Research Article A Convolution-Based Shearlet Transform in Free Metaplectic Domains Tarun K. Garg , 1 Waseem Z. Lone , 2 Firdous A. Shah , 2 and Hatem Mejjaoli 3 1 Department of Mathematics, Satyawati College, University of Delhi, Delhi-110052, India 2 Department of Mathematics, University of Kashmir, South Campus, Anantnag-192101, Jammu and Kashmir, India 3 Department of Mathematics, College of Sciences, Taibah University, P.O. Box 30002, Al Madinah, Al Munawarah, Saudi Arabia Correspondence should be addressed to Firdous A. Shah; fashah@uok.edu.in Received 16 August 2021; Accepted 29 September 2021; Published 22 October 2021 Academic Editor: Ram Jiwari Copyright©2021TarunK.Gargetal.isisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e 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. e 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. 1. Introduction e free metaplectic transformation is a remarkable addition to the class of integral transforms which encompasses several transforms including the Fourier transform, fractional Fourier transform, Fresnel transform, and even basic quadratic phase factor multiplication [1]. In spirit, the FMTis similar to the well- known linear canonical transform (LCT), and for that reason, it is often referred to as the nonseparable LCT. However, unlike the conventional LCT, the FMTis generated via a general 2N × 2N free symplectic matrix M with N(2N + 1) degrees of freedom [2]. e in-built flexible nature of FMT offers a ca- nonical formalism for the representation of several physical systems in a lucid and insightful way. Mathematically, the free metaplectic transform of any f ∈ L 2 (R N ) with respect to the real symplectic matrix M �(A, B; C, D) is given by F M [f](w)� 1 |detB| (1/2)  R N f(x)e iπ w T DB − 1 w− 2w T B T − 1 x+x T B − 1 Ax (  dx, (1) where |detB| ≠ 0. e importance of the arbitrary real parameters involved in (1) lies in the fact that an ap- propriate 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 free- dom, the FMT has been successfully employed in prob- lems 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 [3–6]. e advent of continuous shearlet transforms revo- lutionized the field of multiscale analysis by offering a novel and elegant decomposition of a signal into com- ponents determined by the translations, dilations, and shearing of a single generating function known as the basic shearlet. Unlike the classical wavelets, shearlets are nonisotropic in nature, have compact support, and pro- vide ideally sparse representations with fast decomposi- tion and reconstruction algorithms. However, similar to the wavelets, they are an affine-like system of well-lo- calized waveforms with better directional selectivity. Over the years, the shearlet transforms have achieved mo- mentous success and have emerged as one of the most Hindawi Journal of Mathematics Volume 2021, Article ID 2140189, 23 pages https://doi.org/10.1155/2021/2140189