Research On Quaternion Wavelet Transformation Theory And Its Application In Image Processing

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.