Interpretation and Future
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Hearing and Seeing are two physiologically different means of receiving information. Hearing and the Frequency Domain belong together, but in Vision the natural framework may be Mathematical Morphology. Analyzing sequences for information with linear operators can be naturally associated with the Frequency Domain, since the eigensequences of such operators are periodic. Analyzing them for features like pulses and trends is different. Wavelet Theory has held promises but the constraints have become apparent. Mathematical Morphology, as a framework for analysis and interpretation, has been developed over several years, but has only recently been linked with other attempts at establishing mathematical structure. LU LU -theory developed from the culture of intuitive association with the linear framework (of Electronic Engineers and Statisticians), but is a special part of Mathematical Morphology. Its perspectives yield a rich harvest of strong and intuitively simple results which the biased author risks calling beautiful. This effort is intended to seduce mathematicians, scientists and engineers to explore this beautiful structure for interpretation and usefulness. It is deliberately written with passion rather than complete detail, an impressionistic painting rather than a photograph. The ideas have however grown out of practical problems tackled over more than two decades in several different environments. The simplicity, power and regularity of the practical successes has fed the passion that drove the analysis, for which the author never seemed or felt adequately prepared. In attempting to interpret what LU LU -theory has yielded, the following can be claimed. 1. A framework for analysis and comparison of several useful nonlinear smoothers has been established that contributes to the selection, design and understanding of smoothers for a specific purpose.
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10. Interpretation and Future
2. Conceptual links have been established with the existing (linear) framework for analysis, comparison and design of smoothers, as well as Mathematical Morphology. 3. A concept of “Smoothness” and “Trend” (Local Monotonicity) has been established and linked to an established natural measure of smoothness (Total Variation). 4. A Concept of “Pulse” has been established that is complementary to that of “Trend”. 5. Fundamental operators (Cn and Fn ) have been defined and investigated for establishing a class of (nonlinear-) Multiresolution Analysis that decompose a sequence for such “Trend” and “Pulse” content. 6. The attributes of Neighbor Trend Preservation (ntp) and Full Trend Preservation (ftp) and that of Co-idempotence have been defined and demonstrated to be as useful and important as the well-known concepts of Idempotence and Syntoneness. 7. Heuristically derived procedures (Median Transforms) in Image Processing, which have been known to work well (but lacking theory), have had their success supported by theory and the perceived superiority over Morphological Filters has been shown to be due to lack of fair selection. 8. Heuristi
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