信息稀疏表示算法及其在图像恢复中应用的研究
|
摘要
随着科学技术的发展,以及人们需求的日益提高,在科学与工程诸多领域中出现的数据量呈现海量数据的特征.而研究发现,在人们生活中出现的大量数据都存在冗余、相关性强等特点,因此,如何从数据的特点出发,通过研究其稀疏表示的理论与方法,从而极大减少处理时间的空间复杂度与时间计算复杂度就成为现代数学与信息科学诸多研究领域面临的共同问题.
图像数据是一种最为常见的信息传输与处理用数据,在信息处理领域有着非常重要的地位.本文通过研究图像数据的特点,针对图像恢复这一传统的图像处理领域开展研究,研究了基于信息稀疏表示算法的超分辨图像重构与图像去噪、图像去模糊方法.
1.利用多尺度几何分析中的Bandelet并结合分裂Bregman迭代优化求解算法,建立了一种具有高质量图像恢复性能的超分辨图像重构模型.该方法Bandelet实现图像的高效稀疏表示,利用分裂Bregman方法实现模型的有效优化求解,数值实验表明该算法比已有方法在图像恢复质量方面有一定的提高.
2.建立了基于快速Fourier变换(FFT)的非局部分裂技术的图像去噪、去模糊算法.通过提出改进的图像全变分模型,并在此基础上得到了一种图像去噪、去模糊优化模型,该模型可以通过FFT方法求解,从而大大降低了算法的计算复杂性,并通过试验结果验证了方法的有效性.
With the development of Scinece and Technology, and more and more increasing of the requirement, cloud data is its mainly characteristic in modern science and engineering. As we known, in general, the data as mentioned above are reduancy and dependence, a common question is to consider theory and method for sprase representation of information so as to decrease complexity of space and time used greatly.
Image is a two-dimensional data used information transporation and processing with most commonly used and very improtatntly. In this paper, we study image restoration including super-solution, remove noise and deblurring.
1. Combined bandelet in multiscaling geometry analysis and splitted Bregamn optimization algorithm, a type of super-solution mathematical model is developed. Using Bandelet achieves sparse representation of image, and efficiency splitted Bregman algorithm is used to accelerate th computation speed. Numerical examples shows that the quality of reconstruction image is superior known classical algorithm.
2. Non-Local model via FFT algorithm for image noise removed and deblurring is proposed. By considering improved Total variation model of image, wo obtain optimization method for image noise removed and deblurring. Since FFT algorithm is used the computational complexity is decreased greatly. Experimenations showns the efficiency of the proposed methods.
图像数据是一种最为常见的信息传输与处理用数据,在信息处理领域有着非常重要的地位.本文通过研究图像数据的特点,针对图像恢复这一传统的图像处理领域开展研究,研究了基于信息稀疏表示算法的超分辨图像重构与图像去噪、图像去模糊方法.
1.利用多尺度几何分析中的Bandelet并结合分裂Bregman迭代优化求解算法,建立了一种具有高质量图像恢复性能的超分辨图像重构模型.该方法Bandelet实现图像的高效稀疏表示,利用分裂Bregman方法实现模型的有效优化求解,数值实验表明该算法比已有方法在图像恢复质量方面有一定的提高.
2.建立了基于快速Fourier变换(FFT)的非局部分裂技术的图像去噪、去模糊算法.通过提出改进的图像全变分模型,并在此基础上得到了一种图像去噪、去模糊优化模型,该模型可以通过FFT方法求解,从而大大降低了算法的计算复杂性,并通过试验结果验证了方法的有效性.
With the development of Scinece and Technology, and more and more increasing of the requirement, cloud data is its mainly characteristic in modern science and engineering. As we known, in general, the data as mentioned above are reduancy and dependence, a common question is to consider theory and method for sprase representation of information so as to decrease complexity of space and time used greatly.
Image is a two-dimensional data used information transporation and processing with most commonly used and very improtatntly. In this paper, we study image restoration including super-solution, remove noise and deblurring.
1. Combined bandelet in multiscaling geometry analysis and splitted Bregamn optimization algorithm, a type of super-solution mathematical model is developed. Using Bandelet achieves sparse representation of image, and efficiency splitted Bregman algorithm is used to accelerate th computation speed. Numerical examples shows that the quality of reconstruction image is superior known classical algorithm.
2. Non-Local model via FFT algorithm for image noise removed and deblurring is proposed. By considering improved Total variation model of image, wo obtain optimization method for image noise removed and deblurring. Since FFT algorithm is used the computational complexity is decreased greatly. Experimenations showns the efficiency of the proposed methods.
引文
[1] Mallat S. A Wavelet Tour of Signal Processing [M]. Academic Press, 2nd edition, 1999.
[2]成礼智,王红霞,罗永.小波的理论与应用[M].北京:科学出版社, 2004.
[3] Donoho D L, Flesia A G. Can Recent Innovations in Harmonic Analysis‘Explain’Key Findings in Natural Image Statistics? [J]. Network: Computation in Neural Systems, 2001, 12(3):371-393.
[4] Mallat S and Zhang Z, Matching pursuits with time-frequency dictionaries[J]. IEEE Trans. Signal Process., 1993,41 (10):3397–3415.
[5] Chen S S and Donoho D.A new decomposition method via basis pursuit[J].SIAM J.Sci.Comput.
[6] 1998,20(1):33-61.
[7] Lyu S. Natural Image Statistics for Digital Image Forensics [D]. USA, Department of Computer Science, Dartmouth College, 2005.
[8] Nussbaumer H J, Fast Fourier Transform and Convolution Algorithms [M]. 2nd edition, New York: Springer-Verlag, 1981.
[9] Chui C K. An Introduction to Wavelets [M]. Academic Press, 1992.
[10] Chen S S and Donoho D L. Atomic decomposition by basis pursuit[J]. SIAM Rev. 2001,43 :129–159.
[11] Donoho D L. Orthonormal Ridgelets and Linear Singularities [R]. Department of Statistics, Stanford University, 1998.
[12] Candès E J and Donoho D L. Curvelets - A Surprisingly Effective Nonadaptive Representation for Objects with Edges, Curves and Surfaces [M]. L. L. Schumaker et al. (eds), Vanderbilt University Press, Nashville, TN, 1999.
[13] Donoho D L. Wedgelets: Nearly-minimax estimation of edges[J]. Annals of Stat. 1999,27:859-897.
[14] Pennec E L and Mallat S. Image compression with geometrical wavelets[A]. In Proc. of ICIP’2000[C]. Vancouver, Canada, September ,2000: 66l一664.
[15] Minh N. Do and Martin Vetterli M. Contourlets: a new directional multiresolution image representation[J]. Signal,Systems and Computers, 2002, l: 497–50
[16] Pennec E L and Mallat S. Bandelet Image Approximation and Compression[J]. SIAM Journal of Multiscale Modeling and Simulation.2005,4(3):992-1039.
[17] Osher S, Burger M, et al. An Iterated Regularization Method for Total Variation-Based Image Restoration [J]. Multiscale Modeling & Simulation, 2005, 4(2): 460-489.
[18] Rudin L, Osher S and Fatemi E. Nonlinear total variation based noise removal algorithms[J]. Physica D, 1992,60:259-268.
[19]张慧,成礼智. A-线性Bregman迭代算法[J].计算数学,2010,32(1):1-9.
[20] Cai J F, Osher S and Shen Z W. Split Bregman Methods And Frame Based Image Restorartioin[R]. CAM Report (09-28), University of California Los Angeles ,2009
[21] Cai J F, Chan R H and Shen Z. Linearized Bregman iterations for compressed sensing[J]. Math,Comp., 2009(78):1515-1536.
[22] TGoldstein T, Bresson X, and Osher S. Geometric applications of the split Bregman method: Segmentation and surface reconstruction[R]. CAM Report 09-06, University of California Los Angeles, 2009
[23] Wang Z, Bovik A C. Mean Squared Error: Love it or Leave it? - A New Look at Signal Fidelity Measures [J]. IEEE Signal Processing Magazine, 2009, 26(1): 98-117.
[24] Tichonov A and Arsenin V. Solution of Ill-Posed Problems[M]. New York: Wiley, 1977.
[25] Rudin L and Osher S. Total variation based image restoration with free local constraints[C].Proc. 1st IEEE ICIP,1994, 1:31–35.
[26] Buades A, Coll B and Morel J M. On image denoising methods. SIAM Multiscale Modeling and Simulation.2005, 4(2):490–530.
[27] Gilboa J and Osher S. Nonlocal linear image regularization and supervised segmentation[J]. SIAM Multiscale Modeling and Simulation 2007,6 (2):595–63.
[28] Gilboa J and Osher S.Nonlocal operators with applications to image processing[R]. UCLA CAM report 07-23,2007.
[29] A. Chambolle. An algorithm for total variation minimization and applications{j}. JMIV.2004, 20:89–97.
[30] Liu Y F , Zhang X Q and Osher S.Image Recovery via Nonlocal Operators[R]. Technical Report 35,UCLA 2008.
[31] Yang J, W Yin and. A fast algorithm for edge-preserving variational multichannel image restoration[R]. Tech. Report 08-09,CAAM, Rice University.
[32]庞建新,图像质量客观评价研究[D].合肥,中国科技大学信号与信息处理, 2008.
[2]成礼智,王红霞,罗永.小波的理论与应用[M].北京:科学出版社, 2004.
[3] Donoho D L, Flesia A G. Can Recent Innovations in Harmonic Analysis‘Explain’Key Findings in Natural Image Statistics? [J]. Network: Computation in Neural Systems, 2001, 12(3):371-393.
[4] Mallat S and Zhang Z, Matching pursuits with time-frequency dictionaries[J]. IEEE Trans. Signal Process., 1993,41 (10):3397–3415.
[5] Chen S S and Donoho D.A new decomposition method via basis pursuit[J].SIAM J.Sci.Comput.
[6] 1998,20(1):33-61.
[7] Lyu S. Natural Image Statistics for Digital Image Forensics [D]. USA, Department of Computer Science, Dartmouth College, 2005.
[8] Nussbaumer H J, Fast Fourier Transform and Convolution Algorithms [M]. 2nd edition, New York: Springer-Verlag, 1981.
[9] Chui C K. An Introduction to Wavelets [M]. Academic Press, 1992.
[10] Chen S S and Donoho D L. Atomic decomposition by basis pursuit[J]. SIAM Rev. 2001,43 :129–159.
[11] Donoho D L. Orthonormal Ridgelets and Linear Singularities [R]. Department of Statistics, Stanford University, 1998.
[12] Candès E J and Donoho D L. Curvelets - A Surprisingly Effective Nonadaptive Representation for Objects with Edges, Curves and Surfaces [M]. L. L. Schumaker et al. (eds), Vanderbilt University Press, Nashville, TN, 1999.
[13] Donoho D L. Wedgelets: Nearly-minimax estimation of edges[J]. Annals of Stat. 1999,27:859-897.
[14] Pennec E L and Mallat S. Image compression with geometrical wavelets[A]. In Proc. of ICIP’2000[C]. Vancouver, Canada, September ,2000: 66l一664.
[15] Minh N. Do and Martin Vetterli M. Contourlets: a new directional multiresolution image representation[J]. Signal,Systems and Computers, 2002, l: 497–50
[16] Pennec E L and Mallat S. Bandelet Image Approximation and Compression[J]. SIAM Journal of Multiscale Modeling and Simulation.2005,4(3):992-1039.
[17] Osher S, Burger M, et al. An Iterated Regularization Method for Total Variation-Based Image Restoration [J]. Multiscale Modeling & Simulation, 2005, 4(2): 460-489.
[18] Rudin L, Osher S and Fatemi E. Nonlinear total variation based noise removal algorithms[J]. Physica D, 1992,60:259-268.
[19]张慧,成礼智. A-线性Bregman迭代算法[J].计算数学,2010,32(1):1-9.
[20] Cai J F, Osher S and Shen Z W. Split Bregman Methods And Frame Based Image Restorartioin[R]. CAM Report (09-28), University of California Los Angeles ,2009
[21] Cai J F, Chan R H and Shen Z. Linearized Bregman iterations for compressed sensing[J]. Math,Comp., 2009(78):1515-1536.
[22] TGoldstein T, Bresson X, and Osher S. Geometric applications of the split Bregman method: Segmentation and surface reconstruction[R]. CAM Report 09-06, University of California Los Angeles, 2009
[23] Wang Z, Bovik A C. Mean Squared Error: Love it or Leave it? - A New Look at Signal Fidelity Measures [J]. IEEE Signal Processing Magazine, 2009, 26(1): 98-117.
[24] Tichonov A and Arsenin V. Solution of Ill-Posed Problems[M]. New York: Wiley, 1977.
[25] Rudin L and Osher S. Total variation based image restoration with free local constraints[C].Proc. 1st IEEE ICIP,1994, 1:31–35.
[26] Buades A, Coll B and Morel J M. On image denoising methods. SIAM Multiscale Modeling and Simulation.2005, 4(2):490–530.
[27] Gilboa J and Osher S. Nonlocal linear image regularization and supervised segmentation[J]. SIAM Multiscale Modeling and Simulation 2007,6 (2):595–63.
[28] Gilboa J and Osher S.Nonlocal operators with applications to image processing[R]. UCLA CAM report 07-23,2007.
[29] A. Chambolle. An algorithm for total variation minimization and applications{j}. JMIV.2004, 20:89–97.
[30] Liu Y F , Zhang X Q and Osher S.Image Recovery via Nonlocal Operators[R]. Technical Report 35,UCLA 2008.
[31] Yang J, W Yin and. A fast algorithm for edge-preserving variational multichannel image restoration[R]. Tech. Report 08-09,CAAM, Rice University.
[32]庞建新,图像质量客观评价研究[D].合肥,中国科技大学信号与信息处理, 2008.