# The discrete pulse transform and applications

Thesis (MSc (Mathematical Sciences))--University of Stellenbosch, 2007.

Thesis

Data analysis frequently involves the extraction (i.e. recognition) of parts that are important at the expense of parts that are deemed unimportant. Many mathematical perspectives exist for performing these separations, however no single technique is a panacea as the de nition of signal and noise depends on the purpose of the analysis. For data that can be considered a sampling of a smooth function with added 'well-behaved' noise, linear techniques tend to work well. When large impulses or discontinuities are present, a non-linear approach becomes necessary. The LULU operators, composed using the simplest rank selectors, are non-linear operators that are comparable to the well-known median smoothers, but are computationally e cient and allow a conceptually simple description of behaviour. De ned using compositions of di erent order LULU operators, the discrete pulse transform (dpt) allows the interpretation of sequences in terms of pulses of di erent scales: thereby creating a multi-resolution analysis. These techniques are very di erent from those of standard linear analysis, which renders intuitions regarding their behaviour somewhat undependable. The LULU perspective and analysis tools are investigated with a strong emphasis on practical applications. The LULU smoothers are known to separate signal and noise ef- ciently: they are idempotent and co-idempotent. Sequences are smoothed by mapping them into smoothness classes; which is achieved by the removal, in a consistent manner, of block-pulses. Furthermore, these operators preserve local trend (i.e. they are fully trend preserving). Di erences in interpretation with respect to Fourier and Wavelet decompositions are also discussed. The dpt is de ned, its implications are investigated, and a linear time algorithm is discussed. The dpt is found to allow a multi-resolution measure of roughness. Practical sequence processing through the reconstruction of modi ed pulses is possible; in some cases still maintaining a consistent multi-resolution interpretation. Extensions to two-dimensions is discussed, and a technique for the estimation of standard deviation of a random distribution is presented. These tools have been found to be e ective in the analysis and processing of sequences and images. The LULU tools are an useful alternative to standard analysis methods. The operators are found to be robust in the presence of impulsive and more 'well-behaved' noise. They allow the fast design and deployment of specialized detection and processing algorithms, and are possibly very useful in creating automated data analysis solutions.