1047
[57] T. Tjalkens. Implementation cost of the Huffman-Shannon-Fano code. in: Proceedings of the Data Compression Conference, 2005. Washington DC USA. 2005: IEEE Computer Society, 2005. 123-132
[58] I.H Witten, R. Neal, J.G Cleary. Arithmetic Coding for Data Compression. Communication of the ACM, 1987, 30(6): 520-540
[59] A. Moffat, R. Neal, I.H. Witten. Arithmetic Coding Revisited. ACM Transactions on Information Systems, 1998, 16(3):256-294
[60] A. Lempel, J. Ziv. A Universal Algorithm for Data Compression. IEEE Transactions on Information Theory, 1977, 23(3): 337-343
[61] J.A. Storer, T.G. Szymanski. Data Compression via Textual Substitution. Journal of the ACM, 1982, 29(1): 928-951
[62] A. Lempel, J. Ziv. Compression of Individual Sequences via Variable-Rate Coding. IEEE Transactions on Information Theory, 1978, 24(5): 530-536
[63] E.R Fiala, D.H. Green. Data Compression with Finite Windows. Communications of the ACM, 1989, 32(4): 490-505
[64] R. Williams. An Extremely Fast Ziv-Lempel Data Compression Algorithm, in: J.A. Storer ed. Proceeding of the 1991 Data Compression Conference. Snowbird, UT, USA: IEEE Computer Society Press, 1991. 362-371
[65] T.A. Welch. A Technique for High-performance Data Compression. Computer. 1984, 17(6): 8-19
[66] D. Philips. LZW Data Compression. The Computer Application Journal Circuit Cellar Inc. 1992, 27:36-48
[67] 张凤林,刘思峰.LZW*:一种改进的LZW数据压缩算法.小型微型计算机,2006,27(10):1897-1900
[68] V.S. Miller, M.N. Wegman. Variations on a Theme by Ziv and Lempel. in: Alberto Apostolico, Zvi Galil eds. Proceedings of the NATO Advanced Research Workshop on Combinatorial Algorithms on Words. Berlin: Springer-Verlag, 1985.131-140
[69] A. Netravali, J.O. Limb. Picture coding: a review. Proceedings of the IEEE, 1980, 68(3): 366-406
[70] C.C. Culter. Differential quantization of communication systems. USA, Patent 2 605 361. July 29, 1952
[71] J.R. Jain, A.K. Jain. Displacement measurement and its application in interframe image coding. IEEE Transactions on Communications, 1981, 29(12): 1799-1806
[72] A.N. Netravali, J.D. Robbins. Motion-compensated television coding: Part Ⅰ. Bell System Technical Journal, 1979, 58(3): 631-670
[73] H.G. Musmann, P. Pirsch, H.J. Grallert. Advances in picture coding. Proceedings of the IEEE, 1985, 73(4): 523-548
[74] R. Srinivasan, K. Rao. Predictive Coding Based on Efficient Motion Estimation. IEEE Transactions on Communications, 1985, 33(8): 888-896
[75] Y. Yamashita. Image compression by weighted Karhunen-loeve transform, in: Proceedings of the 13th International Conference on Pattern Recognition. Los Alamitos, CA: IEEE Computer Society Press, 1996. 636-640
[76] A.K. Jain. A Fast Karhunen-Loeve Transforms for a Class of Random Processed. IEEE Transactions on Communications, 1975, 24(9): 1023-1029
[77] R.K. Kouassi, J.C. Devaux, P. Gouton, et al. Application of the Karhunen-Loeve transform for natural color images analysis. Signal, System and Computers, 1997,2(1): 1740-1744
[78] J.B. Martens. Discrete Fourier transform algorithms for real valued sequences. IEEE Transactions on Acoustics, Speech and Signal Processing, 1984, 32(2):390-396
[79] J.A. Johnston. Parallel pipeline fast fourier transformer. IEE Proceedings-F on Communications, Radar and Signal Processing, 1983, 130(6): 564-572
[80] N. Ahmed, T. Natarajan, K.R. Rao. Discrete Cosine Transforms. IEEE Transactions on Communications, 1974, 32: 90-93
[81] W.H. Chen, C.H. Smith, S.C. Fralick. A Fast Computational Algorithm for the Discrete Cosine Transform. IEEE Transactions on Communications, 1977, 25:1004-1009
[82] M.J. Narasimha, A.M. Peterson. On the Computation of the Discrete Cosine Transform. IEEE Transactions on Communications, 1978, 26: 934-936
[83] Nam Ik Cho, S.K. Mitra. Warped Discrete Cosine Transform and Its Application in Image Compression. IEEE Transactions on Circuits and System for Video Technology, 2000, 10(8): 1364-1373
[84] Ⅱ Koo Kim, Nam Ik Cho, S.K. Mitra. Rate-Distortion Optimization of the Image Compression Algorithm Based on the Warped Discrete Cosine Transform. Signal Processing, 2003, 83(9): 1919-1928
[85] J. Shapiro. Embedded image coding using zerotree of wavelet coefficients. IEEE Transactions on Signal Processing, 1993, 41(12): 3445-3462
[86] A. Said, W.A. Pearlman. A new fast and efficient image codec based on set partitioning in hierarchical trees. IEEE Transactions on Circuits Systems Video Technology, 1996, 6(6): 243-250
[87] Zixiang Xiong, K. Ramchandran, M.T. Orchard. Space-frequency quantization for wavelet image coding. IEEE Transactions on Image Processing, 1997, 6(1): 677-693
[88] Zixiang Xiong, Xiaolin Wu. Wavelet image coding using trellis coded space-frequency quantization. Signal Processing Letters, 1999, 6(1): 158-161
[89] I. Daubechies, W. Sweldens. Factoring wavelet transforms into lifting steps. Journal of Fourier Analysis and Applications, 1998, 4(3): 245-267
[90] W. Sweldens. The lifting scheme: a new philosophy in biorthogonal wavelet constructions. in: A.F. Laine, M. Unser eds. Wavelet Applications in Signal and Image Processing Ⅲ. San Diego: SPIE, 1995. 2569:68-79
[91] W. Sweldens. The lifting scheme: A construction of second generation wavelets. SIAM Journal Mathematical Analysis, 1997, 29(2): 511-546
[92] W. Sweldens. The lifting scheme: a custom-design construction of biorthogonal wavelets. Applied and computational Harmonic Analysis, 1996, 3(2): 186-200
[93] A.R. Calderbank, I. Daubechies, W. Sweldens. Wavelet transforms that map integers to integers. Applied and Computational Harmonic Analysis, 1998, 5(3):332-369
[94] M.F. Barnsley, A.D. Sloan. A better way to compress images. Byte. 1988, 13(1):215-223
[95] J.M. Beaumount. Image Data Compression Using Fractal Techniques. BT Technology Journal, 1991, 9(4): 93-109
[96] A.E. Jacquin. A novel fractal block-coding technique for digital images. in: International Conference on Acoustics, Speech, and Signal Processing, 1990,2225-2228
[97] A.E. Jacquin. Fractal image coding: a review. Proceedings of The IEEE, 1993, 81(10): 1451-1465
[98] Y. Fisher. Fractal Image Compression: Theory and Application. New York: Springer Verlag, 1995.30-76
[99] D.M. Monro, S.J. Woolley, Fractal image coding without searching, in: ICIP-94: proceedings. Los Alamitos. Calif.: IEEE Computer Society Press, 1994, 5:557-560
[100] R. Forcheimer, T. Kronander. Image coding from waveform to animation. IEEE Transactions on Acoustics, Speech and Signal Processing, 1989, 37(12):2008-2024
[101] 郑运平,陈传波.一种基于非对称逆布局模型的彩色图像表示方法.软件学报,2007,18(11):2932-2941
[102] Y.P. Zheng, C.B. Chen, W. Huang, et al. A New Non-Symmetry and Anti-Packing Model with K-lines and Its Application to Color Image Representation. in: Hamid R. Arania, Xiangjian He, Tom Hintz, et al eds. Proceedings of the 2007 International Conference on Image Processing, Computer Vision, and Pattern Recognition. Monte Carlo Resort, Las Vegas, Nevada, USA. 2007. Bogart, GA, USA: CSREA Press, 2007. 526-532
[103] C.B. Chen, Y.P. Zheng, M. Sarem, et al. A Novel Algorithm for Multi-Valued Image Representation. in: Jiangsheng Lei, Jingtao Yao, Qingfu Zhang eds. Proceedings of the 3rd International Conference on Natural Computation. Haikou, P. R. C. 2007. Los Alamitos, California, USA: IEEE Computer Society, 2007. 84-89
[104] C.B. Chen, W.J. Hu, L. Wan. Direct Non-Symmetry and Anti-Packing Pattern Representation Model of Medical Images. in: Amino Acids ed. Proceedings of the 1st International Conference on Bioinformatics and Biomedical Engineering. Wuhan, P. R. C. 2007. NY USA: IEEE Computer Society, 2007. 1011-1018
[105] F.G. Heath. Origins of the Binary Code. Scientific American, 1972, 227(2): 76-77
[106] V. Voronin. Holographic representation in image processing tasks. Pattern Recognition and Image Analysis, 2001, 11(1): 265-267
[107] V. Khanna, P. Gupta, J.C. Hwang. Maintaining connected components in quadtree-based representation of images. Iranian Journal of Electrical and Computer Engineering, 2003, 2(1): 53-60
[108] M. Manouvrier, M. Rukoz, G Jomier. Quadtree representations for storage and manipulation of clusters of images. Image and Vision Computing, 2002, 20(7): 513-27
[109] A. Munteanu, J. Cornelis, G. Van Der Auwera, et al. Wavelet image compression -the quadtree coding approach. IEEE Transactions on Information Technology in Biomedicine, 1999,3(3): 176-185
[110] M.H. Gross, O.G Staadt, R. Gatti. Efficient triangular surface approximations using wavelets and quadtree data structures. IEEE Transactions on Visualization and Computer Graphics, 1996, 2(2): 130-143
[111] A. Akimov, A. Kolesnikov, P. Franti. Lossless Compression of Color Map Images by Context Tree Modeling. IEEE Transactions on Image Processing, 2007, 16(1):114-120
[112] E. Bribiesca. A Geometric Structure for 2-Dimensinal Shapes and 3-Dimensional Surfaces. Patten Recognition, 1992, 25(5): 483-496
[113] E. Bribiesca. A New Chain Code. Pattern Recognition, 1999, 32(2): 235-251
[114] W. Pratt. A Bibliography on Television Bandwidth Reduction Studies. IEEE Transactions on Information Theory, 1967, 13(1): 114-115
[115] Y. Liu, S. Ranganath. Wavelet in POCS for image segment representation. Electronics Letters, 2003, 39(19): 1379-1380
[116] K. Huang, Z. Wu, Q. Wang. Image enhancement based on the statistics of visual representation, Image and Vision Computing, 2005, 23(1): 51-57
[117] P. Soo-Chang, C. Fei-Chin. Hierarchical image representation by mathematical morphology subband decomposition. Pattern Recognition Letters, 1995, 16(2):183-192
[118] S. Kumar, G Seetharaman. Hierarchical histograms - a new representation scheme for image-based data retrieval. in: Proceedings: 2000 International Conference on Image Processing. Vancouver Convention & Exposition Centre, Vancouver, British Columbia, Canada: Los Alamitos, Calif, 2000. 1: 69-72
[119] R. Distasi, M. Nappi, S. Vitulano. Heat: Hierarchical entropy approach for texture indexing in image databases. International Journal of Software Engineering and Knowledge Engineering, 2002, 12(5): 501-521
[120] S. Junho, L. Seongjoo, K. Sungsik, et al. Quadtree-based disparity estimation for intermediate view synthesis of stereoscopic image sequences. Optical Engineering,2005, 44(3): 1-12
[121] C. Yung-Kuan, C. Chin-Chen. Block image retrieval based on a compressed linear quadtree. Image and Vision Computing, 2004, 22(5): 391-397
[122] J.J. Laguardia, E. Cueto, M. Doblare. A natural neighbour Galerkin method with quadtree structure. International Journal for Numerical Methods in Engineering,2005, 63(6): 789-812
[123] A. Klinger, C.R. Dyer. Experiments in Picture Representation Using Regular Decomposition. Computer Graphics and Image Process, 1976, 5(1): 68-105
[124] N. Alecandridis, A. Klinger. Picture Decomposition, Tree Data Structure, and Identifying Directional Symmetries as Node Combinations. Computer Graphics and Image Processing, 1978, 8(1): 43-77
[125] M. Shneier. Calculations of Geometric Properties Using Quadtrees. Computer Graphics and Image Processing, 1981, 16(3): 296-302
[126] H. Samet. Connected Component Labeling Using Quadtrees. Communications of the ACM, 1981, 28(3): 487-501
[127] G.M. Hunter, K. Steiglitz. Linear Transformation of Pictures Represented by Quadtrees. Computer Graphics and Image Processing, 1979, 10(3): 289-296
[128] A. Shaffer, H. Samet. Optimal Quadtree Construction Algorithms. Computer Graphics and Image Processing, 1987, 37 (3): 402-419
[129] M.A. Bauer. Set Operation on Linear Quadtrees. Computer Graphic and Image Processing, 1985, 29(2): 248-258
[130] H.H. Atkinson, I. Gargantini, T.R.S. Walsh. Filling by Quadtrees or Octants.Computer Graphics and Image Processing, 1986, 33(2): 138-155
[131] D.J. Abel. A B+--Tree Structure for large Quadtrees. Computer Graphics and Image Processing, 1984, 27(1): 19-31
[132] M.L.P. Van Lierop. Geometrical Transformations on Pictures Represented by leafcodes. Computer Graphics and Image Processing, 1986, 33 (1): 81-98
[133] Tai Sing Lee. Image Representation Using 2D Gabor Wavelets. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1996, 18(10): 959-971
[134] R.M. Mutelo, W.L. Woo, S.S. Dlay. Two Dimensional Orthogonal Wavelet Features for Image Representation and Recognition. in: S. Sanei, J.A. Chambers, J.McWhirter eds. Proceedings of the 15th International Conference on Digital Signal Processing. Wales, UK. 2007. Washington DC USA: IEEE Computer Society, 2007.256-259
[135] W.R. Zettler, J.C. Huffman, D.C.P. Linden. Application of Compactly Supported Wavelets to Image Compression. in: R.J. Moorhead Ⅱ, K.S. Pennington eds. Proceedings of SPIE: Image Processing Algorithms and Techniques. Santa Clara, CA, USA. 1990. Bellingham, WA, USA: SPIE, 1990. 150-160
[136] A. Rosenfeld, J.L. Pfaltz. Sequential Operations in Digital Image Processing. Journal of the ACM, 1966, 13(4): 471-494
[137] R. Lumia, L. Shapiro, O. Zuniga. A New Connected Components Algorithm for Virtual Memory Computers. Computer Vision, Graphics, and Image Process, 1983, 22(2): 287-300
[138] 陈传波.图像分层表示的最优分割和线性二元树.计算机学报,1991,14(7): 505-513
[139] A.J. Robinson. Efficient general-purpose image compression with binary tree predictive coding. Transactions on Image Processing, 1997, 6(4): 601-608
[140] A.J. Robinson, A. Druet, N. Gosset. Video compression with binary tree recursive motion estimation and binary tree residue coding. IEEE Transactions on Image Processing, 2000, 9(7): 1288-1292
[141] A. Shamim, A.J. Robinson. Asymmetric binary tree coding for contour images. Image and Vision Computing, 2003, 21(9): 797-807
[142] 林开颜,吴军辉,徐立鸿.彩色图像分割方法综述.中国图像图形学报A辑,2005,10(1):1-10
[143] A. Bishnu, B.B. Bhattacharya, M.K. Kundu, et al. Euler Vector: A Combinatorial Signature for Gray-Tone Images. in: Bob Werner ed. Proceedings of the International Conference on Information Technology: Coding and Computing. Las Vegas, Nevada, USA. 2002. Washington DC USA: IEEE Computer Society, 2002. 121-126
[144] S.C. Lee, Yiming Wang, E.T. Lee. Compactness Measure of Digital Shapes. in: Nita Patel ed. Proceedings of IEEE Region 5 Conference: Annual Technical and Leadership Workshop. Norman, Oklahoma, USA. 2004. Washington DC USA: IEEE Computer Society, 2004. 103-105
[145] F. Chang, C.J. Chen, C.J. Lu. A linear-time component-labeling algorithm using contour tracing technique. Computer Vision and Image Understanding, 2004, 93(2): 206-220
[146] K. Suzuki, I. Horiba, N. Sugie. Linear-time connected-component labeling based on sequential local operations. Computer Vision and Image Understanding, 2003, 89(1): 1-23
[147] R. Mat-Ali, M. Xue, A. Hachicha, et al. Parallel Quadtree Representation and Moment Invariants Computation of Binary Image for Hierarchical matching on Pyramid Machine. in: E.S. Gelsems, E. Backer eds. Proceedings of the 11th International Conference on Architecture for Vision and Pattern Recognition. Hague, Netherlands. 1992. NW Washington, DC USA: IEEE Computer Society, 1992.87-90
[148] Chin-Hsiung Wu, Shi-Jinn Horng, Pei-Zong Lee. A New Computation of Shape Moments via Quadtree Decomposition. Pattern Recognition, 2001, 34(7): 1319-1330
[149] Luren Yang, Fritz Albregtsen. Fast and Exact Computation of Cartesian Geometric Moments Using Discrete Green's Theorem. Pattern Recognition, 1996, 29(7): 1061-1073
[150] 王中元,胡瑞敏.利用四元树结构计算黑白二值图像周长的方法.计算工程,2005,31(18):169-171
[151] P.G. Howard, F. Kossentini, B. Martins, et al. The emerging JBIG2 standard. IEEE Transactions on Circuit and Systems for Video Technology, 1998, 8(7): 838-848