四元数小波变换理论及其在图像处理中的应用研究
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摘要
小波分析是二十世纪八十年代后期迅速发展起来的一门新兴数学分支，它是在傅里叶变换基础上发展起来的一种新的时频分析方法，在信号分析、图像处理与模式识别等领域中已广泛应用。图像去噪是图像处理中的经典问题之一，数字水印是信息隐藏技术领域的重要分支。目前常用的小波有实离散小波变换（DWT）及复小波变换（CWT）等。四元数小波变换(QWT)是图像处理的一种新的多尺度分析工具，具有良好平移不变性，可以提供不同尺度的一个幅值和三个相位信息。本文主要研究了四元数小波变换的有关理论及其在图像去噪与数字水印中的应用。主要工作总结为如下几个方面：
1.基于四元数代数，希尔伯特变换及传统小波的理论与方法，深入研究了四元数小波的有关概念与性质。首先给出并证明了希尔伯特变换中的有关标准正交基的性质，由此研究了四元数小波变换在空间L~2(R~2)中尺度空间和小波空间中的标准正交基，然后给出了空间L~2(R~2; H)中四元数小波基函数，四元数小波尺度函数的概念，进一步给出了离散四元数小波变换的概念，同时还研究了四元数小波变换的结构及滤波器构造等等。
2.在传统小波图像去噪模型基础上，研究了四元数小波变换在图像去噪中的应用，给出了四元数小波变换域上的三个图像去噪模型与算法：（1）基于四元数小波变换的隐马尔科夫树模型的图像去噪；（2）基于四元数小波变换的非高斯二元分布的贝叶斯统计模型的图像去噪算法；（3）基于四元数小波变换的混合统计模型的图像去噪。实验表明：本文这些方法的去噪效果，无论在峰值信噪比还是在视觉效果上均优于许多经典的去噪算法。
3.给出了基于四元数小波变换域的SAR图像相干斑抑制模型。对于SAR图像，在引进加性模型的基础上，通过四元数小波变换，利用改进的系数分类准则，把系数分为两类：重要系数和非重要系数，提出了改进的Donoho阈值和新的阈值函数，并用它处理重要系数，估计出不含斑的四元数小波变换系数，从而得到抑制了相干斑的SAR图像。对真实SAR图像进行相干斑噪声抑制实验，结果显示：本文的方法在抑斑效果和图像的细节保留上均优于目前的许多方法。
4.给出了一种基于四元数小波变换和奇异值分解相结合的数字图像水印算法。该算法对原始载体图像进行四元数小波变换和奇异值分解，对水印图像进行Arnold变换和奇异值分解，然后把分解的水印嵌入到分解后的原始载体图像中。实验结果表明：该算法对高斯噪声、剪切、JPEG压缩及滤波具有较强的鲁棒性。
Wavelet analysis is a new rapidly developing branch of mathematicsin the1980s, which is a new kind of time-frequency analysis method basedon the Fourier transform. It has been used in signal analysis, imageprocessing, and pattern recognition and so on. Image denoising is one ofthe classic image processing problems and digital watermarking technologyis an important branch of the research field of information hiding. The realdiscrete wavelet transform (DWT) and complex wavelet transform (CWT)are commonly used. Quaternion wavelet transform (QWT) is a new kind ofmultiresolution analysis tools of image processing, and it is nearshift-invariant and can provide amplitude and three phase information indifferent scales. This dissertation mainly study on the theory of quaternionwavelet transform and its application in image de-noising and the digitalwatermarking, the main work summed up in the following aspects:
1. We deeply study the related concepts and properties of quaternionwavelet transform based on the quaternionic algebra, Hilberttransformation and traditional wavelet theory and method. We firstlypresent and prove the properties of standard orthogonal basis about Hilberttransformation, and study the standard orthogonal basis of quaternionwavelet transform in the scale space and wavelet space of L~2(R~2)space. We then propose the concepts of quaternion wavelet base function andscale function in the space L~2(R~2; H). Finally, we put forward the conceptof discrete quaternion wavelet transformation, and study the structure andfilters construction of quaternion wavelet transform and so on.
2. Based on the traditional image de-noising model in wavelet domain,we study the application in image de-noising of quaternion wavelettransform, and gave three de-noising models and algorithms in thequaternion wavelet transform domain:(1) a hidden Markov tree imagede-noising model based on quaternion wavelet transformation (Q-HMT);(2)a image de-noising algorithm based on non-Gaussian bivariate distributionof Bayesian statistical models in quaternion wavelet transformation domain;(3) a mixed statistical image de-noising model in quaternion wavelettransform domain. The experimental results show that our proposedmethods both in peak value signal-to-noise ratio（PSNR）and visual effectare better than many classic de-noising algorithm.
3. SAR image despeckling model based on quaternion wavelettransform is proposed. Additive model is introduced to SAR image, andquaternion wavelet transform is used. We propose an improvedclassification standard where the coefficients are divided into the importantand unimportance coefficients. Moreover, the improved Donoho’sthreshold and new threshold function is used to process the important coefficients, and estimate the original QWT coefficients from noisycoefficients. Thus we get speckle removed of SAR image. Experimentalresults for speckle reduction of real SAR images show that our algorithmoutperforms the other current despeckling algorithms both in the despeckleeffect and reserve of image’s detail.
4. Based on quaternion wavelet transform (QWT) and Singular ValueDecomposition (SVD), a watermarking algorithm is proposed.The originalimage is transformed by QWT and SVD and the watermarking image isprocessed by Arnold transform and SVD. The watermarking imageprocessed is embedded the original image decomposed.Experiment resultsshow that the proposed watermarking algorithm has good robustness forthe attacks of adding Gaussian noises,geometric clipping, JPEGcompression and filtering.
1.基于四元数代数，希尔伯特变换及传统小波的理论与方法，深入研究了四元数小波的有关概念与性质。首先给出并证明了希尔伯特变换中的有关标准正交基的性质，由此研究了四元数小波变换在空间L~2(R~2)中尺度空间和小波空间中的标准正交基，然后给出了空间L~2(R~2; H)中四元数小波基函数，四元数小波尺度函数的概念，进一步给出了离散四元数小波变换的概念，同时还研究了四元数小波变换的结构及滤波器构造等等。
2.在传统小波图像去噪模型基础上，研究了四元数小波变换在图像去噪中的应用，给出了四元数小波变换域上的三个图像去噪模型与算法：（1）基于四元数小波变换的隐马尔科夫树模型的图像去噪；（2）基于四元数小波变换的非高斯二元分布的贝叶斯统计模型的图像去噪算法；（3）基于四元数小波变换的混合统计模型的图像去噪。实验表明：本文这些方法的去噪效果，无论在峰值信噪比还是在视觉效果上均优于许多经典的去噪算法。
3.给出了基于四元数小波变换域的SAR图像相干斑抑制模型。对于SAR图像，在引进加性模型的基础上，通过四元数小波变换，利用改进的系数分类准则，把系数分为两类：重要系数和非重要系数，提出了改进的Donoho阈值和新的阈值函数，并用它处理重要系数，估计出不含斑的四元数小波变换系数，从而得到抑制了相干斑的SAR图像。对真实SAR图像进行相干斑噪声抑制实验，结果显示：本文的方法在抑斑效果和图像的细节保留上均优于目前的许多方法。
4.给出了一种基于四元数小波变换和奇异值分解相结合的数字图像水印算法。该算法对原始载体图像进行四元数小波变换和奇异值分解，对水印图像进行Arnold变换和奇异值分解，然后把分解的水印嵌入到分解后的原始载体图像中。实验结果表明：该算法对高斯噪声、剪切、JPEG压缩及滤波具有较强的鲁棒性。
Wavelet analysis is a new rapidly developing branch of mathematicsin the1980s, which is a new kind of time-frequency analysis method basedon the Fourier transform. It has been used in signal analysis, imageprocessing, and pattern recognition and so on. Image denoising is one ofthe classic image processing problems and digital watermarking technologyis an important branch of the research field of information hiding. The realdiscrete wavelet transform (DWT) and complex wavelet transform (CWT)are commonly used. Quaternion wavelet transform (QWT) is a new kind ofmultiresolution analysis tools of image processing, and it is nearshift-invariant and can provide amplitude and three phase information indifferent scales. This dissertation mainly study on the theory of quaternionwavelet transform and its application in image de-noising and the digitalwatermarking, the main work summed up in the following aspects:
1. We deeply study the related concepts and properties of quaternionwavelet transform based on the quaternionic algebra, Hilberttransformation and traditional wavelet theory and method. We firstlypresent and prove the properties of standard orthogonal basis about Hilberttransformation, and study the standard orthogonal basis of quaternionwavelet transform in the scale space and wavelet space of L~2(R~2)space. We then propose the concepts of quaternion wavelet base function andscale function in the space L~2(R~2; H). Finally, we put forward the conceptof discrete quaternion wavelet transformation, and study the structure andfilters construction of quaternion wavelet transform and so on.
2. Based on the traditional image de-noising model in wavelet domain,we study the application in image de-noising of quaternion wavelettransform, and gave three de-noising models and algorithms in thequaternion wavelet transform domain:(1) a hidden Markov tree imagede-noising model based on quaternion wavelet transformation (Q-HMT);(2)a image de-noising algorithm based on non-Gaussian bivariate distributionof Bayesian statistical models in quaternion wavelet transformation domain;(3) a mixed statistical image de-noising model in quaternion wavelettransform domain. The experimental results show that our proposedmethods both in peak value signal-to-noise ratio（PSNR）and visual effectare better than many classic de-noising algorithm.
3. SAR image despeckling model based on quaternion wavelettransform is proposed. Additive model is introduced to SAR image, andquaternion wavelet transform is used. We propose an improvedclassification standard where the coefficients are divided into the importantand unimportance coefficients. Moreover, the improved Donoho’sthreshold and new threshold function is used to process the important coefficients, and estimate the original QWT coefficients from noisycoefficients. Thus we get speckle removed of SAR image. Experimentalresults for speckle reduction of real SAR images show that our algorithmoutperforms the other current despeckling algorithms both in the despeckleeffect and reserve of image’s detail.
4. Based on quaternion wavelet transform (QWT) and Singular ValueDecomposition (SVD), a watermarking algorithm is proposed.The originalimage is transformed by QWT and SVD and the watermarking image isprocessed by Arnold transform and SVD. The watermarking imageprocessed is embedded the original image decomposed.Experiment resultsshow that the proposed watermarking algorithm has good robustness forthe attacks of adding Gaussian noises,geometric clipping, JPEGcompression and filtering.
引文
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