# A Convolution-Based Shearlet Transform in Free Metaplectic Domains

Authors: Tarun K. Garg; Waseem Z. Lone; Firdous A. Shah; Hatem Mejjaoli
Journal: Journal of Mathematics
 (2021)

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## 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.


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## Body

## 1. Introduction

The 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 FMT is 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 FMT is generated via a general 2N×2N free symplectic matrix M with N2N+1 degrees of freedom [2]. The in-built flexible nature of FMT offers a canonical formalism for the representation of several physical systems in a lucid and insightful way. Mathematically, the free metaplectic transform of any f∈L2ℝN with respect to the real symplectic matrix M=A,B;C,D is given by(1)ℱMfw=1detB1/2∫ℝNfxeiπwTDB−1w−2wTBT−1x+xTB−1Axdx,where detB≠0. The 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 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 [3–6].The 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. Unlike the classical wavelets, shearlets are nonisotropic in nature, have compact support, and provide ideally sparse representations with fast decomposition and reconstruction algorithms. However, similar to the wavelets, they are an affine-like system of well-localized waveforms with better directional selectivity. Over the years, the shearlet transforms have achieved momentous success and have emerged as one of the most effective frameworks for representing multidimensional data ranging over diverse areas of signal and image processing where the detection of directional instincts plays a role [7, 8]. However, shearlet systems are not completely free from cons, and one of the severe limitations is the invariance of the degrees of freedom with regard to the spatial dimension. Therefore, it is quite interesting to revamp the classical shearlet family by inculcating the adaptability to the spatial dimension N. Motivated by this fact, our goal is to combine the merits of the free metaplectic and shearlet transforms into a unique integral transform known as the free metaplectic shearlet transform in arbitrary space dimensions. To achieve our goal, we employ the convolution structure in the free metaplectic domain to construct a new family of shearlets endowed with several extra degrees of freedom determined by the spatial dimension N.The highlights of the article are as follows:To introduce the notion of free metaplectic shearlet transform by invoking the convolution operation in the free metaplectic domain.To study the fundamental properties of the proposed transform including Moyal’s principle, reconstruction formula, and a characterization of range.To examine the pointwise convergence of the reconstruction formula abreast the homogeneous approximation property of the free metaplectic shearlet transform.To formulate the Heisenberg, logarithmic, and Nazarov-type uncertainty inequalities for the free metaplectic shearlet transform.The remainder of the article is structured as follows. Section2 gives an overview of the preliminary aspects of the study encompassing the free metaplectic and shearlet transforms in arbitrary space dimensions. Section 3 is exclusively dedicated to introduce and study the free metaplectic shearlet transform. Section 4 deals with the study of homogeneous approximation property, whereas Section 5 is intended to formulate the Heisenberg and logarithmic-type uncertainty principles for the free metaplectic shearlet transform. The investigation culminates with the discussion of some potential applications of the proposed transform.


## 2. The Free Metaplectic and Shearlet Transforms

In this section, we provide a brief overview of the free metaplectic and shearlet transforms, which facilitates the formulation of the proposed transform.


### 2.1. The Free Metaplectic Transform and Convolution Structure

For typographical convenience, we shall denote a real2N×2N matrix(2)M=ABCD=a11a12…a1Nb11b12…b1Na21a22…a2Nb21b22…b2N⋮⋮⋱⋮⋮⋮⋱⋮aN1aN2…aNNbN1bN2…bNNc11c12…c1Nd11d12…d1Nc21c22…c2Nd21d22…d2N⋮⋮⋱⋮⋮⋮⋱⋮cN1cN2…cNNdN1dN2…dNN,as M=A,B;C,D, where A,B,C, and D are N×N submatrices. Moreover, we recall that the matrix M is said to be a free symplectic matrix if MTJM=J and detB≠0, where J=0,IN;−IN,0 and IN denotes the N-dimensional identity matrix. Furthermore, the submatrices corresponding to free symplectic matrix M=A,B;C,D satisfy(3)ABT=BAT,CDT=DCT,ADT−BCT=IN,or equivalently(4)ATC=CTA,BTD=DTB,ATD−CTB=IN.The transpose and inverse matrices corresponding to the free symplectic matrixM=A,B;C,D are given by MT=AT,CT;BT,DT and M−1=DT,−BT;−CT,AT, respectively. Moreover, we have(5)MM−1=ABCDDT−BT−CTAT=ADT−BCT−ABT+BATCDT−DCT−CBT+DAT=IN00IN.A typical example of a4×4 free symplectic matrix is given below:(6)M=1−1/211/81/841/2−1/43/2−621/2−3/4181−1.Definition 1.
Given a free symplectic matrixM=A,B;C,D, the free metaplectic transformation of any f∈L2ℝN is denoted by ℒMf and is defined as(7)ℒMfw=∫ℝNfxKMx,wdx,where the kernel KMx,w is given by(8)KMx,w=12πN/2detB1/2expiwTDB−1w−2wTBT−1x+xTB−1Ax2,with detB≠0.
The additive property of the free metaplectic transformation (7) is very crucial for its understanding and application and is given by(9)ℒM1ℒM2fxw=ℒM1M2fxw.
Moreover, the Plancherel and inversion formulae corresponding to (7) are given by(10)f,g2=ℒMf,ℒMg2,∀f,g∈L2ℝN,and(11)fx=ℒM−1ℒMfwx=∫ℝNℒMfwKM−1w,xdw,respectively, where M−1=DT,−BT;−CT,AT. Furthermore, the kernel (8) satisfies the following properties:(i)
KM−1w,x=KMx,w¯.(ii)
∫ℝNKMx,wKM−1x,zdx=δz−w.(iii)
∫ℝNKMx,wKM−1z,wdw=δz−x.(iv)
∫ℝNKMx,wKNx,zdx=KMNw,z.
The free metaplectic transformation (7) encompasses several well-known integral transforms such as the Fourier transform (FT), fractional Fourier transform (FrFT), linear canonical transform (LCT), Fresnel transform (FrT) [1] and is depicted in Table 1.
The convolution theory plays a vital role in diverse fields of signal and image processing. We recall the definition of free metaplectic convolution⊛M and some of its basic properties.Table 1
Some special cases of the free metaplectic transformation.
Free symplectic matrixM=A,B;C,DFree metaplectic transformation(i)A=D=0, B=−C=INN-dimensional FT(ii)A=diaga11,…,aNN, B=diagb11,…,bNN, C=diagc11,…,cNN, D=diagd11,…,dNNN-dimensional separable LCT(iii)A=D=diagcosθ1,…,cosθN, B=−C=diagsinθ1,…,sinθNN-dimensional separable FrFT(iv)A=D=INcosθ, B=−C=INsinθN-dimensional nonseparable FrFT(v)A=D=IN, B=diagb11,…,bNN, C=0N-dimensional separable FrT(vi)A=D=IN, C=0N-dimensional nonseparable FrTDefinition 2.
Given a pair of functionsf,g∈L2ℝN, the free metaplectic convolution with respect to the symplectic matrix M=A,B;C,D is defined by(12)f⊛Mgt=∫ℝNfxgt−xexpixTB−1Ax−tTB−1At2dx.
The free metaplectic convolution (12) satisfies the following properties.(i)
Commutative:f⊛gt=g⊛ft.(ii)
Associative:f⊛g⊛ht=f⊛g⊛ht.(iii)
Translation:f⊛gt−k=fx⊛expkTB−1Ax+xTB−1Akgxt.(iv)
Reflection:f⊛g−t=f−x⊛g−xt.(v)
Convolution theorem: for anyf,g∈L2ℝN, equation (12) satisfies(13)ℒMf⊛Mgtw=2πN/2ℒMfwℱgB−1w,where ℱf denotes the N-dimensional Fourier transform of ft given by(14)ℱfw=f^w=12πN/2∫ℝNfxe−ixTwdx.
It is worth noticing that the free metaplectic convolution (12) enjoys both commutative as well as associative properties. Moreover, the elegance of the convolution ⊛M lies in the fact that the convolution theorem is free from any chirp multiplier, which is convenient to implement and can also reduce the effect of the spectral leakage.
We now present an example for the lucid illustration of free metaplectic convolution (12).Example 1.
Consider the following two-dimensional unilateral exponential functions:(15)fx1,x2=exp6x1+5ix22121,x1,x2≥0,0,else.
Also, consider the two-dimensional Gaussian function(16)gx1,x2=1σ2πexp−x12+x222σ2,σ>0.
Denotehz=f⊛Mgz. Then, we have(17)hz=1σ2π2/3∫ℝ+∫ℝ+exp6x1+5ix22121exp−x1−z12+x2−z222σ2×expixTB−1Ax−zTB−1Az2dx1dx2.
For a symplectic matrix(18)M=ABCD=1/61−21/6−5/6−1/61/65/31012/29−31/29−6/29−36/2936/290,we have(19)xTB−1Ax=−36121x1x25/3−1/6−1/6−21/61−5/6−1/6x1x2=−6121x1x25/261/659/61x1x2=−61215x122+20x1x2+x22.
Similarly, one can obtain(20)zTB−1Az=−61215z122+20z1z2+z22.
Substituting the estimates (19) and (20) in (17), we obtain(21)hz=1σ2π2/3∫ℝ+∫ℝ+exp6x1+5ix22121exp−x1−z12+x2−z222σ2×exp−3i1215x122+20x1x2+x22exp3i1215z122+20z1z2+z22dx1dx2=1σ2π2/3exp3i1215z122+20z1z2+z22∫ℝ+∫ℝ+exp6x1+5ix22121×exp−x1−z12+x2−z222σ2exp−3i1215x122+20x1x2+x22dx1dx2=1σ2π2/3exp3i1215z122+20z1z2+z22exp−z12+z222σ2×∫ℝ+exp−36121+15i242+12σ2x12dx1×∫ℝ+exp−25121+3i121+12σ2x22dx2=1σ2π2/3exp3i1215z122+20z1z2+z22exp−z12+z222σ2×π436121+15i242+12σ2−1/225121+3i121+12σ2−1/2.


### 2.2. The Shearlet Transform in Arbitrary Space Dimensions

In this section, we recall the mathematical framework of the shearlet transforms, which serve as a basis for the development of the novel integral transform in arbitrary space dimensions. Formally, we have the following definition of the multidimensional shearlet transform.Definition 3 (see [9]).
For anyf∈L2ℝN, the continuous shearlet transform in arbitrary space dimension is defined by(22)Sℋψfa,s,t=f,ψa,s,t2=∫ℝNfxψa,s,tx¯dx,where(23)ψa,s,tx=detΛa−1/2ψΛa−1Ss−1x−t,⋅4a∈ℝ0,s∈ℝN−1,t∈ℝN,is the shearlet family constituted by the combined action of the scaling DΛa, sharing DSs, and translation Tt operators on the analyzing function ψ∈L2ℝN given by(24)DΛaψx=detΛa−1/2ψΛa−1x,DSsψx=ψSs−1x,andTtψx=ψx−t,respectively, and the matrices involved in (24) are given by(25)Λa=a0N−1T0N−1sgnaa1/NIN−1andSs=1sT0N−1IN−1,where sT=s1,s2,…,sN−1 and sgn⋅ and 0 denote the well-known signum function and the null vector, respectively. The set S=ℝ\0×ℝN−1×ℝN endowed with the operation(26)a,s,t⊙a′,s′,t′=aa′,s+a1−1/Ns′,t+SsΛat′,forms a locally compact group, often called the shearlet group. The left Haar measures on S are given by dv=dadsdt/aN+1. For more details about shearlets and their application, the readers are referred to [10–13] and the references therein.


## 2.1. The Free Metaplectic Transform and Convolution Structure

For typographical convenience, we shall denote a real2N×2N matrix(2)M=ABCD=a11a12…a1Nb11b12…b1Na21a22…a2Nb21b22…b2N⋮⋮⋱⋮⋮⋮⋱⋮aN1aN2…aNNbN1bN2…bNNc11c12…c1Nd11d12…d1Nc21c22…c2Nd21d22…d2N⋮⋮⋱⋮⋮⋮⋱⋮cN1cN2…cNNdN1dN2…dNN,as M=A,B;C,D, where A,B,C, and D are N×N submatrices. Moreover, we recall that the matrix M is said to be a free symplectic matrix if MTJM=J and detB≠0, where J=0,IN;−IN,0 and IN denotes the N-dimensional identity matrix. Furthermore, the submatrices corresponding to free symplectic matrix M=A,B;C,D satisfy(3)ABT=BAT,CDT=DCT,ADT−BCT=IN,or equivalently(4)ATC=CTA,BTD=DTB,ATD−CTB=IN.The transpose and inverse matrices corresponding to the free symplectic matrixM=A,B;C,D are given by MT=AT,CT;BT,DT and M−1=DT,−BT;−CT,AT, respectively. Moreover, we have(5)MM−1=ABCDDT−BT−CTAT=ADT−BCT−ABT+BATCDT−DCT−CBT+DAT=IN00IN.A typical example of a4×4 free symplectic matrix is given below:(6)M=1−1/211/81/841/2−1/43/2−621/2−3/4181−1.Definition 1.
Given a free symplectic matrixM=A,B;C,D, the free metaplectic transformation of any f∈L2ℝN is denoted by ℒMf and is defined as(7)ℒMfw=∫ℝNfxKMx,wdx,where the kernel KMx,w is given by(8)KMx,w=12πN/2detB1/2expiwTDB−1w−2wTBT−1x+xTB−1Ax2,with detB≠0.
The additive property of the free metaplectic transformation (7) is very crucial for its understanding and application and is given by(9)ℒM1ℒM2fxw=ℒM1M2fxw.
Moreover, the Plancherel and inversion formulae corresponding to (7) are given by(10)f,g2=ℒMf,ℒMg2,∀f,g∈L2ℝN,and(11)fx=ℒM−1ℒMfwx=∫ℝNℒMfwKM−1w,xdw,respectively, where M−1=DT,−BT;−CT,AT. Furthermore, the kernel (8) satisfies the following properties:(i)
KM−1w,x=KMx,w¯.(ii)
∫ℝNKMx,wKM−1x,zdx=δz−w.(iii)
∫ℝNKMx,wKM−1z,wdw=δz−x.(iv)
∫ℝNKMx,wKNx,zdx=KMNw,z.
The free metaplectic transformation (7) encompasses several well-known integral transforms such as the Fourier transform (FT), fractional Fourier transform (FrFT), linear canonical transform (LCT), Fresnel transform (FrT) [1] and is depicted in Table 1.
The convolution theory plays a vital role in diverse fields of signal and image processing. We recall the definition of free metaplectic convolution⊛M and some of its basic properties.Table 1
Some special cases of the free metaplectic transformation.
Free symplectic matrixM=A,B;C,DFree metaplectic transformation(i)A=D=0, B=−C=INN-dimensional FT(ii)A=diaga11,…,aNN, B=diagb11,…,bNN, C=diagc11,…,cNN, D=diagd11,…,dNNN-dimensional separable LCT(iii)A=D=diagcosθ1,…,cosθN, B=−C=diagsinθ1,…,sinθNN-dimensional separable FrFT(iv)A=D=INcosθ, B=−C=INsinθN-dimensional nonseparable FrFT(v)A=D=IN, B=diagb11,…,bNN, C=0N-dimensional separable FrT(vi)A=D=IN, C=0N-dimensional nonseparable FrTDefinition 2.
Given a pair of functionsf,g∈L2ℝN, the free metaplectic convolution with respect to the symplectic matrix M=A,B;C,D is defined by(12)f⊛Mgt=∫ℝNfxgt−xexpixTB−1Ax−tTB−1At2dx.
The free metaplectic convolution (12) satisfies the following properties.(i)
Commutative:f⊛gt=g⊛ft.(ii)
Associative:f⊛g⊛ht=f⊛g⊛ht.(iii)
Translation:f⊛gt−k=fx⊛expkTB−1Ax+xTB−1Akgxt.(iv)
Reflection:f⊛g−t=f−x⊛g−xt.(v)
Convolution theorem: for anyf,g∈L2ℝN, equation (12) satisfies(13)ℒMf⊛Mgtw=2πN/2ℒMfwℱgB−1w,where ℱf denotes the N-dimensional Fourier transform of ft given by(14)ℱfw=f^w=12πN/2∫ℝNfxe−ixTwdx.
It is worth noticing that the free metaplectic convolution (12) enjoys both commutative as well as associative properties. Moreover, the elegance of the convolution ⊛M lies in the fact that the convolution theorem is free from any chirp multiplier, which is convenient to implement and can also reduce the effect of the spectral leakage.
We now present an example for the lucid illustration of free metaplectic convolution (12).Example 1.
Consider the following two-dimensional unilateral exponential functions:(15)fx1,x2=exp6x1+5ix22121,x1,x2≥0,0,else.
Also, consider the two-dimensional Gaussian function(16)gx1,x2=1σ2πexp−x12+x222σ2,σ>0.
Denotehz=f⊛Mgz. Then, we have(17)hz=1σ2π2/3∫ℝ+∫ℝ+exp6x1+5ix22121exp−x1−z12+x2−z222σ2×expixTB−1Ax−zTB−1Az2dx1dx2.
For a symplectic matrix(18)M=ABCD=1/61−21/6−5/6−1/61/65/31012/29−31/29−6/29−36/2936/290,we have(19)xTB−1Ax=−36121x1x25/3−1/6−1/6−21/61−5/6−1/6x1x2=−6121x1x25/261/659/61x1x2=−61215x122+20x1x2+x22.
Similarly, one can obtain(20)zTB−1Az=−61215z122+20z1z2+z22.
Substituting the estimates (19) and (20) in (17), we obtain(21)hz=1σ2π2/3∫ℝ+∫ℝ+exp6x1+5ix22121exp−x1−z12+x2−z222σ2×exp−3i1215x122+20x1x2+x22exp3i1215z122+20z1z2+z22dx1dx2=1σ2π2/3exp3i1215z122+20z1z2+z22∫ℝ+∫ℝ+exp6x1+5ix22121×exp−x1−z12+x2−z222σ2exp−3i1215x122+20x1x2+x22dx1dx2=1σ2π2/3exp3i1215z122+20z1z2+z22exp−z12+z222σ2×∫ℝ+exp−36121+15i242+12σ2x12dx1×∫ℝ+exp−25121+3i121+12σ2x22dx2=1σ2π2/3exp3i1215z122+20z1z2+z22exp−z12+z222σ2×π436121+15i242+12σ2−1/225121+3i121+12σ2−1/2.


## 2.2. The Shearlet Transform in Arbitrary Space Dimensions

In this section, we recall the mathematical framework of the shearlet transforms, which serve as a basis for the development of the novel integral transform in arbitrary space dimensions. Formally, we have the following definition of the multidimensional shearlet transform.Definition 3 (see [9]).
For anyf∈L2ℝN, the continuous shearlet transform in arbitrary space dimension is defined by(22)Sℋψfa,s,t=f,ψa,s,t2=∫ℝNfxψa,s,tx¯dx,where(23)ψa,s,tx=detΛa−1/2ψΛa−1Ss−1x−t,⋅4a∈ℝ0,s∈ℝN−1,t∈ℝN,is the shearlet family constituted by the combined action of the scaling DΛa, sharing DSs, and translation Tt operators on the analyzing function ψ∈L2ℝN given by(24)DΛaψx=detΛa−1/2ψΛa−1x,DSsψx=ψSs−1x,andTtψx=ψx−t,respectively, and the matrices involved in (24) are given by(25)Λa=a0N−1T0N−1sgnaa1/NIN−1andSs=1sT0N−1IN−1,where sT=s1,s2,…,sN−1 and sgn⋅ and 0 denote the well-known signum function and the null vector, respectively. The set S=ℝ\0×ℝN−1×ℝN endowed with the operation(26)a,s,t⊙a′,s′,t′=aa′,s+a1−1/Ns′,t+SsΛat′,forms a locally compact group, often called the shearlet group. The left Haar measures on S are given by dv=dadsdt/aN+1. For more details about shearlets and their application, the readers are referred to [10–13] and the references therein.


## 3. A Convolution-Based Shearlet Transform in FMT Domains

In this section, our aim is to introduce a novel integral transform by intertwining the free metaplectic transform (7) and shearlet transform (22). Subsequently, we shall study the fundamental properties of the proposed transform and also formulate the corresponding energy preserving and reconstruction formulae.Definition 4.
For anyf∈L2ℝN, the free metaplectic shearlet transform with respect to an analyzing shearlet ψ∈L2ℝN is defined by(27)SψMfa,s,t=f⊛Mψ˜a,s,0=∫ℝNfxψa,s,tMx¯dx,detB≠0,where(28)ψa,s,tMx=a1−2N/2NψΛa−1Ss−1x−texpitTB−1At−xTB−1Ax2.
We now present an example for the lucid illustration of the proposed transform (27).Example 2.
The process of construction of admissible shearlets is somewhat straightforward. Essentially, any functionψ∈L2ℝ2 such that the corresponding Fourier transform ψ^ is compactly supported away from origin is an admissible shearlet. One of the important examples of such functions is the well-known classical shearlet, defined by [7, 8](29)ψ^w=ψ^ω1,ω2=ψ^1ω1ψ^2ω2ω1,where ψ1∈L2ℝ. Moreover, ψ^1∈C∞ℝ, the class of smooth functions with supp ψ^1⊆−1/2,−1/16∪1/16,1/2 satisfying(30)∑j∈ℤψ^12−jω12=1a.e.ω1∈ℝ.
Also,ψ2∈L2ℝ2 is chosen to be bump function in the sense that(31)∑k=−11ψ^2ω1+k2=1a.e.ω2∈−1,1,with ψ^2∈C∞ℝ and supp ψ^2⊆−1,1. ψ defined by (29) behaves like wavelet along one axis and bump along the other axis.
There are several choices for choosing the functionsψ1 and ψ2 satisfying conditions (30) and (31). Below, we shall construct such a pair of functions. Define a real-valued function α:ℝ⟶ℝ as(32)αx1=0,x1<0,35x14−84x15+70x16−20x17,0≤x1≤1,1,x1>1,.
Note that the function= is symmetric and is the same function used for the construction of Meyer wavelet. Moreover, we define another function β:ℝ⟶ℝ as(33)βx2=sinπ2αx2−1,1≤x2≤2,cosπ2αx2−1,2≤x2≤4,0,otherwise,.
The functionβ is positive, real-valued symmetric with compact support on the interval −4,−1∪1,4. Using the functions α and β, we construct the function ψ1 via the corresponding Fourier transform as(34)ψ^1ω1=α2ω1+β22ω1.
Then, we observe that the functionψ^1 has compact support over −4,−1∪1,4. Next, we define another function ψ2 via the corresponding Fourier transform as(35)ψ^2ω2=α1+ω2,ω2≤0,α1−ω2,ω2>0,.
As a consequence of (27) and Parseval’s formula (10), we have(36)SψMfa,s,t=ℒMf,ℒMψa,s,tM.
We now proceed to computeℒMψa,s,tM as(37)ℒMψa,s,tMw=∫ℝ2a−3/4ψΛa−1Ss−1x−texpitTB−1At−xTB−1Ax2KMx,wdx=a−3/42πdetB1/2expitTB−1At+wTB−1Aw2∫ℝ2ψΛa−1Ss−1x−te−iwTBT−1xdx=a3/42πdetB1/2expitTB−1At+wTB−1Aw2∫ℝ2ψzexp−iwTBT−1SsΛaz+tdz=a3/4KMt,wψ^ΛaTSsTB−1w=a3/4KMt,wψ^a0saab22/detB−b12/detB−b21/detBb11/detBw=a3/4KMt,wψ^ab22/detB−ab12/detBasb22−b21/detBab11−sb12/detBw=a3/4KMt,wψ^1ab22ω1−b12ω2detBψ^2sb22−b21ω1+b11−sb12ω2ab22ω1−b12ω2.
Implementing (37) in (36), we obtain(38)SψMfa,s,t=a3/4∫ℝ2ℒMfwψ^1¯ab22ω1−b12ω2detB×ψ^2¯sb22−b21ω1+b11−sb12ω2ab22ω1−b12ω2KMt,w¯dw.
Equivalently,(39)SψMfa,s,t=a3/4detBe−itTB−1At/2∫ℝ2ℱfxe−itTB−1At/2B−1wψ^1¯ab22ω1−b12ω2detB×ψ^2¯sb22−b21ω1+b11−sb12ω2ab22ω1−b12ω2e−iwTB−1tdw.=a3/4detBe−itTB−1At/2ℱ−1ℱfte−itTB−1At/2B−1wψ^1¯ab22ω1−b12ω2detB×ψ^2¯sb22−b21ω1+b11−sb12ω2ab22ω1−b12ω2BT−1t.
In particular, for the real, free symplectic matrix(40)M=ABCD=0I−I0,we obtain a conventional case(41)Sψfa,s,t=a3/4ℱ−1f^wψ^1¯aω1ψ^2¯1aω2ω1+st.In the following theorem, we assemble some basic properties of the free metaplectic shearlet transform (27) in arbitrary space dimensions.Theorem 1.
For anyf,g∈L2ℝN,α,β∈ℂ and k∈ℝN,λ∈ℝ+, the free metaplectic shearlet transform (27) satisfies the following properties:(i)
Linearity:SψMαf+βga,s,t=αSψMfa,s,t+βSψMga,s,t.(ii)
Anti−linearity:Sαψ+βϕMfa,s,t=α¯SψMfa,s,t+β¯SϕMfa,s,t.(iii)
Translation:SψMTkfa,s,t=expitTB−1Ak−2kTB−1Ak+kTB−1At/2SψMFa,s,t−k,whereFx=exp−ixTB−1Ak+kTB−1Axfx.(iv)
Parity:SψMf−xa,s,t=−SψMfa,s,−t, where ψˇx=ψ−x.(v)
Scaling:SψMfλxa,s,t=1/λSψ˜M′fa,s,λt, where ψ˜x=ψx/λ and M′=A/λ,λB;C,D.For the sake of brevity, we omit the proof of Theorem1.Next, we shall derive a fundamental relationship between FMT (7) and the proposed free metaplectic shearlet transform (27).Lemma 1.
LetSψMfa,s,t be the free metaplectic shearlet transform of any function f∈L2ℝN. Then, the following identity holds:(42)ℒMSψMfa,s,tw=2πN/2a2N−1/2NℒMfwℱψ¯B−1wSsΛa,where ℱ denotes the N-dimensional Fourier transform given by (14).Proof.
The FMT of the free metaplectic shearlet transformSψMfa,s,t is given by(43)ℒMSψMfa,s,tw=ℒMf⊛Mψ˜a,s,0tw=2πN/2ℒMfwℱψ˜a,s,0B−1w.
However, for anyψ∈L2ℝN, we have(44)ℱψ˜a,s,0xB−1w=a1−2N/2N∫ℝNψ−Λa−1Ss−1x¯exp−2πixTB−1wdx=a2N−1/2N∫ℝNψ−z¯exp−2πiSsΛazTB−1wdz=a2N−1/2N∫ℝNψ−z¯exp−2πi−zTΛaTSsTB−1w¯dz=a2N−1/2Nℱψ¯ΛaTSsTB−1w.
Substituting (44) into (43), we obtain(45)ℒMSψMfa,s,tw=2πN/2a2N−1/2NℒMfwℱψ¯ΛaTSsTB−1w.
Consequently, we have(46)SψMfa,s,t=2πN/2a2N−1/2N∫ℝNℒMfwℱψ¯ΛaTSsTB−1wKM−1w,tdw.
This completes the proof of Lemma1.We are now in a position to derive the orthogonality relation for the proposed transform (27).Theorem 2.
LetSψMfa,s,t and SψMga,s,t be the free metaplectic shearlet transforms of any pair of square integrable functions f and g, respectively. Then, we have(47)∫ℝN×ℝN−1×ℝ+SψMfa,s,tSψMga,s,t¯dadsdwaN+1=CψMf,g2,where(48)CψM=2πN∫ℝN−1×ℝ+ℱψΛaTSsTw2dadsaN2−N+1/N<∞,a.e.w∈ℝN,is the admissibility condition associated with ψ.Proof.
Using formula (42), the free metaplectic shearlet transforms of the given pair of functions f,g∈L2ℝN can be expressed as(49)SψMfa,s,t=2πN/2a2N−1/2N∫ℝNℒMfwℱψ¯ΛaTSsTB−1wKM−1w,tdw,SψMga,s,t=2πN/2a2N−1/2N∫ℝNℒMgvℱψ¯ΛaTSsTB−1vKM−1v,tdv.
Consequently, we have(50)∫ℝN×ℝN−1×ℝ+SψMfa,s,tSψMfa,s,t¯dadsdwaN+1=∫ℝN×ℝN−1×ℝ+2πN∫ℝNℒMfwℱψ¯ΛaTSsTB−1wKM−1w,tdw×∫ℝNℒMgvℱψ¯ΛaTSsTB−1vKM−1v,tdv¯dadsdtaN2−N+1/N=∫ℝN−1×ℝ+2πN∫ℝN∫ℝNℒMfwℒMg¯vℱψΛaTSsTB−1v×ℱψ¯ΛaTSsTB−1wdwdv∫ℝNKMv,tKMw,t¯dtdadsaN2−N+1/N=∫ℝN−1×ℝ+2πN∫ℝN∫ℝNℒMfwℒMg¯vℱψΛaTSsTB−1v×ℱψ¯ΛaTSsTB−1wdwdvδv−wdadsaN2−N+1/N=∫ℝNℒMfwℒMg¯w2πN∫ℝN−1×ℝ+ℱψΛaTSsTw2dadsaN2−N+1/Ndw=CψMℒMf,ℒMg2=CψMf,g2.
This completes the proof of Theorem2.Corollary 1.
Forf=g, the orthogonality relation (44) yields(51)SψMfa,s,t22=CψMf22.We now intend to present a formula which allows to recover the input functionf from the corresponding free metaplectic shearlet transform (27).Theorem 3.
LetSψMfa,s,t be the free metaplectic shearlet transform of any f∈L2ℝN with respect to an admissible function ψ∈L2ℝN. Then, f can be reconstructed via(52)fx=1CψM∫ℝN×ℝN−1×ℝ+SψMfa,s,tψa,s,tMxdadsdtaN+1,a.e.,Proof.
Forgx=δx−⋅, we have(53)SψMga,s,t=∫ℝNδx−⋅ψa,s,tM¯xdx=ψa,s,tM¯⋅,and(54)∫ℝNfxδx−⋅dx=f⋅.
Application of orthogonality relation (47) along with (53) and (54) yields(55)∫ℝNfxδx−⋅dx=1CψM∫ℝN×ℝN−1×ℝ+SψMfa,s,tSψMδx−⋅¯a,s,tdadsdtaN+1=1CψM∫ℝN×ℝN−1×ℝ+SψMfa,s,tψa,s,tM⋅dadsdtaN+1.
Equivalently,(56)f⋅=1CψM∫ℝN×ℝN−1×ℝ+SψMfa,s,tψa,s,tM⋅dadsdtaN+1.
This is the required reconstruction formula.In the following theorem, we present a characterization of the range of the proposed transform (27). The result follows as a consequence of the reconstruction formula (52) and the well-known Fubini’s theorem.Theorem 4.
Letψ∈L2ℝN be an admissible shearlet with CψM=1. Then, a function f∈L2ℝN×ℝN−1×ℝ+ is the free metaplectic shearlet transform of a certain signal if and only if it satisfies the reproduction property(57)fa′,s′,t′=∫ℝN×ℝN−1×ℝ+fa,s,tψa,s,tM,ψa′,s′,t′M2dadsdtaN+1.Proof.
Suppose thatf belongs to the range of the free metaplectic shearlet transform (27); that is, f∈SψML2ℝN. Thus, there must exist a square integrable function g∈L2ℝN such that SψMg=f. We now show that f satisfies (57). We proceed as(58)fa′,s′,t′=SψMga′,s′,t′=∫ℝNgxψa′,s′,t′M¯xdx=∫ℝN1CψM∫ℝN×ℝN−1×ℝ+SψMga,s,tψa,s,tMxdadsdtaN+1ψa′,s′,t′M¯xdx=1CψM∫ℝN×ℝN−1×ℝ+SψMga,s,t∫ℝNψa,s,tMxψa′,s′,t′M¯xdxdadsdtaN+1=∫ℝN×ℝN−1×ℝ+fa,s,tψa,s,tM,ψa′,s′,t′M2dadsdtaN+1.
Conversely, suppose that a functionf∈L2ℝN×ℝN−1×ℝ+ satisfies (57). Then, we show that f∈SψML2ℝN, that is, we shall obtain a function g∈L2ℝN such that SψMg=f. Consider the function(59)gx=∫ℝN×ℝN−1×ℝ+fa,s,tψa,s,tMxdadsdtaN+1.
Then, a straightforward computation yields thatg2=f2<∞, so that g∈L2ℝN. Therefore, we have(60)SψMga′,s′,t′=∫ℝNgxψa′,s′,t′M¯xdx=∫ℝN∫ℝN×ℝN−1×ℝ+fa,s,tψa,s,tMxdadsdtaN+1ψa′,s′,t′M¯xdx=∫ℝN×ℝN−1×ℝ+fa,s,t∫ℝNψa,s,tMxψa′,s′,t′M¯xdxdadsdtaN+1=∫ℝN×ℝN−1×ℝ+fa,s,tψa,s,tM,ψa′,s′,t′M2dadsdtaN+1=fa′,s′,t′.
From (28), it follows that SψMg=f. Hence, we conclude that the expression (57) offers a characterization of the range of the free metaplectic shearlet transform (27). This completes the proof of Theorem 4.


## 4. HAP Property Associated with the Proposed Shearlet Transform

In this section, we study the homogeneous approximation property (HAP) for the free metaplectic shearlet transform defined by (27). To begin, we show that the inverse free metaplectic shearlet transform in arbitrary space dimensions has a pointwise convergence result. Second, we show that in the free metaplectic domain, any pair of admissible shearlets satisfies the homogeneous approximation property.For a pair of free metaplectic shearletsψ1,ψ2∈L2ℝN, we denote(61)Cψ1,ψ2M=2πN∫ℝN−1×ℝ+ℱψ1ΛaTSsTB−1wℱψ2ΛaTSsTB−1w¯dadsaN2−N+1/N.Then, for everyp,q,r∈G=a,s,t:a∈ℝ+,s∈ℝN−1,t∈ℝN, we set(62)ΩP1,P2:Q:R≔P1,P2×−Q,QN−1×−R,RN,P2>P1>0,Q>0,R∈ℝ.Moreover, any neighbourhood ofΩP1,P2:Q:R is defined by(63)ΩP1,P2:Q:Rp,q,r=p,q,ra,s,t:p∈P1,P2,q∈−Q,QN−1,r∈−R,RN=pa,q+pN−1/Ns,r+sqapt:p∈P1,P2,q∈−Q,QN−1,r∈−R,RN.For everyψ∈L2ℝN, we define(64)PMa,s,tψx=ψa,s,tMx=a1−2N/2NψΛa−1Ss−1x−texpitTB−1At−xTB−1Ax2.Consequently, the free metaplectic shearlet transform (27) can be rewritten as(65)SψMfa,s,t=f,ψa,s,tM2=f,PMa,s,tψ2,for allf∈L2ℝN.The following lemma deals with the pointwise convergence of the inversion formula (35).Lemma 2.
Letψ1,ψ2∈L2ℝN be an admissible pair of free metaplectic shearlets satisfying the admissibility condition (61) with Cψ1,ψ2M≠0. Assume that f∈L2ℝN such that(66)fP1,P2:Qx=1Cψ1,ψ2M∫P1P2∫−QQ∫ℝNf,PMa,s,tψ12PMa,s,tψ2xdtdsdaaN+1.Then,fP1,P2:Q is uniformly continuous on ℝN and(67)ℒMfP1,P2:Qxw=2πNCψ1,ψ2MℒMfw∫P1P2∫−QQℱψ1¯ΛaTSsTB−1wℱψ2ΛaTSsTB−1wdsdaaN2−N+1/N.Proof.
For anyx∈ℝN, we have(68)∫P1P2∫−QQ∫ℝNf,PMa,s,tψ12PMa,s,tψ2tdtdsdaaN+1≤∫P1P2∫−QQ∫ℝNf,PMa,s,tψ122dt∫ℝNPMa,s,tψ22dt1/2dsdaaN+1=≤∫P1P2∫−QQ∫ℝNf,PMa,s,tψ122dt1/2ψ2dsdaaN+1≤ψ22∫P1P2∫−QQdsdaaN+11/2∫P1P2∫−QQ∫ℝNf,PMa,s,tψ122dtdsdaaN+11/2≤f2ψ22Cψ2MQP2N−P1NP1P2N1/2<∞.
Hence, we conclude that the functionfP1,P2:Q is well defined on ℝN. We now show that fP1,P2:Q is uniformly continuous on ℝN. For any x,x′∈ℝN, we have(69)fP1,P2:Qx−fP1,P2:Qx′=1Cψ1,ψ2M∫P1P2∫−QQ∫ℝNf,PMa,s,tψ12PMa,s,tψ1x−PMa,s,tψ2x′dtdsdaaN+1≤1Cψ1,ψ2M∫P1P2∫−QQ∫ℝNf,PMa,s,tψ122dtdsdaaN+11/2×∫P1P2∫−QQ∫ℝNPMa,s,tψ1x−PMa,s,tψ2x′2dtdsdaaN+11/2≤f2Cψ1MCψ1,ψ2M×∫P1P2∫−QQ∫ℝNPMa,s,tψ1x−PMa,s,tψ2x′2dtdsdaaN+11/2.
From (69), we observe that fP1,P2:Qx−fP1,P2:Qx′⟶0 as x−x′⟶0. Thus, we conclude that fP1,P2:Q is uniformly continuous on ℝN.
Moreover, for any arbitrary functiong∈L1ℝN∩L2ℝN, we have(70)fP1,P2:Q,g2=∫ℝNfP1,P2:Qxgx¯dx=∫ℝN1Cψ1,ψ2M∫P1P2∫−QQ∫ℝNf,PMa,s,tψ12PMa,s,tψ2dtdsdaaN+1gx¯dx=1Cψ1,ψ2M∫P1P2∫−QQ∫ℝNf,PMa,s,tψ12∫ℝNPMa,s,tψ2xgx¯dxdtdsdaaN+1=1Cψ1,ψ2M∫P1P2∫−QQ∫ℝNf,PMa,s,tψ12g,PMa,s,tψ2¯2dtdsdaaN+1=1Cψ1,ψ2M∫P1P2∫−QQ∫ℝN2πN∫ℝNℒMfwℱψ1¯ΛaTSsTB−1wKM−1w,tdw×∫ℝNℒMg¯vℱψ2ΛaTSsTB−1vKM−1v,tdvdsdtdaaN2−N+1/N=2πNCψ1,ψ2M∫P1P2∫−QQ∫ℝNℒMfwℒMg¯wℱψ1¯ΛaTSsTB−1w×ℱψ2ΛaTSsTB−1wdsdwdaaN2−N+1/N=2πNCψ1,ψ2M∫ℝNℒMfwℒMg¯w∫P1P2∫−QQ∫ℝNℱψ1¯ΛaTSsTB−1wℱψ2¯ΛaTSsTB−1wdsdaaN2−N+1/Ndw.
Invoking Parseval’s formula (10) for the FMT, we can deduce that(71)ℒMfP1,P2w=2πNCψ1,ψ2MℒMfw∫P1P2∫−QQℱψ1¯ΛaTSsTB−1wℱψ2ΛaTSsTB−1wdsdaaN2−N+1/N.
This completes the proof of Lemma2.We now study the convergence of the inverse free metaplectic shearlet transform (27).Theorem 5.
Letψ1,ψ2∈L2ℝN be a pair of free metaplectic shearlets satisfying the admissibility condition (61) with Cψ1,ψ2M≠0. Let f∈L1ℝN∩L2ℝN satisfying (66). Then, we have(72)limP1⟶0,P2,Q⟶∞f−fP1,P2:Q∞=0andlimP1⟶0,P2,Q⟶∞f−fP1,P2:Q2=0.Proof.
Using Parseval’s formula (10) for the FMT together with Lemma 2, we obtain(73)f−fP1,P2:Q∞≤f−fP1,P2:Q1=ℒMf−ℒMfP1,P2:Q1=ℒMfw−2πNCψ1,ψ2MℒMfw∫P1P2∫−QQℱψ1¯ΛaTSsTB−1w×ℱψ2ΛaTSsTB−1wdsdaaN2−N+1/N1=∫ℝNℒMfw1−2πNCψ1,ψ2M∫P1P2∫−QQℱψ1¯ΛaTSsTB−1w×ℱψ2ΛaTSsTB−1wdsdaaN2−N+1/Ndw.
Sinceψ1,ψ2 are admissible shearlets, we have(74)∫P1P2∫−QQℱψ1¯ΛaTSsTB−1wℱψ2ΛaTSsTB−1wdsdaaN2−N+1/N≤∫P1P2∫−QQℱψ1¯ΛaTSsTB−1wℱψ2ΛaTSsTB−1wdsdaaN2−N+1/N≤∫P1P2∫−QQℱψ1ΛaTSsTB−1w2dsdaaN2−N+1/N1/2∫P1P2∫−QQℱψ2ΛaTSsTB−1w2dsdaaN2−N+1/N1/2≤∫ℝ+∫ℝℱψ1ΛaTSsTB−1w2dsdaaN2−N+1/N1/2∫ℝ+∫ℝℱψ2ΛaTSsTB−1w2dsdaaN2−N+1/N1/2=Cψ1MCψ2M1/22πN<∞.
Consequently, (73) yields(75)limP1⟶0,P2,Q⟶∞1−2πNCψ1,ψ2M∫P1P2∫−QQℱψ1¯ΛaTSsTB−1wℱψ2ΛaTSsTB−1wdsdaaN2−N+1/N=0.
Finally, using the dominated convergence theorem in (57), we conclude that(76)limP1⟶0,P2,Q⟶∞f−fP1,P2:Q∞=0.
Proceeding in a manner analogous to the above case, we can show that(77)limP1⟶0,P2,Q⟶∞f−fP1,P2:Q2=0.
This completes the proof of Theorem5.For typographical convenience, we shall denoteΩP1′,P2′:Q′:R′p,q,r and ΩP1′,P2′:Q′:R′a′,s′,t′ by G1 and G2, respectively.Theorem 6.
Letψ1,ψ2∈L2ℝN be a pair of free metaplectic shearlets satisfying the admissibility condition (61) with Cψ1,ψ2M≠0. Let f∈L1ℝN∩L2ℝN satisfying (66). Then, for any ε>0, there exist some P2>P1>0,Q>0 and R>0 such that for any p,q,r∈G with 0<P1′<P1,P2≤P2′,Q≤Q′ and R≤R′, we have(78)PMp,q,rfx−1Cψ1,ψ2M∫a,s,t∈G1PMp,q,rf,PMa,s,tψ12PMa,s,tψ2xdadsdtaN+122<ε.Proof.
For anyf∈L1ℝN∩L2ℝN, we have(79)PMp,q,rfx−1Cψ1,ψ2M∫a,s,t∈G1PMp,q,rf,PMa,s,tψ12PMa,s,tψ2xdadsdtaN+122=supg2=1PMp,q,rfx−1Cψ1,ψ2M∫a,s,t∈G1PMp,q,rf,PMa,s,tψ12×PMa,s,tψ2xdadsdtaN+1,g22=supg2=11Cψ1,ψ2M∫a,s,t∉G1PMp,q,rf,PMa,s,tψ12PMa,s,tψ2,g2dadsdtaN+12≤1Cψ1,ψ2M2∫a,s,t∉G2PMp,q,rf,PMa,s,tψ122dadsdtaN+1×supg2=1∫ℝN×ℝN−1×ℝ+g,PMa,s,tψ222dadsdtaN+1=Cψ2MCψ1,ψ2M2∫a,s,t∉G1PMp,q,rf,PMa,s,tψ122dadsdtaN+1=Cψ2MCψ1,ψ2M2∫a,s,t∉G1PMp,q,rf,PMa,s,tψ122dadsdtaN+1=KP1,P2:Q:R.
The desired result (78) follows immediately by choosing P1 sufficiently small and P2,Q,R large enough. This completes the proof of Theorem 6.


## 5. Uncertainty Principles for the Proposed Shearlet Transform

The uncertainty principle is one of the corner stones of harmonic analysis in general and signal processing in particular [14]. This principle in harmonic analysis stems from the uncertainty principle in quantum mechanics which states that the position and the momentum of a particle cannot be both determined explicitly but only in a probabilistic sense with a certain degree of uncertainty. The harmonic analysis version of this principle says that a nontrivial function cannot be sharply localized in both time and frequency domains simultaneously [15, 16]. The essence of the uncertainty principle is that it provides a lower bound for optimal resolution of a signal in both the time and frequency domains. In this section, our aim is to extend the Heisenberg and logarithmic uncertainty principles for the proposed free metaplectic shearlet transform.The classical Heisenberg–Pauli–Weyl uncertainty inequality for anyf∈L2ℝN in the free metaplectic domain is given by(80)∫ℝNtft2dt1/2∫ℝNwℒMfw2dw1/2≥NσminB4π∫ℝNft2dt,where σminB denotes the minimum singular value of the matrix B.Theorem 7.
Letψ∈L2ℝN be an admissible free metaplectic shearlet. Then, for any nontrivial function f∈L2ℝN, the following inequality holds:(81)∫ℝN×ℝN−1×ℝ+tSψMfa,s,t2dadsdtaN+11/2∫ℝNwℒMfw2dw1/2≥NCψMσminB4πf22,where σminB denotes the minimum singular value of the matrix B.Proof.
As a consequence of generalized Heisenberg–Weyl uncertainty principle for the free metaplectic transform [4], we have(82)∫ℝNtSψMfa,s,t2dt1/2∫ℝNwℒMSψMfa,s,tw2dw1/2≥NσminB4π∫ℝNSψMfa,s,t2dt.
Integrating (81) with respect to dsda/aN+1 together with an implication of the well-known Fubini theorem yields(83)∫ℝN−1×ℝ+∫ℝNtSψMfa,s,t2dt1/2∫ℝNwℒMSψMfa,s,tw2dw1/2dsdaaN+1≥NσminB4π∫ℝN×ℝN−1×ℝ+SψMfa,s,t2dadsdtaN+1.
Applying the Cauchy–Schwarz inequality in (82) and then invoking (51), we get(84)∫ℝN×ℝN−1×ℝ+tSψMfa,s,t2dadsdtaN+11/2×∫ℝN×ℝN−1×ℝ+wℒMSψMfa,s,tw2dadsdwaN+11/2≥NCψMσminB4πf22.
As an implication of Lemma1, the second integral on the LHS of (84) can be simplified in the following form:(85)∫ℝN×ℝN−1×ℝ+wℒMSψMa,s,tw2dadsdwaN+1=∫ℝN×ℝN−1×ℝ+w2πN/2a2N−1/2NℒMfwℱψΛaTSsTB−1w2dadsdwaN+1=∫ℝN×ℝN−1×ℝ+2πNwℒMfw2ℱψΛaTSsTB−1w2dadsdwaN2−N+1/N≤2πN∫ℝNwℒMfw2∫ℝN−1×ℝ+ℱψΛaTSsTB−1w2dadsaN2−N+1/Ndw=CψM∫ℝNwℒMfw2dw.
Using (85) in (84), we get(86)∫ℝN×ℝN−1×ℝ+tSψMfa,s,t2dadsdtaN+11/2∫ℝNwℒMfw2dw1/2≥NCψMσminB4πf22.
This completes the proof of Theorem7.Theorem 8.
Letψ∈L2ℝN be an admissible function. Then, for any nontrivial function f∈L2ℝN and 1≤p≤2, the following uncertainty inequality holds:(87)∫ℝN×ℝN−1×ℝ+tSψMfa,s,tpdadsdtaN+11/p∫ℝNwℒMfwpdw1/p≥NCψMσminB4πf22,where σminB denotes the minimum singular value of the matrix B.Proof.
Under the given hypotheses, the dispersions of any nonzero functionf∈L2ℝN in the time and free metaplectic frequency domains satisfy the following inequality:(88)∫ℝNtftpdt1/p∫ℝNwpwℒMfwpdw1/p≥NσminB4π∫ℝNft2dt.
Consequently, we obtain(89)∫ℝNtSψMfa,s,tpdt1/p∫ℝNwℒMSψMfa,s,twpdw1/p≥NσminB4π∫ℝNSψMfa,s,t2dt.
Upon integrating (89) under dsda/aN+1, together with an implication of the well-known Fubini theorem, we obtain(90)∫ℝN−1×ℝ+∫ℝNtSψMfa,s,tpdt1/p∫ℝNwℒMSψMfa,s,twpdw1/pdsdaaN+1≥NσminB4π∫ℝN×ℝN−1×ℝ+SψMfa,s,t2dadsdtaN+1.
Applying the Cauchy–Schwarz inequality in (90) and then invoking (51), we get(91)∫ℝN×ℝN−1×ℝ+tSψMfa,s,tpdadsdtaN+11/p×∫ℝN×ℝN−1×ℝ+wℒMSψMfa,s,twpdadsdwaN+11/p≥NCψMσminB4πf22.
As an implication of Lemma1, the second integral on the LHS of (91) can be simplified in the following form:(92)∫ℝN×ℝN−1×ℝ+wℒMSψMfa,s,twpdadsdwaN+1=∫ℝN×ℝN−1×ℝ+w2πN/2a2N−1/2NℒMfwℱψΛaTSsTB−1wpdadsdwaN+1=∫ℝN×ℝN−1×ℝ+2πpN/2aP2N−1/2NwℒMfwpℱψΛaTSsTB−1wpdadsdwaN+1≤2πpN/2∫ℝNwℒMfwp∫ℝN−1×ℝ+ℱψΛaTSsTB−1w2dadsaN2−N+1/Np/2dw=CψMp/2∫ℝNwℒMfwpdw.
Using (92) in (91), we get the desired result(93)∫ℝN×ℝN−1×ℝ+tSψMfa,s,tpdadsdtaN+11/p∫ℝNwℒMfwpdw1/p≥NCψMσminB4πf22.
This completes the proof of Theorem8.Remark 1.
Forp=2, the uncertainty inequality (87) boils down to Theorem 7, that is,(94)∫ℝN×ℝN−1×ℝ+tSψMfa,s,t2dadsdtaN+11/2∫ℝNwℒMfw2dw1/2≥NCψMσminB4πf22.Theorem 9.
Letψ∈L2ℝN be an admissible function. Then, for any nontrivial function f∈L2ℝN and p>2, the following uncertainty inequality holds:(95)∫ℝN×ℝN−1×ℝ+tpSψMfa,s,t2dadsdtaN+11/p∫ℝNwpℒMfw2dw1/p≥NσminB4πCψM1/pf24/p,where σminB denotes the minimum singular value of the matrix B.Proof.
Using the Holder inequality, we have(96)∫ℝN×ℝN−1×ℝ+tpSψMfa,s,t2dadsdtaN+12/p×∫ℝN×ℝN−1×ℝ+SψMfa,s,t2dadsdwaN+11−2/p=∫ℝN×ℝN−1×ℝ+t2SψMfa,s,t4/pp/2dadsdtaN+12/p×∫ℝN×ℝN−1×ℝ+SψMfa,s,t2−4/p1/1−2/pdadsdtaN+11−2/p≥∫ℝN×ℝN−1×ℝ+t2SψMfa,s,t4/pSψMfa,s,t2−4/pdadsdtaN+1=∫ℝN×ℝN−1×ℝ+t2SψMfa,s,t2dadsdtaN+1,which further yields(97)∫ℝN×ℝN−1×ℝ+tpSψMfa,s,t2dadsdtaN+11/p≥∫ℝN×ℝN−1×ℝ+t2SψMfa,s,t2dadsdt/aN+11/2∫ℝN×ℝN−1×ℝ+SψMfa,s,t2dadsdw/aN+1p−2/2p.
Similarly, from Lemma1, we infer that(98)∫ℝNwpℒMfw2dw1/p≥∫ℝNw2ℒMfw2dw1/2∫ℝNℒMfw2dwp−2/2p=∫ℝNw2ℒMfw2dw1/2CψMp−2/2pCψM∫ℝNℒMfw2dwp−2/2p≥∫ℝNw2ℒMfw2dw1/2CψMp−2/2p∫ℝN×ℝN−1×ℝ+ℒMSψMfa,s,tw2dadsdw/aN+1p−2/2p.
Using Parseval formula of the free metaplectic transform in (98), we obtain(99)∫ℝNwpℒMfw2dw1/p≥∫ℝNw2ℒMfw2dw1/2CψMp−2/2p∫ℝN×ℝN−1×ℝ+SψMfa,s,t2dadsdt/aN+1p−2/2p.
Multiplying (99) and (97) and employing (81) and (51), we obtain(100)∫ℝN×ℝN−1×ℝ+tpSψMfa,s,t2dadsdtaN+11/p∫ℝNwpℒMfw2dw1/p≥∫ℝN×ℝN−1×ℝ+t2SψMfa,s,t2dadsdt/aN+11/2∫ℝNw2ℒMfw2dw1/2CψMp−2/2p∫ℝN×ℝN−1×ℝ+SψMfa,s,t2dadsdw/aN+1p−2/p≥NCψMσminBf2CψMp−2/2p4πCψMf2p−2/p=NσminB4πCψM1/pf24/p.
This completes the proof of Theorem9.In our next theorem, we shall establish the logarithmic uncertainty principle for the proposed transform (27). We denote Schwartz class in L2ℝN by SℝN which is given by(101)SℝN=f∈C∞ℝN:supt∈ℝNtα∂tβft<∞,where C∞ℝN is the class of smooth functions, α,β are multiindices, and ∂t denotes the usual partial differential operator.Theorem 10.
LetSψMfa,s,t be the free metaplectic shearlet transform of any f∈SℝN. Then, we have(102)∫ℝN×ℝN−1×ℝ+lntSψMfa,s,t2dadsdtaN+1+CψM∫ℝNlnwBT−1ℒMfw2dw≥Γ′N/4ΓN/4−lnπCψMf22,whenever the LHS of (102) is defined.Proof.
The logarithmic uncertainty principle for the FMT is given by(103)∫ℝNlntft2dt+∫ℝNlnwBT−1ℒMf2dw≥Γ′N/4ΓN/4−lnπ∫ℝNft2dt,for all f∈SℝN, where Γ⋅ denotes the well-known Euler gamma function. Identify SψMfa,s,t as a function of the translation parameter t and replace f∈SℝN in (103) by SψMfa,s,t as(104)∫ℝNlntSψMfa,s,t2dt+∫ℝNlnwBT−1ℒMSψMfa,s,tw2dw≥Γ′N/4ΓN/4−lnπ∫ℝNSψMfa,s,t2dt.
Integrating (104) with respect to the measure dads/aN+1, we have(105)∫ℝN×ℝN−1×ℝ+lntSψMfa,s,t2dadsdtaN+1+∫ℝN×ℝN−1×ℝ+lnwBT−1ℒMSψMfa,s,tw2dadsdwad+1≥Γ′N/4ΓN/4−lnπ∫ℝN×ℝN−1×ℝ+SψMfa,s,t2dadsdtaN+1.
By implementing the orthogonality relation in (105), we obtain(106)∫ℝN×ℝN−1×ℝ+lntSψMfa,s,t2dadsdtaN+1+∫ℝN×ℝN−1×ℝ+lnwBT−1ℒMSψMfa,s,tw2dadsdwaN+1≥Γ′N/4ΓN/4−lnπCψMf22.
We now simplify the second integral of (106) with the aid of Lemma 1 as(107)∫ℝN×ℝN−1×ℝ+lnwBT−1ℒMSψMfa,s,tdadsdwad+1=∫ℝN×ℝN−1×ℝ+lnwBT−12πN/2a2N−1/2NℒMfwℱψΛaTSsTB−1w2dadsdwaN+1=∫ℝN×ℝN−1×ℝ+lnwBT−12πNℒMfwℱψΛaTSsTB−1w2dadsdwaN2−N+1/N=∫ℝNlnwBT−1ℒMfw22πN∫ℝN−1×ℝ+ℱψΛaTSsTB−1w2dadsaN2−N+1/Ndw=CψM=∫ℝNlnwBT−1ℒMfw2dw.
Plugging (107) into (106), we obtain the desired inequality for the continuous free metaplectic shearlet transform as(108)∫ℝN×ℝN−1×ℝ+lntSψMfa,s,t2dadsdtaN+1+CψM∫ℝNlnwBT−1ℒMfw2dw≥Γ′N/4ΓN/4−lnπCψMf22.
This completes the proof of Theorem10.Towards the end of this section, we shall derive Nazarov’s uncertainty principle which takes into consideration the notion of support of the function instead of the dispersion as used in the Heisenberg–Pauli–Weyl inequality. In this direction, we have the following theorem.Theorem 11.
LetSψMfa,s,t be the free metaplectic shearlet transform of any signal f∈L2ℝN and T1,T2 be two finite measurable subsets of ℝN. Then, the following inequality holds:(109)CeCminT1‖T2,T11/nWT2,WT1T21/n∫ℝN\T1∫ℝN−1×ℝ+SψMfa,s,t2dadsdtaN+1+CψM∫ℝN\T2BTℒMfw2dw≥CψM∫ℝNft2dt,where WT1 is the mean width of T1 and T1 denotes the Lebesgue measure of T1.Proof.
IfT1 and T2 are two finite Lebesgue measurable subsets of ℝN and f∈L2ℝN, then Nazarov’s uncertainty principle for the free metaplectic transformation reads as [4](110)CeCminT1‖T2,T11/nWT2,WT1T21/n∫ℝN\T1ft2dt+∫ℝN\T2BTℒMfw2dw≥∫ℝNft2dt,where WT1 is the mean width of T1 and T1 denotes the Lebesgue measure of T1.
IdentifySψMfa,s,t as a function of the translation parameter t and replace f∈L2ℝN in (110) by SψMfa,s,t∈L2ℝN as(111)CeCminT1‖T2,T11/nWT2,WT1T21/n∫ℝN\T1SψMfa,s,t2dt+∫ℝN\T2BTℒMSψMfa,s,tw2dw≥∫ℝNSψMfa,s,t2dt.
Integrating (111) with respect to the measure dads/aN+1, we obtain(112)CeCminT1‖T2,T11/nWT2,WT1T21/n∫ℝN\T1∫ℝN−1×ℝ+SψMfa,s,t2dadsdtaN+1+∫ℝN\T2BT∫ℝN−1×ℝ+ℒMSψMfa,s,tw2dadsdwaN+1≥∫ℝN×ℝN−1×ℝ+SψMfa,s,t2dadsdwaN+1.
Combination of orthogonality relation (32) and Lemma 1 gives the desired relation(113)CeCminT1‖T2,T11/nWT2,WT1T21/n∫ℝN\T1∫ℝN−1×ℝ+SψMfa,s,t2dadsdtaN+1+CψM∫ℝN\T2BTℒMfw2dw≥CψM∫ℝNft2dt.
This completes the proof of Theorem11.


## 6. Potential Applications

As the free metaplectic shearlet transform relies on an arbitrary2N×2N free symplectic matrix MA,B;C,D with N2N+1 degrees of freedom, it can achieve richer representations of signals which reveal their characteristics beyond the Fourier domain. By varying the matrix parameters involved in the formulation of the free metaplectic shearlet transform, the concentration of the shearlet spectrum can be optimized in the free metaplectic domain. As such, the proposed transform is expected to be of substantial importance in diverse applications, such as nonstationary signal representation, detection, parameter estimation, and filter design. In the context of signal representation, the free metaplectic shearlet transform can efficiently handle the LFM signals, which constitutes one of the most important nonstationary signals and is widely used in radar and sonar systems. Besides, in the realm of edge detection, parameter estimation, and filter design, the free metaplectic shearlet transform is expected to be of vital importance. For example, suppose that the received multidimensional signal fx comprises the desired signal sx and the unwanted signal (noise) nx together. Then, we can derive an optimal free metaplectic domain wherein the shearlet spectrum of the signal fx has the maximum local energy concentration. Based on this highly localized shearlet spectrum, one can realize the detection and the parameter estimation of the desired signal sx.



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*Source: 2140189-2021-10-22.xml*