INTEGRO-DIFFERENTIAL EQUATIONS-BASED MULTISCALE WAVELET SHRINKAGE AND MULTI-DIMENSIONAL CONTINUOUS SHEARLET TRANSFORM
Authors: Fatima Khan
Published: June 2024
Abstract
<p>In recent years, the study of wavelets has gained substantial attention, revealing their profound relevance in various branches of pure and applied mathematics. Wavelet theory has proven to be a valuable tool across multiple scientific disciplines, engineering, and other fields. Notably, wavelets have played a pivotal role in developing essential technologies such as the FBI's digital fingerprint image compression standard in 1993 and JPEG2000, the current industry standard for image compression. They have facilitated the analysis of fundamental linear operators within harmonic analysis and partial differential equations, as demonstrated in the Calderon-Zygmund theory. In the realm of image processing, wavelet shrinkage emerges as a crucial technique to address core issues, including image compression, noise reduction, feature extraction, and object recognition. One fascinating aspect of wavelet shrinkage is its diverse mathematical foundations. It finds inspiration in various mathematical domains, encompassing partial differential equations, the calculus of variations, harmonic analysis, and statistics. This interdisciplinary nature has invigorated the application of wavelets in the realm of engineering and technology. Moreover, pioneering work by Donoho and others has delved into wavelet shrinkage methods within the context of minimax estimation. Their investigations have revealed that wavelet shrinkage provides asymptotically optimal estimates for noisy data, surpassing the performance of linear estimators. This opens new horizons for utilizing wavelets in diverse engineering applications.</p>
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