Abstract
The free metaplectic transformation (FMT) is a multidimensional integral transform that encompasses a broader range of integral transforms, from the classical Fourier to the more recent linear canonical transforms. The aim of this study is to introduce a novel shearlet transform by employing the free metaplectic convolution structures. Besides obtaining the orthogonality relation, inversion formula, and range theorem, we also study the homogeneous approximation property for the proposed transform. Towards the culmination, we formulate the Heisenberg and logarithmic-type uncertainty principles associated with the free metaplectic shearlet transform.
Highlights
Where |detB| ≠ 0. e importance of the arbitrary real parameters involved in (1) lies in the fact that an appropriate choice of the parameters can be used to inculcate a sense of rotation and shift into both the time and frequency axes, resulting in an efficient analysis of the chirp-like signals which are ubiquitous in nature and in man-made systems
Due to these extra degrees of freedom, the free metaplectic transformation (FMT) has been successfully employed in problems demanding several controllable parameters arising in various branches of science and engineering, such as harmonic analysis, reproducing kernel Hilbert spaces, optical systems, quantum mechanics, sampling, image processing, and so on [3,4,5,6]
E advent of continuous shearlet transforms revolutionized the field of multiscale analysis by offering a novel and elegant decomposition of a signal into components determined by the translations, dilations, and shearing of a single generating function known as the basic shearlet
Summary
We provide a brief overview of the free metaplectic and shearlet transforms, which facilitates the formulation of the proposed transform. A typical example of a 4 × 4 free symplectic matrix is given below:. E Free Metaplectic Transform and Convolution Structure. Given a free symplectic matrix M (A, B; C, D), the free metaplectic transformation of any f ∈ L2(RN) is denoted by LM[f] and is defined as LM[f](w). E additive property of the free metaplectic transformation (7) is very crucial for its understanding and application and is given by. We recall the definition of free metaplectic convolution ⊛ M and some of its basic properties. Given a pair of functions f, g ∈ L2(RN), the free metaplectic convolution with respect to the symplectic matrix M (A, B; C, D) is defined by. E free metaplectic convolution (12) satisfies the following properties. E Shearlet Transform in Arbitrary Space Dimensions. For more details about shearlets and their application, the readers are referred to [10,11,12,13] and the references therein
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