Non-decimated wavelet transform in statistical assessment of scaling: Theory and applications

Thumbnail Image
Kang, Min Kyoung
Vidakovic, Brani
Associated Organization(s)
Supplementary to
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.
Date Issued
Resource Type
Resource Subtype
Rights Statement
Rights URI