基于量子测量的框架构造
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摘要
框架理论是继小波理论发展之后发展起来的一个新的研究方向,在信号处理领域有着重要的应用。量子信号处理作为新兴的处理方法,以其并行化的特性为信号处理提供了不少新的思路。本文的主要目的是在量子信息基本原理的分析和研究基础上,为信号处理分析,特别是框架分析构造,寻找新的模型。
认知图是一种通过表示和推理概念间的因果关系来模拟系统运转的系统建模工具。本文将量子元的概念引入认知图中,首次提出了量子认知图的概念。其优势在于可以充分利用各方统计数据,更好地模拟真实系统。
本文对量子信息理论的另一重要应用,是将量子测量和量子层析理论与框架理论联系起来,利用量子理论,建立了信号矩阵的表达方式,提出了一种基于量子层析的无相位信息的框架重构算法。文中给出了两种典型的可以由幅值信息直接重构的框架形式SICPOVM和最大无偏基,并讨论了最大无偏基框架的传输鲁棒性。讨论说明该类框架在某一些特定系数丢失的情况下,仍然能够重构信号。在最后,本文对量子信息理论在认知图和框架理论的发展方向进行了展望。
Frame theory plays an important role in the theory of signal processing besides the wavelet theory.Quantum signal processing provides new ideas for the method of processing signal.Our objective of this paper is new model for signal processing,especially frame expansion analysis,based on the study of basic theory of the quantum imformation.
Cognitive Maps is an important graphic means of representing causal relationship between concepts and analyzing inference patterns.We for first time propose a Quantum Cognitive Map Model which introduce quantum information theory into Cognitive Maps.The advantage of the model is more information preserving and better approaching the real world.
We also apply the quantum information theory into the frame theory.Our goal is to propose the frame reconstruction method without phase information.First,theory of quantum measurement and quantum tomogram is related to the frame theory.Then,we pose the concept of signal matrix,and presents two typical frames needed including SICPOVM and Maximal Mutually Unbiased bases. Finally,the robustness of this frames is discussed.
认知图是一种通过表示和推理概念间的因果关系来模拟系统运转的系统建模工具。本文将量子元的概念引入认知图中,首次提出了量子认知图的概念。其优势在于可以充分利用各方统计数据,更好地模拟真实系统。
本文对量子信息理论的另一重要应用,是将量子测量和量子层析理论与框架理论联系起来,利用量子理论,建立了信号矩阵的表达方式,提出了一种基于量子层析的无相位信息的框架重构算法。文中给出了两种典型的可以由幅值信息直接重构的框架形式SICPOVM和最大无偏基,并讨论了最大无偏基框架的传输鲁棒性。讨论说明该类框架在某一些特定系数丢失的情况下,仍然能够重构信号。在最后,本文对量子信息理论在认知图和框架理论的发展方向进行了展望。
Frame theory plays an important role in the theory of signal processing besides the wavelet theory.Quantum signal processing provides new ideas for the method of processing signal.Our objective of this paper is new model for signal processing,especially frame expansion analysis,based on the study of basic theory of the quantum imformation.
Cognitive Maps is an important graphic means of representing causal relationship between concepts and analyzing inference patterns.We for first time propose a Quantum Cognitive Map Model which introduce quantum information theory into Cognitive Maps.The advantage of the model is more information preserving and better approaching the real world.
We also apply the quantum information theory into the frame theory.Our goal is to propose the frame reconstruction method without phase information.First,theory of quantum measurement and quantum tomogram is related to the frame theory.Then,we pose the concept of signal matrix,and presents two typical frames needed including SICPOVM and Maximal Mutually Unbiased bases. Finally,the robustness of this frames is discussed.
引文
[Anon] (1994). "Nikon, National Labs Joint in Measurement System." Photonics Spectra 28(12): 32-32.
Abdelkefi, F. (2007). Sigma-Delta quantization of geometrically uniform finite frames. 32nd IEEE International Conference on Acoustics, Speech and Signal Processing, Honolulu, HI, Ieee.
Abdelkefi, F. (2008). "Performance of Sigma-Delta Quantizations in Finite Frames." Ieee Transactions on Information Theory 54(11): 5087-5101.
Aguilar, J. (2002). Adaptive random fuzzy cognitive maps. Advances in Artificial Intelligence - Iberamia 2002, Proceedings. Berlin, Springer-Verlag Berlin. 2527: 402-410.
Aguilar, J. (June 2005). "A survey about Fuzzy Cognitive maps papers." International Journal of Computational Cognition Vol 3(No.2).
Alexandr A, E. a. D. V. "Quantum neural networks."
Balan, R., P. Casazza, et al. (2006). "On signal reconstruction without phase." Applied and Computational Harmonic Analysis 20(3): 345-356.
Balan, R., P. G. Casazza, et al. (2007). "A new identity for Parseval frames." Proceedings of the American Mathematical Society 135(4): 1007-1015.
Baoyu, L. Z. (2003). "A study of quantum neural networks." IEEE int.Conf.Neural Networks&Signal Processing.
Benedetto, J. J. and M. Fickus (2003). "Finite normalized tight frames." Advances in Computational Mathematics 18(2-4): 357-385.
Benedetto, J. J., A. M.Powell, et al. (2004). Sigma-Delta (Sigma Delta) quantization and finite frames. IEEE International Conference on Acoustics Speech and Signal Processing, Montreal, CANADA.
Bernardini, R. and R. Rinaldo (2005). "Efficient reconstruction from frame-based multiple descriptions." Ieee Transactions on Signal Processing 53(8): 3282-3296.
Calderbank, A. R., P. G. Casazza, et al. (2009). constructing fusion frames with desired parameters, http://www.home.uos.de/kutyniok/papers/FFConstruct.pdf.
Calderbank, R., P. G. Casazza, et al. (2009) "fusion frames existence and construction."
Casazza, P. G. (2004). "Custom building finte frames wavelets,frames and operator theory." Amer Math Soc: 61-68.
Casazza, P. G., M. Fickus, et al. (2009). "Constructing tight fusion frames." Applied and Computational Harmonic Analysis.
Casazza, P. G. and J. Kovacevic (2003). "Equal-norm tight frames with erasures." Advances in Computational Mathematics 18(2-4): 387-430.
Casazza, P. G. and G. Kutyniok (2003) "Frames of subspaces."
Casazza, P. G., G. Kutyniok, et al. (2008). "Fusion frames and distributed processing." Applied and Computational Harmonic Analysis 25(1): 114-132.
Casazza, P. G. and M. T. Leon (2006). "Existence and construction of finite frames." Concr Appl Math 4(3): 277-289.
Casazza, P. G. and N. Leonhard (2006). Classes of finite equal norm Parseval frames. AMS-SIAM Meeting on Frames and Operator Theory in Analysis and Signal Processing held at the Joint Mathematics Meeting, San Antonio, TX, Amer Mathematical Soc.
Cvetkovic, Z. (2003). "Resilience properties of redundant expansions under additive noise and quantization." Ieee Transactions on Information Theory 49(3): 644-656.
Daubechie, I. (1990). "The wavelet transform,time-frequency localization and signal analysis." IEEE Trans Inform Theory 39: 961-1005.
Daubechies, I. (1992). Ten lectures on wavelets, SIAM.
Donoho, D. L. (1995). "De-Noising by Soft-Thresholding." Ieee Transactions onInformation Theory 41(3): 613-627.
Eldar, Y. C. and G. D. Forney (2002). "Optimal tight frames and quantum measurement." Ieee Transactions on Information Theory 48(3): 599-610.
Eldar, Y. C. and A. V. Oppenheim (2002). "Quantum signal processing." Ieee Signal Processing Magazine 19(6): 12-32.
Forouzanfar, M., H. A. Moghaddam, et al. (2008). "Locally adaptive multiscale Bayesian method for image denoising based on bivariate normal inverse Gaussian distributions." International Journal of Wavelets Multiresolution and Information Processing 6(4): 653-664.
Gabor.D (1946). "Theory of communication." J.IEE 93: 429-457.
Hagiwara, M. (1992). Extended fuzzy cognitive maps. Fuzzy Systems, 1992., IEEE International Conference on.
Heil, C. and D. Walnut (1989). "Continuous and discrete wavelet transforms." SIAM Review 31: 628-666.
I.Daubechies, A.Grossman, et al. (1986). "Painless nonorthogonal expansions." J.Math.Phys 27: 1271-1283.
Koo, H., N. I. Cho, et al. (2008). Image denoising based on a statistical model for wavelet coefficients. 33rd IEEE International Conference on Acoustics, Speech and Signal Processing, Las Vegas, NV, Ieee.
Kosko, B. (1986). "Fuzzy Cognitive Maps " Int.Journal of Man-Machine Studies 24: 65-75.
Kovacevic, J. and A. Chebira (2007). "Life beyond bases: The advent of frames (part I)." Ieee Signal Processing Magazine 24(4): 86-104.
Kovacevic, J., P. L. Dragotti, et al. (2002). "Filter bank frame expansions with erasures." Ieee Transactions on Information Theory 48(6): 1439-1450.
Liu, R. S. a. Z.-Q. (Oct1999). "A Contextual Fuzzy Cognitive Map Framework for Geographic Information Systems." Ieee Transactions on Fuzzy Systems Vol.7(No.5).
Liu, Y. M. a. Z.-Q. (2000). "On Causal Inference in Fuzzy Cognitive Maps." Ieee Transactions on Fuzzy Systems 8(1).
Liu, Z.-Q. (1999). "Contextual Fuzzy Cognitive Map for Decision Support in Geographic Information Systems." Ieee Transactions on Fuzzy Systems 7(5).
Nielsen, M. A. and I. L. Chuang (2000). Quantum computation and quantum information. Cambridge, Cambridge University Press.
Obata, T. and M. Hagiwara (1997). "Neural Cognitive Maps." 0-7803-4053-1/97 IEEE.
R.J.Duffin and A.C.Schaeffer (1952). "A class of nonharmonic Fourier series." Trans.Amer.Math.Soc 72: 341-366.
Robert, R. and W. Shanyne (2002). "Isometric tight frameS." The Journal of Linear Algebra 9: 122-128.
Shen, L. X., M. Papadakis, et al. (2006). "Image denoising using a tight frame." Ieee Transactions on Image Processing 15(5): 1254-1263.
Stylios C D, G. P. P. (2000). "Fuzzy cognitive maps: a soft computing technique for intelligent control[A]." IEEE International Symposium on intelligent control[c](Italy Patras): 97-19.
Tseng, C. C. and T. M. Hwang (2003). Quantum digital image processing algorithms. Proc of the 16th IPPR Conference on Computer Vision,Graphics and Image Processing: 827-834.
Weaver, J. B., Y. S. Xu, et al. (1991). "Filtering Noise from Images with Wavelet Transforms." Magnetic Resonance in Medicine 21(2): 288-295.
Wootters, W. K. (2006). "Quantum measurements and finite geometry." Foundations of Physics 36(1): 112-126.
Xu, Z. M., A. Makur, et al. (2007). On the robustness of filter bank frame to quantization and erasures. 32nd IEEE International Conference on Acoustics, Speech and Signal Processing, Honolulu, HI, Ieee.
Yuuki Nakamiya, M. K., Osamu,Takahashi,Shigeo Sato,Koji Nakajima (2005). "Basic property of a Quantum Neural Network Composed of Kane's Qubits." Proceedings of International Joint Conference on Neural Networks: 1104-1107.
辛友明(2007). "一种最优的紧框架构造算法."华东师范大学学报(自然科学版) 3: 58-61.
张彬,许延发, et al. (2007). "基于小波框架的红外/可见光图像融合."光学技术32(3): 334-336.
Abdelkefi, F. (2007). Sigma-Delta quantization of geometrically uniform finite frames. 32nd IEEE International Conference on Acoustics, Speech and Signal Processing, Honolulu, HI, Ieee.
Abdelkefi, F. (2008). "Performance of Sigma-Delta Quantizations in Finite Frames." Ieee Transactions on Information Theory 54(11): 5087-5101.
Aguilar, J. (2002). Adaptive random fuzzy cognitive maps. Advances in Artificial Intelligence - Iberamia 2002, Proceedings. Berlin, Springer-Verlag Berlin. 2527: 402-410.
Aguilar, J. (June 2005). "A survey about Fuzzy Cognitive maps papers." International Journal of Computational Cognition Vol 3(No.2).
Alexandr A, E. a. D. V. "Quantum neural networks."
Balan, R., P. Casazza, et al. (2006). "On signal reconstruction without phase." Applied and Computational Harmonic Analysis 20(3): 345-356.
Balan, R., P. G. Casazza, et al. (2007). "A new identity for Parseval frames." Proceedings of the American Mathematical Society 135(4): 1007-1015.
Baoyu, L. Z. (2003). "A study of quantum neural networks." IEEE int.Conf.Neural Networks&Signal Processing.
Benedetto, J. J. and M. Fickus (2003). "Finite normalized tight frames." Advances in Computational Mathematics 18(2-4): 357-385.
Benedetto, J. J., A. M.Powell, et al. (2004). Sigma-Delta (Sigma Delta) quantization and finite frames. IEEE International Conference on Acoustics Speech and Signal Processing, Montreal, CANADA.
Bernardini, R. and R. Rinaldo (2005). "Efficient reconstruction from frame-based multiple descriptions." Ieee Transactions on Signal Processing 53(8): 3282-3296.
Calderbank, A. R., P. G. Casazza, et al. (2009). constructing fusion frames with desired parameters, http://www.home.uos.de/kutyniok/papers/FFConstruct.pdf.
Calderbank, R., P. G. Casazza, et al. (2009) "fusion frames existence and construction."
Casazza, P. G. (2004). "Custom building finte frames wavelets,frames and operator theory." Amer Math Soc: 61-68.
Casazza, P. G., M. Fickus, et al. (2009). "Constructing tight fusion frames." Applied and Computational Harmonic Analysis.
Casazza, P. G. and J. Kovacevic (2003). "Equal-norm tight frames with erasures." Advances in Computational Mathematics 18(2-4): 387-430.
Casazza, P. G. and G. Kutyniok (2003) "Frames of subspaces."
Casazza, P. G., G. Kutyniok, et al. (2008). "Fusion frames and distributed processing." Applied and Computational Harmonic Analysis 25(1): 114-132.
Casazza, P. G. and M. T. Leon (2006). "Existence and construction of finite frames." Concr Appl Math 4(3): 277-289.
Casazza, P. G. and N. Leonhard (2006). Classes of finite equal norm Parseval frames. AMS-SIAM Meeting on Frames and Operator Theory in Analysis and Signal Processing held at the Joint Mathematics Meeting, San Antonio, TX, Amer Mathematical Soc.
Cvetkovic, Z. (2003). "Resilience properties of redundant expansions under additive noise and quantization." Ieee Transactions on Information Theory 49(3): 644-656.
Daubechie, I. (1990). "The wavelet transform,time-frequency localization and signal analysis." IEEE Trans Inform Theory 39: 961-1005.
Daubechies, I. (1992). Ten lectures on wavelets, SIAM.
Donoho, D. L. (1995). "De-Noising by Soft-Thresholding." Ieee Transactions onInformation Theory 41(3): 613-627.
Eldar, Y. C. and G. D. Forney (2002). "Optimal tight frames and quantum measurement." Ieee Transactions on Information Theory 48(3): 599-610.
Eldar, Y. C. and A. V. Oppenheim (2002). "Quantum signal processing." Ieee Signal Processing Magazine 19(6): 12-32.
Forouzanfar, M., H. A. Moghaddam, et al. (2008). "Locally adaptive multiscale Bayesian method for image denoising based on bivariate normal inverse Gaussian distributions." International Journal of Wavelets Multiresolution and Information Processing 6(4): 653-664.
Gabor.D (1946). "Theory of communication." J.IEE 93: 429-457.
Hagiwara, M. (1992). Extended fuzzy cognitive maps. Fuzzy Systems, 1992., IEEE International Conference on.
Heil, C. and D. Walnut (1989). "Continuous and discrete wavelet transforms." SIAM Review 31: 628-666.
I.Daubechies, A.Grossman, et al. (1986). "Painless nonorthogonal expansions." J.Math.Phys 27: 1271-1283.
Koo, H., N. I. Cho, et al. (2008). Image denoising based on a statistical model for wavelet coefficients. 33rd IEEE International Conference on Acoustics, Speech and Signal Processing, Las Vegas, NV, Ieee.
Kosko, B. (1986). "Fuzzy Cognitive Maps " Int.Journal of Man-Machine Studies 24: 65-75.
Kovacevic, J. and A. Chebira (2007). "Life beyond bases: The advent of frames (part I)." Ieee Signal Processing Magazine 24(4): 86-104.
Kovacevic, J., P. L. Dragotti, et al. (2002). "Filter bank frame expansions with erasures." Ieee Transactions on Information Theory 48(6): 1439-1450.
Liu, R. S. a. Z.-Q. (Oct1999). "A Contextual Fuzzy Cognitive Map Framework for Geographic Information Systems." Ieee Transactions on Fuzzy Systems Vol.7(No.5).
Liu, Y. M. a. Z.-Q. (2000). "On Causal Inference in Fuzzy Cognitive Maps." Ieee Transactions on Fuzzy Systems 8(1).
Liu, Z.-Q. (1999). "Contextual Fuzzy Cognitive Map for Decision Support in Geographic Information Systems." Ieee Transactions on Fuzzy Systems 7(5).
Nielsen, M. A. and I. L. Chuang (2000). Quantum computation and quantum information. Cambridge, Cambridge University Press.
Obata, T. and M. Hagiwara (1997). "Neural Cognitive Maps." 0-7803-4053-1/97 IEEE.
R.J.Duffin and A.C.Schaeffer (1952). "A class of nonharmonic Fourier series." Trans.Amer.Math.Soc 72: 341-366.
Robert, R. and W. Shanyne (2002). "Isometric tight frameS." The Journal of Linear Algebra 9: 122-128.
Shen, L. X., M. Papadakis, et al. (2006). "Image denoising using a tight frame." Ieee Transactions on Image Processing 15(5): 1254-1263.
Stylios C D, G. P. P. (2000). "Fuzzy cognitive maps: a soft computing technique for intelligent control[A]." IEEE International Symposium on intelligent control[c](Italy Patras): 97-19.
Tseng, C. C. and T. M. Hwang (2003). Quantum digital image processing algorithms. Proc of the 16th IPPR Conference on Computer Vision,Graphics and Image Processing: 827-834.
Weaver, J. B., Y. S. Xu, et al. (1991). "Filtering Noise from Images with Wavelet Transforms." Magnetic Resonance in Medicine 21(2): 288-295.
Wootters, W. K. (2006). "Quantum measurements and finite geometry." Foundations of Physics 36(1): 112-126.
Xu, Z. M., A. Makur, et al. (2007). On the robustness of filter bank frame to quantization and erasures. 32nd IEEE International Conference on Acoustics, Speech and Signal Processing, Honolulu, HI, Ieee.
Yuuki Nakamiya, M. K., Osamu,Takahashi,Shigeo Sato,Koji Nakajima (2005). "Basic property of a Quantum Neural Network Composed of Kane's Qubits." Proceedings of International Joint Conference on Neural Networks: 1104-1107.
辛友明(2007). "一种最优的紧框架构造算法."华东师范大学学报(自然科学版) 3: 58-61.
张彬,许延发, et al. (2007). "基于小波框架的红外/可见光图像融合."光学技术32(3): 334-336.