M-J集的物理意义与增强图像生成算法的研究
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摘要
大多数的图像生成算法，都是按照从上到下从左到右的顺序扫描图像，计算每个像素点的颜色值。在图像的生成过程中，我们无法了解图像的整体特征，或者只能了解到图像的部分特征。这些传统的生成方法对于用户自主选择图像质量、修改生成参数的交互式应用中，是很不适用的。在本文中作者对现有的隔行扫描、四叉树、Baptista图像生成方法进行了分析，提出了一种新的基于交互式应用的图像生成方法：首先根据Baptista的思想对图像进行平均取样，然后基于待插像素点的邻域像素值分布的几何特征，对像素点进行评估以决定像素点的值。实验证明，采用此生成算法，仅提供较少的数据量，即可生成较清晰的图像。在减少数据传输量的同时，提高了图像的生成质量。
本文的另一部分是基于一个典型的Langevin问题——在双势井和变化的磁场中并受一恒冲量断续作用的运动带电粒子的动力学分析，利用频闪采样法，给出了描述粒子速度变化规律的复差分方程。选取适当的磁场强度和时间间隔(采样周期)，将这一差分方程简化为用来构造广义M-J(Mandelbrot-Julia)集的复映射，并基于粒子的动力学特征探讨了广义M-J集的物理意义。结果发现：(1)广义M-J集的分形结构特征可形象地反映出粒子速度的变化规律；(2)选取的时间间隔有、无意义，决定了广义M-J集的分形结构是否具有连续性；(3)广义M-J集的演化——即粒子速度的变化规律依赖于相角主值范围的不同选取；(4)若改变磁场强度和时间间隔的选取，如选取一随机波动的磁场，则此时广义J集可能会出现内部被填充的结构特征，即在速度空间中粒子的不稳定周期轨道的闭包出现“爆炸”现象。
本文首先对M-J集和增强图像生成算法所涉及的分形理论和图像处理概念进行了阐述。本文的重点是第四章和第五章，分别对M-J集在Langevin问题中的应用和交互式应用中的彩色图像生成算法进行了研究和探讨，本文的最后对实验的结果进行了分析和总结。本文关于M-J集的相关内容已经在SCI核心收录期刊《物理学报》上发表，并被SCI与EI收录。
Most image generation algorithms scan image pixels according to top-to-bottom and left-to-right scanning; calculate the value of every pixel. It gives no idea of how the final image will be while it is being generated, or only part of image attribute can be known. Those traditional scanning is not a good method to interactive application where the user can choose image quality freely and interfere in the rendering modifying the generation parameters. In this paper the author analyzed interlaced image, quadtree and Baptista image generation method, and then presented a new interactive image generation method: Firstly, Sampling evenly for image pixels on the base of Baptista's thinking, and evaluating the pixel's value according to image pixel distribution's geometry character of pixel's neighborhood. The experimental results show that the interpolation algorithm can reconstruct high resolution images using less data. The quality of image generation is improved, while the quantity of data transmission is reduced greatly.
Based on a dynamics research of the typical Langevin problem, i.e., a moving charged particle under the continuous influence of a constant impulse in a double-well potential and a time-dependent magnetic field, using the stroboscopic sampling, we propose complex difference equations which can describe the change rule of particle's velocity. By selecting appropriate magnetic intensity and time intervals (sampling period), we reduce the difference equations to complex mapping which is used to construct the generalized M-J sets. Based on the particle's dynamics characteristic, we discuss the physical meaning of the generalized M-J sets. The author found that: (1) The fractal structure of the generalized M-J sets may visually reflect the change rule of particle's velocity; (2) Whether the selected time intervals is significative determine whether the fractal structure of the generalized M-J sets has the continuity; (3) The evolution of the generalized M-J sets, i.e., the change rule of particle's velocity, depends on the different choices of the principal range of the phase angle; (4) If we change the choices of the magnetic intensity and time intervals, for example, choose a random fluctuant magnetic field, the generalized M-J sets may present the interior-filling structure feature, i.e., "explosion" phenomena appears in the closure of the particle's instable periodic orbits in the velocity space.
Firstly, paper explains some related fractal theory and concept of image processing about M-J Sets and incremental image generation. The emphasis of this paper is section four and section five, research and discuss over M-J Sets' application in Langevin problem and interactive color image generation algorithm. Finally, paper summarizes the whole paper. Correlative content about M-J Sets in this paper is published in Acta Physica Sinica, which is indexed by SCI and EI.
本文的另一部分是基于一个典型的Langevin问题——在双势井和变化的磁场中并受一恒冲量断续作用的运动带电粒子的动力学分析，利用频闪采样法，给出了描述粒子速度变化规律的复差分方程。选取适当的磁场强度和时间间隔(采样周期)，将这一差分方程简化为用来构造广义M-J(Mandelbrot-Julia)集的复映射，并基于粒子的动力学特征探讨了广义M-J集的物理意义。结果发现：(1)广义M-J集的分形结构特征可形象地反映出粒子速度的变化规律；(2)选取的时间间隔有、无意义，决定了广义M-J集的分形结构是否具有连续性；(3)广义M-J集的演化——即粒子速度的变化规律依赖于相角主值范围的不同选取；(4)若改变磁场强度和时间间隔的选取，如选取一随机波动的磁场，则此时广义J集可能会出现内部被填充的结构特征，即在速度空间中粒子的不稳定周期轨道的闭包出现“爆炸”现象。
本文首先对M-J集和增强图像生成算法所涉及的分形理论和图像处理概念进行了阐述。本文的重点是第四章和第五章，分别对M-J集在Langevin问题中的应用和交互式应用中的彩色图像生成算法进行了研究和探讨，本文的最后对实验的结果进行了分析和总结。本文关于M-J集的相关内容已经在SCI核心收录期刊《物理学报》上发表，并被SCI与EI收录。
Most image generation algorithms scan image pixels according to top-to-bottom and left-to-right scanning; calculate the value of every pixel. It gives no idea of how the final image will be while it is being generated, or only part of image attribute can be known. Those traditional scanning is not a good method to interactive application where the user can choose image quality freely and interfere in the rendering modifying the generation parameters. In this paper the author analyzed interlaced image, quadtree and Baptista image generation method, and then presented a new interactive image generation method: Firstly, Sampling evenly for image pixels on the base of Baptista's thinking, and evaluating the pixel's value according to image pixel distribution's geometry character of pixel's neighborhood. The experimental results show that the interpolation algorithm can reconstruct high resolution images using less data. The quality of image generation is improved, while the quantity of data transmission is reduced greatly.
Based on a dynamics research of the typical Langevin problem, i.e., a moving charged particle under the continuous influence of a constant impulse in a double-well potential and a time-dependent magnetic field, using the stroboscopic sampling, we propose complex difference equations which can describe the change rule of particle's velocity. By selecting appropriate magnetic intensity and time intervals (sampling period), we reduce the difference equations to complex mapping which is used to construct the generalized M-J sets. Based on the particle's dynamics characteristic, we discuss the physical meaning of the generalized M-J sets. The author found that: (1) The fractal structure of the generalized M-J sets may visually reflect the change rule of particle's velocity; (2) Whether the selected time intervals is significative determine whether the fractal structure of the generalized M-J sets has the continuity; (3) The evolution of the generalized M-J sets, i.e., the change rule of particle's velocity, depends on the different choices of the principal range of the phase angle; (4) If we change the choices of the magnetic intensity and time intervals, for example, choose a random fluctuant magnetic field, the generalized M-J sets may present the interior-filling structure feature, i.e., "explosion" phenomena appears in the closure of the particle's instable periodic orbits in the velocity space.
Firstly, paper explains some related fractal theory and concept of image processing about M-J Sets and incremental image generation. The emphasis of this paper is section four and section five, research and discuss over M-J Sets' application in Langevin problem and interactive color image generation algorithm. Finally, paper summarizes the whole paper. Correlative content about M-J Sets in this paper is published in Acta Physica Sinica, which is indexed by SCI and EI.
引文
(1) 李后强，王富强．分形理论及其在分子科学中的应用．第一版．北京：科学出版社，1993．1～2
(2) 王兴元．广义M-J集的分形机理．第一版．大连：大连理工大学出版社．2002．1～50
(3) Jacquin. Fractal image coding based on a fractal theory of iterated contractive image transformation. IEEE Trans. on Image Processing, 1992, 1(1):18～30
(4) Chong Sze Tong, Man Wong. Adaptive approximate nearest neighbor search for fractal image compression. IEEE Trans. on Image Processing, 2002, 11(6):605～615
(5) K. Belloulata, J. Konrad. Fractal image compression with region-based functionality. IEEE Trans. on Image Processing, 2002, 11(4):351～362
(6) 陈毅松，王继成，张福炎．分形图像编码的快速细粒度迭代解码．计算机学报．2002，25(3)：269～275
(7) Lai Cheung Ming, Lam Kin Man, Siu Wan Chi. a fast fractal image coding based on kick out and zero contrast conditions. IEEE Trans. on Image Processing, 2003, 12(11): 1398～1403
(8) Gerry Melnikov, Aggelos K. Katsaggelos. A jointly optimal fractal/DCT compression scheme, IEEE Trans. on Multimedia, 2002, 4(4):413～422
(9) Taekon Kim, Robert E. Van Dyck, David J.Miller. Hybrid fractal zerotree wavelet image coding. Signal Processing:Image Communication, 2002, 17(4):347～360
(10) M.D. Geoffrey. A wavelet-based analysis of fractal image compression. IEEE Trans. on Image Processing, 1998, 7(2):141～152
(11) J.M. Mas Ribé s, B. Simon, B. Macq. Combined Kohonen neural networks and discrete cosine transform method for iterated transformation theory. Signal Processing:Image Communication, 2001, 16(7):643～656
(12) K.T. Sun, S.J. Lee, P.Y. Wu, Neural network approaches to fractal image compression and decompression. Neurocomputing, 2001, 41(1-4):91～107
(13) D. Dasgupta, G. Hernandez, F. Nino. An evolutionary algorithm for fractal coding of binary images. IEEE Trans. on Evolutionary Computation, 2000, 4(2):172～181
(14) Xiaolin Wu, Jiang Wen, Wing Hung Wong. Conditional entropy coding of VQ indexes for image compression. IEEE Trans. on Image Processing, 1999, 8(8):1005～1013
(15) Chang Su Kim, Rin Chul Kim, Sang Uk Lee. A fractal vector quantizer for image coding. IEEE Trans. on Image Processing, 1998, 7(11):1598～1602
(16) Chong Sze Tong, Minghong Pi. Analysis of a hybrid fractal-predictive-coding compression scheme. Signal Processing:Image Communication, 2003, 18(6):483～495
(17) R.G. Van Schyndel, A.Z. Tirkel, C.F. Osborne. A digital watermark, Proc. of ICIP 94. 1994, 1(2):86～90.
(18) 孙家广．计算机图形学．第三版．北京：清华大学出版社，1998．463～467
(19) 何斌，马天予，王运坚，朱红莲．Visual C++数字图像处理．第一版．北京：人民邮电出版社，2001．1～3
(20) 章毓晋．图象工程．第一版．北京：清华大学出版社，2000．2～3
(21) 贾永红．数字图像处理．第一版．武汉：武汉大学出版社，2003．8～9
(22) 杨海军，梁德群，毕胜．基于图象方向性信息测度的图象象素分类．中国图象图形学报，2001，6(5)：429～433
(23) Karu K, Jain A K, Bolle R M. Is there any texture in the image. Pattern Recognition, 1996, 29 (8): 1437～1446
(24) Ran X, Favardin N. A perceptually motivated three-component image model. IEEE Trans. Image Processing, 1995, 4(3) :430～447
(25) Chou W S. Classify image pixels into shaped smooth and textured points. Pattern Recognition, 1999, 32 (9):1697～1706
(26) Rossetti Baptista, Humberto. A method for incremental image generation. Computers and Graphics, 1999, 23(3):449～454
(27) 曾文曲，王向阳．分形理论及分形理论的计算机模拟．第二版．沈阳：东北大学出版社，2001．144～145
(28) 王兴元，孟庆业．基于Langevin问题探讨广义M-J集的物理意义．物理学报，2004，53(2)：388～395
(29) Kenneth R. Castleman. Digital Image Processing.第一版．北京：电子工业出版社，2002. 46～47
(30) Wang Xingyuan. Research on the relation of chaos activity characteristics of the cardiac system with the evolution of species. Chinese science bulletin, 2002, 47(24): 2042～2048
(31) Wang Xingyuan, Liu Xiangdong, Zhu Weiyong. Analysis of c-plane fractal images from z←z~α+c forα＜O. Fractals, 2000, 8(3):307～314
(32) Wang Xingyuan. Fractal structures of the non-boundary region of the generalized Mandelbrot set. Progress in Natural Science, 2001, 11(9):693～700
(33) Chen Ning, ZhuWeiyong. Bud-Sequence conjecture on M fractal image and M-J conjecture between C and Z planes fromz←z~w+c(W=α+iβ). Computers and Graphics, 1998, 22(4): 537～546
(34) 王兴元，刘向东，朱伟勇．由复映射z←z~α+c(α＜0)所构造的广义M-J集的研究．数学物理学报，1999，19(1)：73～79
(35) 王兴元，朱伟勇．IFS吸引子的计算机模拟．计算物理，2000，17(1)：407～413
(36) 王兴元，朱伟勇．正实数阶广义J集的嵌套拓扑分布定理．东北大学学报(自然科学版)，1999，20(5)：489～492
(37) 王兴元，顾树生．负实数阶广义J集的演化．中国图象与图形学报，2001，6A(5)：491～495
(38) 王兴元．复映射z←z~w+c(w∈C)的广义Mandelbrot和Julia组合集．东北大学学报(自然科学版)．2001，22(6)：611～614
(39) 王兴元，刘向东，顾树生，朱伟勇．正实数阶广义M集的嵌套拓扑分布定理．东北大学学报(自然科学版)，2000，21(2)：155～158
(40) 彭金生，李高翔．近代量子光学导论．第一版．北京：科学出版社，1996．98～102
(2) 王兴元．广义M-J集的分形机理．第一版．大连：大连理工大学出版社．2002．1～50
(3) Jacquin. Fractal image coding based on a fractal theory of iterated contractive image transformation. IEEE Trans. on Image Processing, 1992, 1(1):18～30
(4) Chong Sze Tong, Man Wong. Adaptive approximate nearest neighbor search for fractal image compression. IEEE Trans. on Image Processing, 2002, 11(6):605～615
(5) K. Belloulata, J. Konrad. Fractal image compression with region-based functionality. IEEE Trans. on Image Processing, 2002, 11(4):351～362
(6) 陈毅松，王继成，张福炎．分形图像编码的快速细粒度迭代解码．计算机学报．2002，25(3)：269～275
(7) Lai Cheung Ming, Lam Kin Man, Siu Wan Chi. a fast fractal image coding based on kick out and zero contrast conditions. IEEE Trans. on Image Processing, 2003, 12(11): 1398～1403
(8) Gerry Melnikov, Aggelos K. Katsaggelos. A jointly optimal fractal/DCT compression scheme, IEEE Trans. on Multimedia, 2002, 4(4):413～422
(9) Taekon Kim, Robert E. Van Dyck, David J.Miller. Hybrid fractal zerotree wavelet image coding. Signal Processing:Image Communication, 2002, 17(4):347～360
(10) M.D. Geoffrey. A wavelet-based analysis of fractal image compression. IEEE Trans. on Image Processing, 1998, 7(2):141～152
(11) J.M. Mas Ribé s, B. Simon, B. Macq. Combined Kohonen neural networks and discrete cosine transform method for iterated transformation theory. Signal Processing:Image Communication, 2001, 16(7):643～656
(12) K.T. Sun, S.J. Lee, P.Y. Wu, Neural network approaches to fractal image compression and decompression. Neurocomputing, 2001, 41(1-4):91～107
(13) D. Dasgupta, G. Hernandez, F. Nino. An evolutionary algorithm for fractal coding of binary images. IEEE Trans. on Evolutionary Computation, 2000, 4(2):172～181
(14) Xiaolin Wu, Jiang Wen, Wing Hung Wong. Conditional entropy coding of VQ indexes for image compression. IEEE Trans. on Image Processing, 1999, 8(8):1005～1013
(15) Chang Su Kim, Rin Chul Kim, Sang Uk Lee. A fractal vector quantizer for image coding. IEEE Trans. on Image Processing, 1998, 7(11):1598～1602
(16) Chong Sze Tong, Minghong Pi. Analysis of a hybrid fractal-predictive-coding compression scheme. Signal Processing:Image Communication, 2003, 18(6):483～495
(17) R.G. Van Schyndel, A.Z. Tirkel, C.F. Osborne. A digital watermark, Proc. of ICIP 94. 1994, 1(2):86～90.
(18) 孙家广．计算机图形学．第三版．北京：清华大学出版社，1998．463～467
(19) 何斌，马天予，王运坚，朱红莲．Visual C++数字图像处理．第一版．北京：人民邮电出版社，2001．1～3
(20) 章毓晋．图象工程．第一版．北京：清华大学出版社，2000．2～3
(21) 贾永红．数字图像处理．第一版．武汉：武汉大学出版社，2003．8～9
(22) 杨海军，梁德群，毕胜．基于图象方向性信息测度的图象象素分类．中国图象图形学报，2001，6(5)：429～433
(23) Karu K, Jain A K, Bolle R M. Is there any texture in the image. Pattern Recognition, 1996, 29 (8): 1437～1446
(24) Ran X, Favardin N. A perceptually motivated three-component image model. IEEE Trans. Image Processing, 1995, 4(3) :430～447
(25) Chou W S. Classify image pixels into shaped smooth and textured points. Pattern Recognition, 1999, 32 (9):1697～1706
(26) Rossetti Baptista, Humberto. A method for incremental image generation. Computers and Graphics, 1999, 23(3):449～454
(27) 曾文曲，王向阳．分形理论及分形理论的计算机模拟．第二版．沈阳：东北大学出版社，2001．144～145
(28) 王兴元，孟庆业．基于Langevin问题探讨广义M-J集的物理意义．物理学报，2004，53(2)：388～395
(29) Kenneth R. Castleman. Digital Image Processing.第一版．北京：电子工业出版社，2002. 46～47
(30) Wang Xingyuan. Research on the relation of chaos activity characteristics of the cardiac system with the evolution of species. Chinese science bulletin, 2002, 47(24): 2042～2048
(31) Wang Xingyuan, Liu Xiangdong, Zhu Weiyong. Analysis of c-plane fractal images from z←z~α+c forα＜O. Fractals, 2000, 8(3):307～314
(32) Wang Xingyuan. Fractal structures of the non-boundary region of the generalized Mandelbrot set. Progress in Natural Science, 2001, 11(9):693～700
(33) Chen Ning, ZhuWeiyong. Bud-Sequence conjecture on M fractal image and M-J conjecture between C and Z planes fromz←z~w+c(W=α+iβ). Computers and Graphics, 1998, 22(4): 537～546
(34) 王兴元，刘向东，朱伟勇．由复映射z←z~α+c(α＜0)所构造的广义M-J集的研究．数学物理学报，1999，19(1)：73～79
(35) 王兴元，朱伟勇．IFS吸引子的计算机模拟．计算物理，2000，17(1)：407～413
(36) 王兴元，朱伟勇．正实数阶广义J集的嵌套拓扑分布定理．东北大学学报(自然科学版)，1999，20(5)：489～492
(37) 王兴元，顾树生．负实数阶广义J集的演化．中国图象与图形学报，2001，6A(5)：491～495
(38) 王兴元．复映射z←z~w+c(w∈C)的广义Mandelbrot和Julia组合集．东北大学学报(自然科学版)．2001，22(6)：611～614
(39) 王兴元，刘向东，顾树生，朱伟勇．正实数阶广义M集的嵌套拓扑分布定理．东北大学学报(自然科学版)，2000，21(2)：155～158
(40) 彭金生，李高翔．近代量子光学导论．第一版．北京：科学出版社，1996．98～102