Directional time-frequency analysis via continuous frames111Research partially supported by a Start-Up Grant for Collaboration between EuroTech Universities funded by the Presidential Office, Technische Univeristät München, Germany
Grafakos and Sansing GS () have shown how to obtain directionally sensitive time-frequency decompositions in based on Gabor systems in the key tool is the “ridge idea,” which lifts a function of one variable to a function of several variables. We generalize their result by showing that similar results hold starting with general frames for both in the setting of discrete frames and continuous frames. This allows to apply the theory for several other classes of frames, e.g., wavelet frames and shift-invariant systems. We will consider applications to the Meyer wavelet and complex B-splines. In the special case of wavelet systems we show how to discretize the representations using -nets.
keywords:Discrete and continuous frames, Gabor system, ridge function, Radon transform, directionally sensitive time-frequency decomposition, shift-invariant system, Meyer wavelet, complex B-spline, discretization of the sphere
Msc: 42C15, 42C40, 65D07
Expansions of functions or signals as superpositions of basic building blocks with desired properties is one of the main tools in signal analysis. The expansions can be either in terms of an integral, a discrete sum, or a combination of both.
Many real-world signals depend on more than one variable. Depending on the type of expansion one is interested in, there are various ways to obtain such expansions. If an orthonormal basis for is given, one can obtain an orthonormal basis for via a simple tensor product, but this is highly inefficient. Some of the standard methods to obtain expansions in e.g., wavelet frames of Gabor frames, have similar versions in but they might not be optimal in order to detect features or special properties of the signal at hand. Other expansions are born in typically for e.g., caplets HR (), ridgelets C (); CD () and shearlets GL (); all of these can be considered as higher-dimensional wavelet-type systems with additional structure.
A different approach (parallel to the ridgelet construction) for Gabor systems was proposed by Grafakos and Sansing GS (). Starting with Gabor systems in , they developed a directionally sensitive Gabor-type expansion in using ridge functions. Two approaches were discussed in GS (): a discrete one, based on Gabor frames for and a semi-discrete version based on continuous Gabor systems generated by two non-perpendicular functions.
In this paper, we extend the main results in GS () in various ways. First, we observe that the above mentioned non-orthogonality places GS () in the setting of continuous frames, originally developed by Ali et al. AAG1 () resp. by Kaiser Ka (). Using techniques from frame theory, we then prove that the results in GS () have parallel versions starting with general frames for both in the discrete and the continuous setting. The results are applied to the Meyer wavelet and complex B-splines. In the special case of wavelet systems we show how to discretize the representations using -nets.
In the rest of the introduction, we will introduce some notation and state the necessary facts about ridge functions and (continuous) frames. Then, in Section 2 we present the generalizations of the results in GS (). Semi-discrete representations of functions in are investigated in Section 3, where we also apply the results to the Meyer wavelet and to complex B-splines. In the final Section 4, we obtain fully discrete representations for wavelet-type systems on bounded domains, by replacing the integral over the unit sphere by an appropriately chosen -net.
Some remarks concerning the notation: Since we deal with functions in and lift them to functions in , we need to consider inner products and the Fourier transform on different spaces. In general, for functions we define the Fourier transform by
where denotes the canonical inner product on We extend the Fourier transform to a unitary operator on in the usual way. The inverse Fourier transform of a function will be denoted by Also, for functions , , we use the notation
whenever the right hand side converges. The unit sphere in will be denoted by and the Schwartz space of rapidly decreasing functions on by
1.1 Ridge functions and the Radon transform
Let us now introduce the “ridge procedure” that lifts functions of one variable to functions of several variables. Ridge functions were originally introduced by Pinkus P (). Our starting point is to extend the ordinary differential operator on to certain non-differentiable functions. In fact, given define the differential operator acting on functions , by
this definition clearly also makes sense for a large class of non-differentiable functions.
In the entire note, we use the following terminology, which relates functions (written with lower case letters), the corresponding ridge functions (written with a subscript), the action of the differential operator on the given function (written with capital letters), and the associated ridge function (written with capital letters and a subscript).
Consider any function
For define the ridge function on by
For define the weighted ridge function by
Given the Radon transform of a function (in the direction ) is defined by
The Radon transform can be extended to a bounded operator from to see, e.g., (N, , p. 16 ff.). We also note that the Fourier slice theorem relates the (one-dimensional) Fourier transform of the Radon transform of a function to the (-dimensional) Fourier transform of by the formula
The following lemma shows a close relation between ridge functions and the Radon transform.
1.2 Continuous frames
In this section we review some of the known results about general continuous frames, as well as their concrete manifestations within Gabor analysis and wavelet theory.
Let be a complex Hilbert space and a measure space with a positive measure . A continuous frame is a family of vectors for which the following hold:
(i) For all , the mapping is a measurable function on .
(ii) There exist constants such that
The continuous frame is tight if we can choose
For every continuous frame, there exists at least one dual continuous frame, i.e., a continuous frame such that each has the representation
the integral in (1.8) should be interpreted in the weak sense, i.e., as
If is a continuous tight frame with bound then is a dual continuous frame.
Continuous frames generalize the more widely known (discrete) frames. In fact, in the case where () is a countable set equipped with the counting measure, Definition 1.1 yields the classical frames. Continuous frames were introduced independently by Ali et al. AAG1 () and Kaiser Ka (). We will present the most important concrete cases below; for constructions of (discrete) frames, see the monographs daubechies (); G (); CBN ().
There are also several well known examples of continuous frames for available in the literature. In order to introduce these, consider the translation, modulation, and scaling–operators on defined by
Let . Then
Proposition 1.2 has an immediate and well known consequence concerning the construction of continuous tight Gabor frames and dual pairs. The result shows that it is very easy to construct such frames, especially with windows belonging to the Schwartz space
For any the Gabor system is a continuous tight frame for with respect to equipped with the Lebesgue measure, with frame bound
For any functions for which the Gabor systems and are dual continuous frames.
A wavelet system has the form for a suitable function We say that satisfies the admissibility condition if
The admissibility condition gives the following result, see, e.g., (daubechies, , Prop. 2.4.1):
Assume that is admissible. Then, for all functions ,
Again, Proposition 1.4 immediately leads to a construction of a tight frame:
If is admissible, then is a continuous frame for with respect to equipped with the Haar measure with frame bound
Note that the admissibility condition is easy to satisfy, even with generators In fact, all functions with vanishing mean
satisfy the admissibility condition, see, e.g., lmr94 ().
2 Decompositions via continuous frames
We first show that any pair of continuous dual frames for consisting of functions in leads to an integral representation of functions for which This generalizes Theorem 3 in GS (). Note that Theorem 3 in GS () does not use the term continuous frame, but just the technical condition in the particular context of Gabor analysis this means exactly that the functions generate continuous dual Gabor frames, as we saw in Corollary 1.3.
Let and be dual continuous frames for consisting of functions in Then, for such that
Proof. Calculating the left-hand-side using Lemma 1.1 yields
so we arrive at
Note that because and are dual continuous frames for also and are dual continuous frames. The term above in the parantheses is exactly the frame decomposition with respect to these frames of the function evaluated at the point thus
Inserting this yields that
From here on, we can just refer to the proof of Theorem 3 in GS () in order to complete the proof. In fact, by the Fourier slice theorem, inserting this in (2.2), and splitting the integral over into integrals over and a few changes of variables yield that (2.2) equals as desired.
We have already sen in Section 1 that it is easy to construct continuous tight wavelet frames for that are generated by functions thus, it is easy to give applications of Theorem 2.1. However, for the purpose of applications our ultimate goal is to provide discrete realizations of the theory, so we will not consider concrete cases here.
3 Semi-discrete representations
The integral representation in Theorem 2.1 involves integrals over as well the sphere as the set Letting be a discrete set equipped with the counting measure, we can of course apply the result to discrete frames as well; in this case we obtain what we will call a semi-discrete representation of functions only involving an integral over and a sum over the discrete index set. Nevertheless, we will follow the approach by Grafakos and Sensing, see Theorem 5 in GS (), where a semi-discrete representation is derived in the Gabor case, independently of the continuous case. The reason for doing this is that the technical conditions are slightly weaker in this approach, leading to a representation that is valid for a larger class of functions.
In order to prove the next theorem, we need a result that is stated as part of Lemma 2, GS ().
Given a function , we have that
for almost every .
Let and let be a countable index set.
Let denote a frame for with frame bounds and define the associated functions and as in Definition 1.0. Then
Assuming that and are dual frames for both consisting of functions in then
Proof. By Lemma 1.1, applied to the function we have that
where the last equality following by partial integration and the assumption Now, by the frame assumption on
As shown in GS (),
As for the proof of (ii), the frame decomposition associated with the frames and and applied to the function yields that
it follows that
Let be a smooth function of sigmoidal shape required to satisfy for , for , and . An example of such a function is for instance the polynomial , for .
The classical Meyer wavelet is defined in the Fourier domain by
It is well known that is a Schwartz function and that
For , the Meyer wavelet and the ridge function with , are plotted in Figure 1.
Complex B-splines are a natural extension of the classical Curry-Schoenberg B-splines. They were introduced in forster06 () and then studied and extended in a series of papers, see, e.g., FGMS (); FMUe (); forstermasso10 (); multivarSplines (). Given with the complex B-spline is defined in the Fourier domain by
Setting , , one notices that
implying that complex B-splines reside on the main branch of the complex logarithm and are thus well-defined.
Compared with the classical cardinal B-splines, complex B-splines possess an additional modulation and phase factor in the frequency domain:
The existence of these two factors allows the extraction of additional information from sampled data. In fact, the spectrum of a complex B-spline consists of the spectrum of a real-valued B-spline combined with a modulating and a damping factor. The presence of an imaginary part causes the frequency components on the negative and positive real axis to be enhanced with different signs. This has the effect of shifting the frequency spectrum towards the negative or positive frequency side, depending on the sign of the imaginary part. This allows for an approximate single band analysis which is not possible with any real valued function, but necessary for certain phase retrieval tasks in signal processingforster06 (); fivegoodreasons (). Figure 2 gives an example of a complex B-spline in time domain as well as in frequency domain.
For the complex B-splines belong to the Sobolev spaces (with respect to the -Norm and with weight ).
Complex B-splines generate a multiresolution analysis of . In particular, is a Riesz basis for .
To construct the corresponding orthonormalized wavelet , we apply a high pass filter to the complex B-spline scaling function . To this end, denote by
the autocorrelation filter. The convergence of the series is proved in forster06 (). Then
is an orthonormal scaling function. For an example, see Figure 3.
The scaling filter is denoted by
The associated orthonormal wavelet is given by
Although complex B-splines do not belong to the function class , for the existence of integrals of the type (1.1) that involve functions as defined in (1.4), it is not necessary that . It suffices that and belong to or that additionally one of them belongs to a Sobolev space with , i.e., an appropriate subspace of . Choosing for so that these integrals exist. Therefore, we can define
4 Discrete Representations
In this section, we consider the cube,
and functions . We will present a discretization of the sphere which ultimately leads to a complete discrete representation of functions . This discretization was also considered in C () and is based on the concept of an –net. It is one of several existing discretization methodologies (other choices include the methods in conradprestin (), ward (), and brauchart ()). Let us recall the definition of a finite –net.
Let be a metric space and a discrete set . Given any the set is called an –net for if
, where denotes the closed ball of radius centered at .
An -net is called finite if is a finite set.
Note that since the sphere is compact, hence totally bounded, an –net exists for for all see suther ().
We employ the following discretization procedure for ; see also C ().
Choose an , and discretize the scale parameter by the sequence , where and is selected appropriately.
For , set .
Let be an -net of and require that following condition holds: There exist positive constants and so that for all and all
Note that for , , and thus . It can be proved that the number of points in the -net satisfies the following bounds:
Next, we list the standing assumptions for this section. General setup: Let and assume that
, for some , and .
In particular, we remark that
if condition (i) is satisfied, then satisfies the admissibility condition (1.10);
condition (ii) is satisfied if
For the proof of our result we need (C, , Theorem 4), which we state here in our notation:
(C, , Theorem 4) Assume that the function satisfies the following two conditions:
, for some , and .
Then there exists so that for any , we can find two constants (depending on , , , and ) so that, for any ,
We will now show that under the general setup and with the discretization of the unit sphere in term of the –net introduced above, there exists a discrete frame for .
Let be as in the general setup and let . Then there exists a so that (4.1) holds for any given i.e., the orthogonal projection of onto forms a frame for .
Proof. Let be defined as in (1.4). Then,