Filters and Matrix Factorization

We give a number of explicit matrix-algorithms for analysis/synthesis in multi-phase filtering; i.e., the operation on discrete-time signals which allow a separation into frequency-band components, one for each of the ranges of bands, say N , starting with low-pass, and then corresponding filtering in the other band-ranges. If there are N bands, the individual filters will be combined into a single matrix action; so a representation of the combined operation on all N bands by an N x N matrix, where the corresponding matrix-entries are periodic functions; or their extensions to functions of a complex variable. Hence our setting entails a fixed N x N matrix over a prescribed algebra of functions of a complex variable. In the case of polynomial filters, the factorizations will always be finite. A novelty here is that we allow for a wide family of non-polynomial filter-banks. Working modulo N in the time domain, our approach also allows for a natural matrix-representation of both down-sampling and up-sampling. The implementation encompasses the combined operation on input, filtering, down-sampling, transmission, up-sampling, an action by dual filters, and synthesis, merges into a single matrix operation. Hence our matrixfactorizations break down the global filtering-process into elementary steps. To accomplish this, we offer a number of adapted matrix factorizationalgorithms, such that each factor in our product representation implements in a succession of steps the filtering across pairs of frequency-bands; and so it is of practical significance in implementing signal processing, including filtering of digitized images. Our matrix-factorizations are especially useful in the case of the processing a fixed, but large, number of bands

Complex-valued wavelet lifting and applications

Signals with irregular sampling structures arise naturally in many fields. In applications such as spectral decomposition and nonparametric regression, classical methods often assume a regular sampling pattern, thus cannot be applied without prior data processing. This work proposes new complex-valued analysis techniques based on the wavelet lifting scheme that removes ‘one coefficient at a time’. Our proposed lifting transform can be applied directly to irregularly sampled data and is able to adapt to the signal(s)’ characteristics. As our new lifting scheme produces complex-valued wavelet coefficients, it provides an alternative to the Fourier transform for irregular designs, allowing phase or directional information to be represented. We discuss applications in bivariate time series analysis, where the complex-valued lifting construction allows for coherence and phase quantification. We also demonstrate the potential of this flexible methodology over real-valued analysis in the nonparametric regression context

Wavelet Based Feature Extraction and Dimension Reduction for the Classification of Human Cardiac Electrogram Depolarization Waveforms

An essential task for a pacemaker or implantable defibrillator is the accurate identification of rhythm categories so that the correct electrotherapy can be administered. Because some rhythms cause a rapid dangerous drop in cardiac output, it is necessary to categorize depolarization waveforms on a beat-to-beat basis to accomplish rhythm classification as rapidly as possible. In this thesis, a depolarization waveform classifier based on the Lifting Line Wavelet Transform is described. It overcomes problems in existing rate-based event classifiers; namely, (1) they are insensitive to the conduction path of the heart rhythm and (2) they are not robust to pseudo-events. The performance of the Lifting Line Wavelet Transform based classifier is illustrated with representative examples. Although rate based methods of event categorization have served well in implanted devices, these methods suffer in sensitivity and specificity when atrial, and ventricular rates are similar. Human experts differentiate rhythms by morphological features of strip chart electrocardiograms. The wavelet transform is a simple approximation of this human expert analysis function because it correlates distinct morphological features at multiple scales. The accuracy of implanted rhythm determination can then be improved by using human-appreciable time domain features enhanced by time scale decomposition of depolarization waveforms. The purpose of the present work was to determine the feasibility of implementing such a system on a limited-resolution platform. 78 patient recordings were split into equal segments of reference, confirmation, and evaluation sets. Each recording had a sampling rate of 512Hz, and a significant change in rhythm in the recording. The wavelet feature generator implemented in Matlab performs anti-alias pre-filtering, quantization, and threshold-based event detection, to produce indications of events to submit to wavelet transformation. The receiver operating characteristic curve was used to rank the discriminating power of the feature accomplishing dimension reduction. Accuracy was used to confirm the feature choice. Evaluation accuracy was greater than or equal to 95% over the IEGM recordings