Discrete Directional Gabor Frames
We develop a theory of discrete directional Gabor frames for functions defined on the -dimensional Euclidean space. Our construction incorporates the concept of ridge functions into the theory of isotropic Gabor systems, in order to develop an anisotropic Gabor system with strong directional sensitivity. We present sufficient conditions on a window function and a sampling set for the corresponding directional Gabor system to form a discrete frame. Explicit estimates on the frame bounds are developed. A numerical implementation of our scheme is also presented, and is shown to perform competitively in compression and denoising schemes against state-of-the-art multiscale and anisotropic methods, particularly for images with significant texture components.
Given a square integrable function and constants , the associated Gabor system (also known as Weyl–Heisenberg system) generated by and the lattice , , is defined by
In 1946, Dennis Gabor proposed to study such systems for their usefulness in the analysis of information conveyed by communication channels [Gabor46]. The resulting theory led to many applications ranging from auditory signal processing, to pseudodifferential operator analysis, to uncertainty principles. The edited volumes by Benedetto and Frazier [benedetto1993] and by Feichtinger and Strohmer [FS1, FS2], as well as Gröchenig’s treatise [Grochenig], provide detailed treatments of various aspects of this rich and beautiful theory.
Among many interesting recent developments in time-frequency analysis, we want to consider the set of ideas which expands the notion of traditional Gabor systems by including directional information. Directional content is especially important in many applications, such as those in remote sensing or medical imaging, where interesting features often propagate in a particular direction. Thus, it is not surprising that this avenue of research achieved significant success in the closely related field of wavelet analysis. Contourlets [DoV03], curvelets and ridgelets [acha_curvelets1, acha_curvelets2, CG02], bandlets [LePM], wedgelets [Don99], and shearlets [GKL06, LLKW, shearlet_acha, shearlet_edge], are just a few examples of directional multiscale constructions introduced to address the problems of identifying edges and directions in various forms of imagery.
Counter to a common belief that in image analysis wavelet-based techniques are superior to other approaches, Gabor methods play a significant role in several important areas, including fingerprint recognition, texture analysis, and vasculature detection. These examples of data are not covered by typical assumptions about existence of a sparse, curvilinear set of singularities, and thus present a new type of difficulties. To overcome these difficulties in the context of Gabor analysis, an effort similar to that in the directional wavelet theory has been undertaken. Directional Gabor filters have been proposed as a model for multichannel neuronal behavior by Daugman [D1, D2] and Watson [Watson], and to solve problems in texture analysis by Porat and Zeevi [PZ]. Wang et al. proposed to use directional Gabor filters in character recognition [Wang]. Hong, Jain, and Wan proposed to use Gabor ridge filter banks for fingerprint enhancement [Hong]. Gabor methods have also been employed to achieve state-of-the-art performance in image denoising, compared to multiscale anisotropic methods, such as shearlets and curvelets [Gabor_Framelets]. Directional Fourier methods have also been successfully deployed for edge detection [Czaja+Wickerhauser, CW2].
On the theoretical side, Grafakos and Sansing proposed the concept of directional time-frequency analysis in [Grafakos_Sansing]. In this paper, they introduce directional sensitivity in the time-frequency setting by considering projections onto the elements of the unit sphere. The Radon transform arises naturally in this context and it enables continuous and semi-continuous reconstruction formulas. Giv proposed a related variant of short-time Fourier transform [Giv] and established its orthogonality relations, as well as some operator theoretic properties. The relationship with the classical short-time Fourier transform and the quasi-shift invariance of the new scheme has been analysed in [AceskaGiv]. Results from [Grafakos_Sansing] have been extended to more general continuous representations in Sobolev spaces in [CFM]. Gabor systems have also been analyzed from the viewpoint of shearlet theory [gabor_shearlets].
In the present paper, we develop a theory of discrete directional Gabor frames, related to the concept of Gabor ridge systems. In Section 2 we review some of the necessary features of time frequency analysis. In Section 3 we characterize certain minimal constraints on the spaces of functions that can be efficiently represented by directional Gabor systems. An elementary example is provided in Section 4, and Section 5 is concerned with the derivation of sufficient conditions for the existence of discrete directional Gabor systems. We close with some numerical examples in Section 6.
We begin with a review of some relevant background in frame theory and Gabor analysis.
Let be a Hilbert space. A discrete set is called a (discrete) frame for if there exist constants such that:
The optimal choices of are the frame bounds. A frame is said to be tight if . It is Parseval if .
Let be a separable Hilbert space, a locally compact Hausdorff space equipped with a positive Radon measure such that supp(. A set is a continuous frame with respect to for if there exist constants such that:
Definition 2.2 generalizes Definition 2.1 by allowing for a continuous indexing of the frame elements. In the study of continuous frames, the discretization problem arises naturally: when can a continuous frame be discretized to acquire a discrete frame? That is, when can the continuous indexing set be replaced with a discrete indexing set, while still retaining the frame property? A prominent abstract approach to the discretization problem involves the coorbit space theory of Feichtinger and Gröchenig [atomic_decomp_FG], [Fornasier_Rauhut]. This method considers the representations induced by the group structure of the indexing set . Thus, coorbit space theory is most effective when the continuous frame is parametrized by a group. This is the case for continuous wavelet systems and continuous shearlet systems [shearlet_coorbit], which can be parametrized by the affine and shearlet groups, respectively.
There is also substantial interplay between discrete and continuous representations in the context of Gabor theory, that is, between discrete Gabor frames and the short-time Fourier transform. We are interested in studying this interplay in the context of Gabor systems that have a degree of directional sensitivity incorporated. The anisotropic generalization of Gabor systems developed by Grafakos and Sansing is continuous, and the theory they develop proves certain systems are continuous representations for particular function spaces. However, a discrete theory is desired, both out of pure mathematical interest and for applications purposes. Some partial results towards this exist, such as Theorem 2.13. However, a regime for full discretization has not yet been proved. This article develops sufficient conditions for full discretization; see Theorem 5.23.
We now consider some relevant background on classical Gabor theory and the continuous theory of directional Gabor systems. Unless stated otherwise, denotes the usual inner product.
For and , is the function defined by
Let . The Fourier transform of is the function given by
Note that the Fourier transform is uniquely extendable to a unitary operator on ; this result is Plancherel’s Theorem [BenedettoCzaja]. The inverse Fourier transform of is denoted .
Let . The short-time Fourier transform of a function with respect to the window function is the function given by
The short-time Fourier transform is also called a “sliding window Fourier transform,” [Grochenig] or “voice transform,” for its connections to speech processing [STFT_speech]. The window function may be chosen with desired regularity and localization properties. An important property of the short-time Fourier transform is its invertibility. Indeed, the original function, , is recovered from by integrating against translations and modulations of any such that : for such a ,
where convergence is pointwise almost everywhere. In particular, if , we see the short-time Fourier transform is a continuous frame.
The Radon transform, in particular the Fourier slice theorem, shall play a crucial role in the theory of discrete directional Gabor frames. Recall that denotes the space of Schwartz functions on [Hormander].
Let . The Radon transform of is the function given by the formula
The notation shall be used in what follows. The operator can be extended to a continuous operator from to uniformly in . Since the Fourier transform is naturally defined on , it is reasonable to consider its relation to the Radon transform. The Fourier slice theorem describes this relationship.
(Fourier slice theorem) Let . Then the Fourier transform of and are related in the following way:
The continuous theory of Grafakos and Sansing involves weighting the frame elements in the frequency domain. When the weights are polynomials, this corresponds to differentiation in the time domain. Indeed, this motivates the following definition.
For , the differential operator of order for functions is given by
This definition naturally extends to multi-indices for functions .
We now come to the main definition of [Grafakos_Sansing].
Let be a real-valued, not identically zero window function. For , , we define
The weighted Gabor ridge functions generated by are functions given by
The convention of using lower case letters to refer to the generator and capital letters to refer to the weighted Gabor ridge function was established in [Grafakos_Sansing], and shall be maintained throughout this section. We note that ridge functions have seen significant application in the theories of function approximation and neural networks [ridge_Candes], [ridge_Chui].
It is clear that the weighted Gabor ridge function is constant along any hyperplane for constant . Along the direction , these functions modulate like a one dimensional Gabor function. The weighting in the Fourier domain permits reconstruction of a signal from the coefficients
Weighted Gabor ridge functions yield a continuous reproducing formula that is analogous to (2.1), with an additional integral to account for the directional character of the ridge function construction [Grafakos_Sansing].
Let be two window functions such that . Suppose and . Then,
Note that without the weights in the Fourier domain, the reconstruction formula does not hold. Indeed, let , and the adjoint of the Radon transform:
We have the following result [Grafakos_Sansing].
Let for some . Then for with , the following identity holds:
Thus, instead of reconstructing itself, we reconstruct the weighted back-projection . Indeed, one may compute that for a suitable constant ,
so that we do not recover perfectly in general. Here, the fractional power of the Laplacian is understood in the sense of a pseudodifferential operator.
There is also a Parseval-type formula for Gabor ridge functions, Corollary 1 in [Grafakos_Sansing]:
Let and let be not identically . There exists a constant , depending only on , such that
Note that Theorem 2.12 expresses the fact that the Gabor ridge system
forms a continuous representation for .
In the case of Gabor ridge functions, the continuous system is indexed by the set . In general, this set does not admit a non-trivial group structure. Indeed, the only spheres which admit non-trivial Lie structures are and [Hofmann]. Thus, the use of co-orbit theory to prove a discretization appears challenging. We shall prove sufficient conditions for the discretization of Gabor ridge systems, but using techniques other than coorbit theory.
As a starting point, consider the following semi-discrete representing formula for Gabor ridge systems, which is Theorem 5 in [Grafakos_Sansing].
There exist and such that for all , we have have:
where the constants depend only on . Moreover, for this choice of , we have
The representation (2.2) is called semi-discrete because there are two discrete sums, but also an integral over . We are interested in a full discretization of the Gabor ridge system. Grafakos and Sansing did not present such a theory in [Grafakos_Sansing], though they suggested the co-orbit theory of Feichtinger and Gröchenig might be used. We pursue a different approach, based on the work of Hernández, Labate, and Weiss [HLW].
3 Finding a Space of Functions to Represent
We begin by noting that, once discretized, the frequency weights used in the weighted Gabor ridge construction are not needed. Indeed, the frequency weights are needed in the continuous regime to perform a continuous change of variables to address the fact that in general. Moreover, when considering the discrete implementation of our construction, the weights will have the effect of imposing additional averaging of the experimental data. Our target image class is textures, which will be adversely affected by such averaging. Hence, there is also numerical significance attached to not considering the frequency weights at present. Thus, we consider frame elements of the form
for a function . We seek a discrete system of the form
along with a space of functions for which this set will be a discrete frame. The indexing set must be discrete with respect to the natural topology of . The nomenclature used to describe such a system shall be discrete directional Gabor frame.
A first, somewhat naive, approach to developing a discrete directional Gabor frame is to start with the semi-discrete representation, Theorem 2.13, and choose a fixed, discrete set of points on the unit circle at which to sample. This would essentially be replacing the integral in (2.2) with a finite sum. To investigate this approach, we consider systems of the form
where is an arbitrary uniformly discrete set, and is an arbitrary finite set. Note that since is compact, is finite if and only if it is discrete. We show that such a system cannot be a frame for any reasonable space of functions. This proof is based on ideas found in Chapter 4 of [Lax_Zalcman].
Let be finite and let . Then there exists with support in such that and
These equalities follow from Parseval’s identity and the Fourier slice theorem, respectively. Let be the lines . Let be a polynomial vanishing on these lines, and let have support in . Then vanishes on each of the lines ; in particular,
Let and define , where is a normalization constant so that . By construction, is supported in , and , so that
where . It follows that , as desired.
Let , let be an arbitrary discrete set, and let be a fixed finite set. Let be fixed. Then the system
is not a frame for any subspace of which contains . In fact, it cannot even be Bessel for such a space.
By Lemma 1 in [Grafakos_Sansing], . Then:
By Proposition 3.14, we may find , , such that
Noting that this forces
the result is proved. ∎
Thus, infinitely many directions from must be included in any discrete directional Gabor frame representing Schwartz functions supported on some open set about the origin.
This leads us to consider what space of functions can be represented with a discrete directional Gabor frame. The following result demonstrates that we cannot have a discrete directional Gabor frame for any function space that contains , regardless of sampling set.
Let be any discrete directional Gabor system. Then there exists a sequence such that
In particular, cannot be a frame for any function space containing ; it cannot even be Bessel.
We will show that may be chosen with the property that
for a fixed . By rotating as needed, we may assume without loss of generality that . Then we compute:
Now, let us choose such that
where is chosen such that is uniformly bounded in , and such that
The are chosen such that for all , and
which gives the desired result.∎
So, it is clear that in order to discretize a Gabor ridge system to acquire a discrete directional Gabor frame, infinitely many directions on the sphere must be incorporated, and our space of functions must be chosen very specifically. Indeed, we must shrink the space of functions not to include all of . In the spirit of classical sampling, we will consider functions supported in a fixed compact domain.
4 An Elementary Example
We show it is possible to produce a discrete directional Gabor system representing a space of functions with restricted support. This example is based on manipulating directional Gabor systems to act like Fourier series.
Let , the indicator function on . Let be such that the mapping given by is a bijection. Set
Then we have
for all where .
We remark that the penultimate line is simply the sum of the Fourier coefficients of . This example shows the existence of a tight discrete directional Gabor frame for the subspace of consisting of functions supported in . Note that in this proof, we make no use of the translations in the system . Indeed, the window is supported on , and we consider functions supported only on the ball . It is of interest if this parameter set can be made non-trivial, to produce more interesting examples of discrete directional Gabor frames.
5 Sufficient Conditions for a Discrete System
In this section, we develop sufficient conditions on the window function and sampling set for the discrete directional Gabor system
to be a frame. We will consider a particular class of indexing sets, parametrized by real numbers that determine the localization of the window function . We will thus refer to our index sets as , to denote this parametrization.
Let be the subset of with support in .
We prove sufficient conditions for the existence of discrete directional Gabor frames using tools developed by Hernández, Labate, and Weiss [HLW]. This theory uses absolutely convergent almost periodic Fourier series to provide necessary and sufficient conditions for certain functions to generate discrete frames. Among the classes of functions they characterize are classical discrete Gabor and discrete wavelet systems. In the process of developing sufficient conditions for discrete directional Gabor frames, we show their methods are partially extensible to certain anisotropic systems.
Recall that .
Let , . Then for a fixed
By Lemma 1 in [Grafakos_Sansing] and the Fourier slice theorem,
Periodizing this integral, we see
To transition from the penultimate to the ultimate line, we applied Parseval’s formula to the function
To go from the penultimate to ultimate line, we sum over and exploit the periodization. Since is Schwartz class, we may interchange the integral and sum. This gives the result.
Let be as in Theorem 4.17, i.e., let be such that the mapping given by is a bijection. Let where . While many such sets exists, one simple construction is
for arbitrary . Note that by construction, for , which implies also does not split as a direct product. Moreover, with defined generally as above, the sampling set is indeed discrete.
The set is a discrete subset of .
It suffices to show that is a discrete subset of . Let and be the smallest positive real number so that . First suppose . Let . Notice that is finite and . So we can choose a neighborhood so that . Let . Now, suppose . This implies and so . Thus . Since when , then we must have . Hence .
It remains to prove the case when . Since is a bijection between and , there is precisely one so that . We set . Since every satisfies except when , it follows that . Therefore, is discrete.
We are now prepared to give a sufficient condition for a discrete directional Gabor frame. A key ingredient in the proof is a multidimensional version of Kadec’s -Theorem [2D_Kadec].
Let and suppose there exists such for all , the associated satisfies
Then the collection is a Riesz basis for , with frame bounds .
We also require a result on perturbing Fourier frames to acquire Bessel systems. The following result appears in [Duffin_Schaeffer] in the case ; we require the result for arbitrary . The result is probably known, but we give a proof for completeness.
Suppose is such that there exists such that for all ,
Let and be such that . Then
where for any . In particular, if , then .
Since , is supported in . So, by Schwartz’s Paley-Wiener Theorem [Hormander], is analytic on . Consider its power series expansion about a fixed
where is a multi-index. Then for any , an application of the Cauchy-Schwarz inequality yields:
Recalling the correspondence between polynomial multiplication and the Fourier transform, , and thus . We may thus sum over as follows: