Characterisation of wave front sets by the Stockwell transform

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Abstract

We characterise the wave front sets via the Stockwell transform defined as a combination of the short-time Fourier transformation and a special class of rotations. The main results of the first part are necessary and sufficient criteria related to the directional smoothness of a tempered distribution in the cases when the space dimension n equals 1, 2, 4 or 8. In the second part, we extend the results to arbitrary nZ+.

Introduction

The aim of this article is to characterise the wave front sets of tempered distributions by the Stockwell transform (ST). We consider separately the classical wave front set and the Sobolev-type wave front set, both of them introduced by Hörmander [9], [10]. Since limitations concerning the dimension n of Rn related to the continuity of a certain change of variables for the ST imply that n should be equal to 1,2,4 or 8, we first characterise the wave fronts for this specific instances of n and then treat the general case by applying these results on suitable partitions of the set {1,2,3,...,n} combined with appropriate diffeomorphisms on Rn.

In order to motivate the use of the ST, we recall some facts related to various well-known transforms used for the characterisations of the wave front sets. The short-time Fourier transform (STFT) and the wavelet transform (WT) are commonly used in signal and image processing for the decomposition of signals in order to obtain more precise local information (cf. [4], [5], [6], [8], [9], [11], [22]).

The STFT provides the spectrum localised by the window function. This is well developed method used and applied in various fields. However, significant barrier in application of the STFT is the fact that the fixed window function has to be predefined, which leads to a poor time-frequency resolution and, in general, the absence of a sufficiently good reconstruction algorithm [5]. The WT is used to overcome some of the shortcomings of the STFT. With the dilatation and translation of the window function, the WT has better phase modulation in the spectral domain. However, the self-similarity caused by the translation and the overlap in the frequency domain becomes non-avoidable since they do not permit straightforwardly the transfer of scale information into proper frequency information [8].

The Stockwell transform (ST) [21] also decomposes a signal into temporal and frequency components. In contrast to the WT, the ST exhibits a frequency-invariant amplitude response and covers the whole temporal axis creating full resolutions for each designated frequency. It is invertible, and recovers the exact phase and the frequency information without reconstructing the signal. The problem with the ST is its redundancy. But, there have been different strategies in order to improve the performance and the application of the ST, cf. [20], [21], [23], [24].

In [12], [13], [15], [16], [19], integral transforms such as wavelet transforms, shearlet transform, Radon and short-time Fourier transform are used for the description of the wave front sets (cf. [9]). The results of [13] were improved in [16] through the use of a parameterised wavelet transform. More deep results are obtained in [3] where the wavelet transform was composed with a suitable dilatation group. In this way the authors characterise the wave front set related to the variety of this dilation group and the dual action of the group. Although the wavelet transform gives the identification of the singular support of a function, it does not give information about a global geometry of the singular point. For this purpose Kutyniok and Labate [12] define the continuous shearlet transform which, besides localisation of singular points, also gives the orientation of singularities of the distribution. Candès and Donoho [2] proposed another directional wavelet transform called continuous curvelet transform which gives good resolution if we assume that the singularity is a curve.

In this paper we follow some of the ideas of [16] and make characterisation of the wave front sets by the ST combining the good features of the short-time Fourier transform and the wavelet transform through the use of special rotations. The ST can be viewed as a frequency dependent STFT or a phase corrected wavelet transform.

In the recent article [7], this transform has been defined on the space of distributions, and an asymptotic analysis of distributions via the Stockwell transform has been performed.

The paper is organised as follows. After short preliminaries, we introduce and study in Section 1 the properties of the matrix-valued maps A:ξ=(ξ1,ξ2,,ξn)RnAξGL(n,R), for n=1,2,4,8. The Stockwell transform in the space of tempered distributions is given in Section 2. Section 3 is devoted to the characterisations of the classical and the Sobolev-type wave fronts when the dimension is 1, 2, 4 or 8. We illustrate the content of Section 4 by the following example. If we decompose the space dimension n as a linear combination of dimensions 1,2,4 and 8, for example 17=2×8+1, our results of Section 3 gives us wave fronts withinR17×(R8{0}×R8{0}×R{0}) (instead of R17×(R17{0}). This problem is overcome in Section 4 (in Proposition 4.5).

We fix the constants in the Fourier transform as follows: F(f)(ω)=fˆ(ω)=(2π)n2Rneitωf(t)dt, fL1(Rn); thus, its inverse is given by F1(f)(t)=F(f)(t), fL1(Rn). As standard, GL(n,R) stands for the general linear group of degree n over R, i.e. the group of all n×n invertible matrices of real numbers. As customary, O(n) and SO(n) are the orthogonal and special orthogonal group of degree n respectively.

Next, we define the regular points of the cotangent bundle related to an fS(Rn). A conic neighbourhood of a point ξRn{0} is a set Γ(ξ)Rn such that Γ(ξ) contains a ball B(ξ,ϵ)={τRn;|τξ|<ϵ} for some ϵ>0 and, for any p in Γ(ξ) and any α>0, αp belongs to Γ(ξ).

Definition 1.1

For a distribution fD(Rn), a point (x,ξ)Rn×(Rn\{0}) is called a regular directed point of f if and only if there exist: (i) a function ϕD(Rn) with ϕ(x)=1 and (ii) a closed conic neighbourhood Γ(ξ)Rn of ξ, such that|fϕˆ(ξ)|C(1+|ξ|)N for allξΓ(ξ),NN.

From the definition, we note that the decay condition imposed with (1.1) effectively only concerns the behaviour of large frequencies. The wave front set WF(f) is the complement of the set of regular directed points.

Note that there are several more approaches to the concept of wave front sets, and with that different definitions are proposed [1], [19].

In this section we study the matrix-valued maps A:ξ=(ξ1,ξ2,,ξn)RnAξGL(n,R), for n=1,2,4,8, where Aξ are defined as

  • n=1, Aξ1=[ξ];

  • n=2, Aξ1=[ξ1ξ2ξ2ξ1];

  • n=4, Aξ1=[ξ1ξ2ξ3ξ4ξ2ξ1ξ4ξ3ξ3ξ4ξ1ξ2ξ4ξ3ξ2ξ1];

  • n=8,Aξ1=[ξ1ξ2ξ3ξ4ξ5ξ6ξ7ξ8ξ2ξ1ξ4ξ3ξ6ξ5ξ8ξ7ξ3ξ4ξ1ξ2ξ7ξ8ξ5ξ6ξ4ξ3ξ2ξ1ξ8ξ7ξ6ξ5ξ5ξ6ξ7ξ8ξ1ξ2ξ3ξ4ξ6ξ5ξ8ξ7ξ2ξ1ξ4ξ3ξ7ξ8ξ5ξ6ξ3ξ4ξ1ξ2ξ8ξ7ξ6ξ5ξ4ξ3ξ2ξ1].

Recall [16], the matrix Rξ=|ξ|1Aξ1SO(n), ξRn\{0}, with Aξ given as above, satisfies the following properties:

  • 1.

    Rn\{0}ξRξO(n) is continuous;

  • 2.

    Rξ(ξ/|ξ|)=e1, where e1=(1,0,,0), ξRn\{0}.

Remark 1.2

Note that matrices satisfying the above properties are not uniquely determined. All properties and all the results of the paper will hold if we multiply some of the columns of Aξ1 by −1 except the first one.

Remark 1.3

Our limitation only to the cases n=1,2,4,8 comes from the requirement Rn\{0}ξRξO(n) to be continuous (see [16, Prop. 1]). This is essential for our work, because when transforming sets by changes of variables that involves these matrices, discontinuities are not allowed.

Remark 1.4

Lemmas 5.7, 5.8, and Proposition 5.9 of [17] are related to the ones already proved in [16]. We summarise below the claims that we need for the sake of completeness; in [16] instead of e1, the direction en is considered as privileged.

Using the results of [16] and [17], we have the next assertions.

Proposition 1.5

There exists a C map ξA˜ξ, Rn\{0}GL(n,R), such thatAξ1(τ)=A˜τ1(ξ),for allξ,τRn\{0}, and |ξ|1A˜ξ1O(n), ξRn\{0}. In particular, the map ξA˜ξ1, Rn\{0}GL(n,R), is also C.

Proof

Assume first n=8. For each ξR8\{0}, defineBξ=[ξ1ξ2ξ3ξ4ξ5ξ6ξ7ξ8ξ2ξ1ξ4ξ3ξ6ξ5ξ8ξ7ξ3ξ4ξ1ξ2ξ7ξ8ξ5ξ6ξ4ξ3ξ2ξ1ξ8ξ7ξ6ξ5ξ5ξ6ξ7ξ8ξ1ξ2ξ3ξ4ξ6ξ5ξ8ξ7ξ2ξ1ξ4ξ3ξ7ξ8ξ5ξ6ξ3ξ4ξ1ξ2ξ8ξ7ξ6ξ5ξ4ξ3ξ2ξ1]. Notice that detBξ=|ξ|8 and hence ξBξ, R8\{0}GL(8,R), is smooth. Define A˜ξ=Bξ1, ξR8\{0}. By direct inspection one verifies that A˜ξ1 satisfies (1.2) which completes the proof of the proposition when n=8. When n is 1, 2 or 4, the proof directly follows by taking Bξ to be the upper left n×n submatrix of (1.3). 

Remark 1.6

Since O(n) is properly embedded submanifold of GL(n,R), the maps ξ|ξ|1Aξ1 and ξ|ξ|1A˜ξ1, Rn\{0}O(n), are also C.

As in the proof of [17, Lemma 5.7],Aξt(τ)=|ξ|2Aξ1(τ),for allτRn,ξRn\{0}. Indeed,Aξt(τ)=((|ξ|1|ξ|Aξ1)1)t(τ)=(1|ξ|(1|ξ|Aξ1)t)t(τ)=1|ξ|2Aξ1(τ). Moreover, Rξ(ξ/|ξ|)=e1 implies Aξt(ξ)=e1.

The following result has been proved in [18, Theorem 3.1(a)]; for the sake of completeness, we supply a (very simple) proof of it.

Proposition 1.7

Let n{1,2,4,8}, τRn\{0}, and ω=|ξ|2Aξ1(τ), ξRn\{0}. Then|ω|n|dω|=|ξ|n|dξ|.

Proof

As ω=A˜τ1(ξ/|ξ|2), the absolute value of the Jacobian det(ωj/ξk) is the product of |detA˜τ1|=|τ|n (by Proposition 1.5) with the absolute value of the Jacobian of ξξ/|ξ|2. But the latter is |ξ|2n|det(I2|ξ|2[ξ1ξn]t[ξ1ξn])| which, by the Sylvester's determinant theorem, equals |ξ|2n. As |ω|=|τ|/|ξ| (since |ξ|1Aξ1O(n)), the result follows. 

Section snippets

The Stockwell transform

In [17], Riba has defined the multidimensional Stockwell transform under the assumption that the window function gL1(Rn)L2(Rn) satisfies the admissibility conditionCg=Rn|gˆ(ωe1)|2dω|ω|n<, where e1=(1,0,0,...,0), and the condition Rng(t)dt=1. We will assume as in [16] that the window function satisfies:

  • 1.

    gˆD(Rn), gˆ0 (and thus gS(Rn));

  • 2.

    Ω=suppgˆ does not contain 0.

Up to the end of this section we assume n=1,2,4,8.

For the window g and 1m<, the Stockwell transform (ST) of a distribution fS

Characterisation of the wave front set via the Stockwell transform

We continue to assume n{1,2,4,8}. This is a crucial assumption for Subsections 3.1 and 3.2.

The main results of the paper give the principles of directional smoothness, by providing criteria for regular directed points using the Stockwell transform. Unlike the short-time Fourier transform, where the wave front set does not depend on the used window [15], [19], the wave front set by means of the wavelet transform depends on the chosen wavelet [3], [16]. To resolve this problem, as in [16] we

Generalisation of the Stockwell wave front set: the case of arbitrary n

In this section we generalise the definition of wave front sets via the Stockwell transform in dimensions different from n=1,2,4 and 8.

Let nZ+ and I={I1,,Ik}, kZ+, be a partition of the set {1,,n} such that for each j, nj=|Ij| is either 1, 2, 4 or 8. Let gjS(Rnj), j=1,,k, satisfy the following:

  • (1)

    F(gj)D(Rnj) and F(gj)(ξj)0, for all ξj=(ξ1j,,ξnjj)Rnj, for all j=1,,k;

  • (2)

    there exists r(0,1) such that suppF(gj)Bnj(e1j,r), j=1,,k, where e1jRnj is the point whose first coordinate is 1 and

Acknowledgments

The authors gratefully acknowledge the partial support by the Macedonian and the Serbian Academy of Sciences and Arts, through the bilateral project Microlocal Analysis and Application.

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