基于PDTDFB和HMM的图像去噪研究
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摘要
图像去噪是图像处理方面的一个经典问题,近年来该问题主要采用多尺度几何分析理论进行研究,其中将多尺度几何分析理论与隐马尔可夫模型(HMM)相结合是该领域中的一个研究热点。与传统方法相比,基于多尺度变换域的隐马尔可夫树(HMT)模型能够充分挖掘和描述多尺度变换域系数在尺度间、尺度内和方向间的统计相关性,有效解决传统方法去噪后图像边缘模糊问题。因此,基于多尺度变换域的隐马尔可夫树模型在图像去噪领域具有广阔的应用前景。
论文在研究金字塔对偶树方向滤波器组(PDTDFB)变换和隐马尔可夫模型的基础上,重点研究了图像经PDTDFB变换之后,其系数在尺度间和子带间表现出的相关性特性以及隐马尔可夫模型对这种相关性的统计特性,提出了两种新的基于PDTDFB变换的隐马尔可夫树(HMT)模型用于图像去噪。主要工作如下:
①根据PDTDFB变换系数的相关性特征和分布,提出了基于PDTDFB复系数的实部和虚部的双树PDTDFB-HMT去噪模型。利用两状态混合高斯模型描述PDTDFB系数的分布,对PDTDFB变换的高频子带系数的建立HMT模型,运用EM算法对HMT模型训练获得含噪声图像的模型参数。通过与噪声值的差进而获得不含噪声图像的模型参数。然后运用贝叶斯准则估计出不含噪声图像PDTDFB系数的后验均值估计值,最后用PDTDFB逆变换重建信号,得到最终的去噪图像。
②基于模的大小对于图像很小平移的不敏感性,在给定的位置和尺度上可对图像产生更为准确的估计,提出了基于PDTDFB复系数的模的PDTDFB-HMT去噪模型。首先对噪声图像进行PDTDFB变换,然后对高频子带系数进行取模运算并计算幅角关系,建立基于模的PDTDFB-HMT,去噪过程与双树PDTDFB-HMT类似,最后根据模和幅角的转换关系恢复系数,利用PDTDFB逆变换重建信号,得到最终的去噪图像。
在对高斯白噪声的仿真实验中,上述二种方法与其他几种典型的去噪算法相比,在客观评价标准上,峰值信噪比(PSNR)有不同程度的提高,较好地去除了高斯白噪声;在主观评价标准上,较好地保留了图像的边缘和细节信息。尤其是基于模的PDTDFB-HMT方法,对图像边缘的保持尤为显著,具有较好的去噪效果。但是由于HMM的统计运算量很大,导致算法执行效率偏低,有待进一步改进提高。
Image denoising is a classical subject in image processing. In recent years, image denoising algorithm based multi-scale geometric analysis becomes more and more popular in this field. Multi-scale geometric analysis combined with HMM is a research hot spot at present. Comparison with the traditional denoising method, the HMM based on multi-scale transform domain can fully capture and describe the statistical correlation of the multi-scale transform coefficients in inter-scale and inter-direction, and effectively solve the blur problem of image edge. Therefore it has promising applications to image de-noising.
In this paper, after studying the elementary concept of PDTDFB transform and HMM, the dependencies between PDTDFB coefficients and the statistical properties of HMM are paid more attention. Finally two new denoising algorithms based on PDTDFB and Hidden Markov Tree (HMT) are proposed. The main contributions of this paper are summarized as follows.
①Performing PDTDFB transform in noise image, then use HMT model to characterize the dependencies of PDTDFB coefficients, thus a new HMT based on the real part and imaginary part of PDTDFB coefficient is modeled. The 2-state Gaussian mixture model is used to approximate the image PDTDFB coefficients margin distribution and the matrix of state transform can capture the dependencies of the PDTDFB coefficients across the scales, EM method is used to train the HMT to get the parameters of HMT of the noise image, and then through subtracting the noise value to get the HMT parameters of original image. Bayesian is used to estimate the PDTDFB coefficients of original image, finally reconstruct the denoised image.
②The size of module is not sensitive to small shift in image, which can lead to more accurate estimation in given location and scale, thus the PDTDFB-HMT model based on the model of coefficient is proposed. After performing PDTDFB transform according to the decompose level and direction in the noise image, and calculate module of PDTDFB coefficient and argument,then model the HMT to characterize the module of PDTDFB coefficients. The middle process is similar to dual-tree PDTDFB-HMT, at last recover PDTDFB coefficient according to the relationship between module and argument, reconstruct of signal with PDTDFB inverse transform to get final denoised image.
With regard to Gaussian white noise, a theoretical analysis and simulation results show, compared with several other typical denoising algorithms, whether the objective evaluation or subjective evaluation of image denoising, the two new methods raise PSNR at different degrees and effectively remove Gaussian white noise. They can remain edge information and details of original image, especially the method of PDTDFB-HMT based on module. However, the algorithms should be improved to reduce the time cost.
论文在研究金字塔对偶树方向滤波器组(PDTDFB)变换和隐马尔可夫模型的基础上,重点研究了图像经PDTDFB变换之后,其系数在尺度间和子带间表现出的相关性特性以及隐马尔可夫模型对这种相关性的统计特性,提出了两种新的基于PDTDFB变换的隐马尔可夫树(HMT)模型用于图像去噪。主要工作如下:
①根据PDTDFB变换系数的相关性特征和分布,提出了基于PDTDFB复系数的实部和虚部的双树PDTDFB-HMT去噪模型。利用两状态混合高斯模型描述PDTDFB系数的分布,对PDTDFB变换的高频子带系数的建立HMT模型,运用EM算法对HMT模型训练获得含噪声图像的模型参数。通过与噪声值的差进而获得不含噪声图像的模型参数。然后运用贝叶斯准则估计出不含噪声图像PDTDFB系数的后验均值估计值,最后用PDTDFB逆变换重建信号,得到最终的去噪图像。
②基于模的大小对于图像很小平移的不敏感性,在给定的位置和尺度上可对图像产生更为准确的估计,提出了基于PDTDFB复系数的模的PDTDFB-HMT去噪模型。首先对噪声图像进行PDTDFB变换,然后对高频子带系数进行取模运算并计算幅角关系,建立基于模的PDTDFB-HMT,去噪过程与双树PDTDFB-HMT类似,最后根据模和幅角的转换关系恢复系数,利用PDTDFB逆变换重建信号,得到最终的去噪图像。
在对高斯白噪声的仿真实验中,上述二种方法与其他几种典型的去噪算法相比,在客观评价标准上,峰值信噪比(PSNR)有不同程度的提高,较好地去除了高斯白噪声;在主观评价标准上,较好地保留了图像的边缘和细节信息。尤其是基于模的PDTDFB-HMT方法,对图像边缘的保持尤为显著,具有较好的去噪效果。但是由于HMM的统计运算量很大,导致算法执行效率偏低,有待进一步改进提高。
Image denoising is a classical subject in image processing. In recent years, image denoising algorithm based multi-scale geometric analysis becomes more and more popular in this field. Multi-scale geometric analysis combined with HMM is a research hot spot at present. Comparison with the traditional denoising method, the HMM based on multi-scale transform domain can fully capture and describe the statistical correlation of the multi-scale transform coefficients in inter-scale and inter-direction, and effectively solve the blur problem of image edge. Therefore it has promising applications to image de-noising.
In this paper, after studying the elementary concept of PDTDFB transform and HMM, the dependencies between PDTDFB coefficients and the statistical properties of HMM are paid more attention. Finally two new denoising algorithms based on PDTDFB and Hidden Markov Tree (HMT) are proposed. The main contributions of this paper are summarized as follows.
①Performing PDTDFB transform in noise image, then use HMT model to characterize the dependencies of PDTDFB coefficients, thus a new HMT based on the real part and imaginary part of PDTDFB coefficient is modeled. The 2-state Gaussian mixture model is used to approximate the image PDTDFB coefficients margin distribution and the matrix of state transform can capture the dependencies of the PDTDFB coefficients across the scales, EM method is used to train the HMT to get the parameters of HMT of the noise image, and then through subtracting the noise value to get the HMT parameters of original image. Bayesian is used to estimate the PDTDFB coefficients of original image, finally reconstruct the denoised image.
②The size of module is not sensitive to small shift in image, which can lead to more accurate estimation in given location and scale, thus the PDTDFB-HMT model based on the model of coefficient is proposed. After performing PDTDFB transform according to the decompose level and direction in the noise image, and calculate module of PDTDFB coefficient and argument,then model the HMT to characterize the module of PDTDFB coefficients. The middle process is similar to dual-tree PDTDFB-HMT, at last recover PDTDFB coefficient according to the relationship between module and argument, reconstruct of signal with PDTDFB inverse transform to get final denoised image.
With regard to Gaussian white noise, a theoretical analysis and simulation results show, compared with several other typical denoising algorithms, whether the objective evaluation or subjective evaluation of image denoising, the two new methods raise PSNR at different degrees and effectively remove Gaussian white noise. They can remain edge information and details of original image, especially the method of PDTDFB-HMT based on module. However, the algorithms should be improved to reduce the time cost.
引文
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[2]张兆礼,赵春晖,梅晓丹.现代图像处理技术及Matlab实现[M].北京:人民邮电出版社, 2001, 35-60.
[3]张宇,王希勤,彭应宁.自适应中心加权的改进均值滤波算法[J].清华大学学报, 1999, 39(9): 76-78.
[4]蔡靖,杨晋生,丁润涛.分类均值-加权中值混合滤波器[J].天津大学学报, 1999, 32(5): 551-554.
[5] Prasongsook Walailak, Rangsanseri Yuttapong, Thitiamajshima Wiyada. Image denoising by wavelet-domain Wiener filtering[A]. Proceeding of the International Conference on Telecommunications[C]. 2002.
[6] S. V. Vaseghi, Advanced Signal Processing and Noise Reduction 2nd Edition[M]. New York: John Wiley&Sons, 2001.
[7] Donoho D. L. , Johnstone I. M. . Ideal spatial adaptation via wavelet shrinkage[J]. Biometrica, 1994, 81(12): 425-455.
[8] Coifman R. R. , Donoho D. L. . Translation-invariant denoising[R]. In: Antoine A., Oppenheim G.eds. . Wavelets in Statistics. New York: Springer, 1995, 125-150
[9] Chang S. G. , Yu B. , Vetterli M. . Spatially adaptive wavelet thresholding with context modeling for image denoising[J]. IEEE Transactions on Image Processing, 2000, 9(9): 1522-1531.
[10] Donoho D. L. . De-noising by soft-thresholding[J]. IEEE Transactions on Information Theory, 1995, 41(3): 613-627.
[11] Crouse M. S. , Nowak R. D. , Baraniuk R. G. . Wavelet-based statistical signal processing using hidden Markov models[J]. IEEE Transactions on Signal Processing, 1998, 46(4): 886-902.
[12] Choi H. , Romberg J. , Baraniuk R. , et al. Hidden Markov tree modeling of complex wavelet transforms[A]. pro. ICASSP[C]. InstanbuI, 2000, 6-9.
[13] Po D. D. Y. , Do M. N. . Directional multiscale modeling of images using the contourlet transform [J]. IEEE Transactions on Image Processing, 2006, 15(6): 1610-1622.
[14] Fan G. L. , Xia X. G. . Improved hidden Markov models in the wavelet-domain[J]. IEEE Transaction on Signal Processing, 2001, 49 (1): 115-120.
[15] Romberg J. K. , Choi H. , Baraniuk R. G. . Bayesian treest ructured image modeling using wavelet-domain hidden Markov models[J]. IEEE Transactions on Image Processing, 2001, 10(7): 1056-1068.
[16]焦李成,侯彪等.图像多尺度几何分析理论与应用[M].西安:西安电子科技大学出版社. 2008, (7): 280-389.
[17] Truong T. Nguyen, Soontorn Oraintara. The Shiftable Complex Directional Pyramid, PartI: Theoretical Aspects[J]. IEEE Transactions on Signal Processing, 2008, 56(10): 4651-4660.
[18] Truong T. Nguyen, Soontorn Oraintara. The Shiftable Complex Directional Pyramid, PartII: Implementation and Applications[J]. IEEE Transactions on Signal Processing. 2008, 56(10): 4661-4672.
[19] Peyre G. , Mallat S. . Second Generation Bandelets and their Application to Image and 3D Meshes Compression[J]. Mathematics Analysis, MIA’04, Pairs, 2004.
[20] Pennec E. Le. , Mallat S. . Sparse Geometric Image Representations with Bandelets[J]. IEEE Transactions on Image Processing, 2005, 14(4): 423-438.
[21] Candes E. J. , Donoho D. L. . Curvelets a surprisingly effective nonadaptive representation for objects with edges[C]. In Cohen A, Rabut C, Schumaker L L(Eds), Saint-malo: Vanderbilt University Press, 2000, 105-200.
[22] M. N. Do, M. Vetterli. Contourlets: a directional multiresolution image representation [C]. Proceedings of International Conference on Image Processing, 2002, 357-360.
[23] M. N. Do, M. Vetterli. The contourlet transform: an efficient directional multiresolution agree Presentation[J]. IEEE Transactions on Image Processing, 2005, 14(12):2091-2106.
[24]李伟,杨晓蕙,石光明,焦李成.基于几何多尺度方向窗的小波图像去噪[J].西安电子科技大学学报(自然科学版), 2006, 33(5): 682-686.
[25] P. J. Burt, E. H. Adelson. The Laplacian Pyramid as a Compact Image Code[J]. IEEE Transactions on Communications. 1983, 31(4): 532-540.
[26] R. H. Bamberger, M. J. T. Smith. A Filter Bank for the Directional Decomposition of Images: Theory and Design[J]. IEEE Transactions on Signal Processing. 1992, 40(4): 882-893.
[27] N. G. Kingsbury. The dual-tree complex wavelet transform: A new efficient tool for image restoration and enhancement[A]. Proc of European Signal Processing[C], Rhodes, 1998, 319-322.
[28]周景超,戴汝为,肖柏华.图像质量评价研究综述[J].计算机科学, 2008, 35( 7): 41-46.
[29]余松煌,张文军,孙军.现代图像处理、压缩与识别技术[M].北京:科学出版社, 1998, 43-59.
[30] S. I. Park, M. J. Smith, R. M. Mersereau. Improved structure of maximally decimateddirectional filter banks for spatial image analysis[J]. IEEE Transactions on Image Processing, 2004, 13(11): 1424–1431.
[31] M. N. Do, M. Vetterli.“Contourlets”in Beyond Wavelets[J]. Studies in Computational Mathematics, 2003, (10) : 83–105.
[32] An P. N. Vo, Soontorn Oraintara, Truong T. Nguyen. Using Phase and Magnitude Information of the Complex Directional Filter Bank for Texture Image Retrieval[J]. ICIP(4) 2007, 61-64.
[33] Do M. N. . Directional multiresolution image representations[J]. Ph. D. thesis, Swiss Federal Institute of Technology, Lausanne, Switzerland, Dec. 2001.
[34] A. Karasaridis, E. Simoncelli. A filter design technique for steerable pyramid image transforms[C]. Proceedings of ICASSP, May, Atlanta, GA. 1996, 7–10.
[35] Kingsbury N. . Image processing with complex wavelets[J]. Philosophical Transactions Royal Society London A, 1999, 357(9): 2543-2560.
[36] Do M. N. , Vetterli M. .The contourlet transform: an efficient directional multiresolution image representation[J]. IEEE Transactions on Image Processing, 2005, 14(6): 760-769.
[37] Baum L. E. , Petrie T. . Statistical inference for Probabilistic functions of finite state Markov chains[J]. Annual Mathematics Statistics, 1966, (37): 1554-1563.
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