Construction of non-stationary biorthogonal Wavelets based on cardinal Chebyshev B-spline

Abstract

In this paper, we propose a novel method for designing non-stationary biorthogonal wavelets that leverages Bezout equations to compute wavelet filter coefficients. Our approach integrates cardinal Chebyshev B-splines of varying degrees (second, third, and fourth) as bases for constructing these wavelets, providing explicit formulations for both low-pass and high-pass filters at all levels. By employing this method, we streamline the design process, making it more efficient and accessible while accommodating diverse processing requirements. The utilization of Bezout equations offers a systematic framework for deriving wavelet filter coefficients, enhancing the reproducibility and reliability of the design process. This systematic approach ensures that the resulting wavelets possess desired properties such as vanishing moments, smoothness, and optimal localization. Moreover, by incorporating cardinal Chebyshev B-splines, we can effectively capture intricate signal and image features, addressing challenges encountered in traditional wavelet design methods. Our method not only simplifies the design process but also provides a comprehensive solution by explicitly formulating filters and dual filters across all levels. This comprehensive approach ensures that the resulting wavelets meet the specific demands of applications in signal and image processing, including compression, denoising, and pattern recognition. These advancements hold significant implications for various domains, including information technology, healthcare, and engineering, where efficient and reliable processing techniques are crucial. Moving forward, it is imperative to refine algorithms, assess parameter robustness, and conduct thorough empirical validations across diverse datasets and application scenarios. This will ensure the efficacy and scalability of our proposed method and pave the way for further advancements in non-stationary wavelet design and its applications. By continuously improving and validating our approach, we can foster innovation and enhance practical outcomes in a wide range of fields.