Characterisation of wave front sets by the Stockwell transform
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 related to the continuity of a certain change of variables for the ST imply that n should be equal to 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 combined with appropriate diffeomorphisms on .
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 , for . 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 and 8, for example , our results of Section 3 gives us wave fronts within (instead of ). This problem is overcome in Section 4 (in Proposition 4.5).
We fix the constants in the Fourier transform as follows: , ; thus, its inverse is given by , . As standard, stands for the general linear group of degree n over , i.e. the group of all invertible matrices of real numbers. As customary, and are the orthogonal and special orthogonal group of degree n respectively.
Next, we define the regular points of the cotangent bundle related to an . A conic neighbourhood of a point is a set such that contains a ball for some and, for any p in and any , αp belongs to .
Definition 1.1 For a distribution , a point is called a regular directed point of f if and only if there exist: (i) a function with and (ii) a closed conic neighbourhood of ξ, such that
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 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 , for , where are defined as
- •
, ;
- •
, ;
- •
, ;
- •
,
Recall [16], the matrix , , with given as above, satisfies the following properties:
- 1.
is continuous;
- 2.
, where , .
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 by −1 except the first one. Remark 1.3 Our limitation only to the cases comes from the requirement 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 , the direction is considered as privileged.
Using the results of [16] and [17], we have the next assertions.
Proposition 1.5 There exists a map , , such that and , . In particular, the map , , is also .
Proof Assume first . For each , define Notice that and hence , , is smooth. Define , . By direct inspection one verifies that satisfies (1.2) which completes the proof of the proposition when . When n is 1, 2 or 4, the proof directly follows by taking to be the upper left submatrix of (1.3). □
Remark 1.6 Since is properly embedded submanifold of , the maps and , , are also .
As in the proof of [17, Lemma 5.7], Indeed, Moreover, implies .
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 , , and , . Then
Proof As , the absolute value of the Jacobian is the product of (by Proposition 1.5) with the absolute value of the Jacobian of . But the latter is which, by the Sylvester's determinant theorem, equals . As (since ), the result follows. □
Section snippets
The Stockwell transform
In [17], Riba has defined the multidimensional Stockwell transform under the assumption that the window function satisfies the admissibility condition where , and the condition . We will assume as in [16] that the window function satisfies:
- 1.
, (and thus );
- 2.
does not contain 0.
Up to the end of this section we assume .
For the window g and , the Stockwell transform (ST) of a distribution
Characterisation of the wave front set via the Stockwell transform
We continue to assume . 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 and 8.
Let and , , be a partition of the set such that for each j, is either 1, 2, 4 or 8. Let , , satisfy the following:
- (1)
and , for all , for all ;
- (2)
there exists such that , , where 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.
References (24)
- et al.
Continuous curvelet transform. I. Resolution of the wavefront set
Appl. Comput. Harmon. Anal.
(2005) Directional short-time Fourier transform
J. Math. Anal. Appl.
(2013)- et al.
A smooth introduction to the wavefront set
J. Phys. A, Math. Theor.
(2014) - et al.
Resolution of the wavefront set using general continuous wavelet transforms
J. Fourier Anal. Appl.
(2016) Foundations of Time-Frequency Analysis
(2001)- et al.
Directional short-time Fourier transform of distributions
J. Inequal. Appl.
(2016) - et al.
Tauberian theorems for the Stockwell transform of Lizorkin distributions
Appl. Anal.
(2020) Wavelets. An Analysis Tool
(1995)The Analysis of Linear Partial Differential Operators I
(1983)Lectures on Nonlinear Hyperbolic Differential Equations
(1997)