Non-decimated Wavelet Transform in Statistical Assessment of Scaling: Theory and Applications
Kang, Min Kyoung
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In this thesis, we introduced four novel methods that facilitate the scaling estimation based on NDWT. Chapter 2 introduced an NDWT matrix which is used to perform an NDWT in one or two dimensions. The use of matrix significantly decreased the computation time when 2-D inputs of moderate size are transformed under MATLAB environment, and such reduction of computation time was augmented when the same type of NDWT is performed repeatedly. With 2-D inputs, an NDWT matrix yielded a scale-mixing NDWT, which is more compressive than the standard 2-D NDWT. The retrieval of an original signal after the transform was possible with a weight matrix. An NDWT matrix can handle signals of non-dyadic sizes in one or two dimensions. The proposed NDWT matrix was used for the transforms in Chapters 3-5. Chapter 3 introduced a method for scaling estimation based on a non-decimated wavelet spectrum. A distinctive feature of NDWT, redundancy, enables us to obtain local spectra and improves the accuracy of scaling estimation. For simulated signals with known $H$ values, the method yields estimators of $H$ with lower mean squared errors. We characterized mammographic images with the proposed scaling estimator and anisotropy measures from non-decimated wavelet spectra for breast cancer detection, and obtained the best diagnostic accuracy in excess of 80\%. Some real-life signals are known to possess a theoretical value of the Hurst exponent. Chapter 4 described a Bayesian scaling estimation method that utilizes the value of a theoretical scaling index as a mean of prior distribution and estimates $H$ with MAP estimation. The accuracy of estimators from the proposed method is robust to small misspecification of the prior mean. We applied the method to a turbulence velocity signal and yielded an estimator of $H$ close to the theoretical value. Chapter 5 proposed two methods based on NDWT for robust estimation of Hurst exponent $H$ of 1-D self-similar signals. The redundancy of NDWT, which improved the accuracy of estimation, introduced autocorrelations within the wavelet coefficients. With the two proposed methods, we alleviated the autocorrelation in three ways: taking the logarithm prior to taking the median, relating Hurst exponent to the median instead of mean of the model distribution, and resampling the coefficients.