Compactly Supported Curvelet-Type Systems
We study a flexible method for constructing curvelet-type frames. These curvelet-type systems have the same sparse representation properties as curvelets for appropriate classes of smooth functions, and the flexibility of the method allows us to give a constructive description of how to construct curvelet-type systems with a prescribed nature such as compact support in direct space. The method consists of using the machinery of almost diagonal matrices to show that a system of curvelet molecules which is sufficiently close to curvelets constitutes a frame for curvelet-type spaces. Such a system of curvelet molecules can then be constructed using finite linear combinations of shifts and dilates of a single function with sufficient smoothness and decay.
Second-generation curvelets were introduced by Candès and Donoho, who also proved that curvelets give an essentially optimal sparse representation of images (functions) that are except for discontinuities along piecewise -curves . It follows that efficient compression of such images can be archived by thresholding their curvelet expansions. Curvelets form a multiscale system with effective support that follows a parabolic scaling relation . Moreover, they also provide an essentially optimal sparse representation of Fourier integral operators  and an optimal sparse and well organized solution operator for a wide class of linear hyperbolic differential equations . However, curvelets are band-limited, and contrary to wavelets it is an open question whether compactly supported curvelet-type systems exist.
In this paper we study a flexible method for generating curvelet-type systems with the same sparse representation properties as curvelets (when sparseness is measured in curvelet-type sequence spaces). The method uses a perturbation principle which was first introduced in , further generalized in , and refined for frames in . We give a constructive description of how to construct curvelet-type system consisting of finite linear combinations of shifts and dilates of a single function with sufficient smoothness and decay. This gives the flexibility to construct curvelet-type systems with a prescribed nature (see Section 6) such as compact support in direct space. For the sake of convenience the construction will only be done in , but it can easily be extended to . The main results can be found in Sections 4 and 5.
The curvelet-type sequence spaces we use are associated with curvelet-type spaces which were introduced in . Here was constructed by applying a curvelet-type splitting of the frequency space to a general construction of decomposition spaces, thereby obtaining a natural family of smoothness spaces for which curvelets constitute frames (see Section 2). Originally, this construction of decomposition spaces based on structured coverings of the frequency space was introduced by Feichtinger and Gröbner  and Feichtinger . For example, the classical Triebel-Lizorkin and Besov spaces correspond to dyadic coverings of the frequency space (see ).
The outline of the paper is as follows. In Section 2 we define second-generation curvelets and curvelet-type spaces. Furthermore, we introduce curvelet molecules which will be the building blocks for our compactly supported curvelet-type frames. Next, in Section 3 we use the properties of curvelet molecules to show that the “change of frame coefficient” matrix is almost diagonal if the curvelet molecules have sufficient regularity. With the machinery of almost diagonal matrices, we can then in Section 4 show that curvelet molecules which are close enough to curvelets constitute frames for the curvelet-type spaces. Finally, in Section 5 we give a constructive description of how to construct these curvelet molecules from finite linear combinations of shifts and dilates of a single function with sufficient smoothness and decay. We conclude the paper with a short discussion in Section 6 of the possible functions which can be used to construct the curvelet molecules.
2. Second-Generation Curvelets
We begin this section with a brief definition of curvelets and curvelet molecules which will later be used to construct curvelet-type frames. Furthermore, we define the curvelet-type spaces for which curvelets constitute frames. For a much more detailed discussion of the curvelet construction, we refer the reader to [1, 3], and for decomposition spaces, of which the curvelet-type spaces are a subclass, we refer to [7, 8].
Let be an even window that is supported on such that its -periodic extension obeys Define for and . Next, with the angular window in place, let obey with supported in a neighborhood of the origin. We then define Notice that the support of is contained in a rectangle given by where is determined uniquely for a minimal and ( depends weakly on , see [1, Section 2.2]). With and the system is an orthonormal basis for .
We let , and by duality extend it uniquely from to . Finally, we define where is rotation by the angle , and as coarse-scale elements we define , where and is sufficiently small. The system is called curvelets, . It can be shown that curvelets constitute a tight frame for (see [1, Section 2.2]).
To later construct curvelet-type frames, we need a system of functions which share the essential properties of curvelets. As we will see, curvelet molecules, which were introduced in  and used there to study hyperbolic differential equations, have all the properties we need. For , we define , and for suitably differentiable functions we define .
Definition 2.1. A family of functions is said to be a family of curvelet molecules with regularity , , if for they may be expressed as where and all functions satisfy the following.(i)For there exist constants independent of such that (ii)There exist constants independent of such that The coarse-scale molecules, , must take the form , where satisfies (2.8).
It can be shown that curvelets constitute a family of curvelet molecules with regularity for any (see [3, page 1489]).
To define the curvelet-type spaces which together with the associated sequence spaces will characterize the sparse representation properties of curvelets, we need a suitable partition of unity.
Definition 2.2. Let for , where was defined in (2.3). A bounded admissible partition of unity (BAPU) is a family of functions satisfying:(i); (ii);(iii).
Definition 2.3. Let be a BAPU and . For and , we define as the set of distributions satisfying
We also need the sequence spaces associated with the curvelet-type spaces. For the sake of convenience, we write instead of when the index set is clear from the context.
Definition 2.4. For and , we define the sequence space as the set of sequences satisfying where the -norm is replaced with the -norm if .
Notice that the sequence spaces are special cases of as we have .
Next, we introduce frames for and use the notation when there exist two constants , depending only on “allowable” parameters, such that .
Definition 2.5. We say that a family of functions in the dual of is a frame for if for all we have The following is called the frame expansion of when it exists: in the sense of , where is the frame operator , .
Proposition 2.6. Assume that , and . For any finite sequence , one has Furthermore, is a frame for with frame operator ,
Notice that frame expansions for two frames and have the same sparseness when measured in the associated sequence space if and also constitute frames for , Hence, to get a curvelet-type system with the same sparse representation properties as curvelets , it suffices to prove that constitutes a frame for .
3. Almost Diagonal Matrices
To generate curvelet-type frames in the following sections we introduce the machinery of almost diagonal matrices in this section. Almost diagonal matrices where used in  on Besov spaces, and here we find an associated notion of almost diagonal matrices on . The goal is to find a definition so that the composition of two almost diagonal matrices gives a new almost diagonal matrix and almost diagonal matrices are bounded on .
To help us define almost diagonal matrices we use a slight variation of the pseudodistance introduced in  which was constructed in . For this we need the center of in direct space, , and the “direction” of .
Definition 3.1. Given a pair of indices and , we define the dyadic-parabolic pseudodistance as where
The dyadic-parabolic distance was studied in detail in , and from there we can deduce the following properties.(i)For there exists such that (ii)For there exists such that (iii)For and there exists such that (iv)Let and be two families of curvelet molecules with regularity , . Then there exists such that
These properties lead us to the following definition of almost diagonal matrices on .
Definition 3.2. Assume that , and . Let and . A matrix is called almost diagonal on if there exists such that
Remark 3.3. Note that by using (3.5), we get that the composition of two almost diagonal matrices on gives a new almost diagonal matrix on .
We are now ready to show the most important property of almost diagonal matrices; they act boundedly on the curvelet-type spaces.
Proposition 3.4. If is almost diagonal on , then is bounded on .
Proof. We only prove the result for as the result for follows in a similar way with replaced by . Let , and assume for now that . We begin with looking at the -norm of . By using Minkowski’s inequality, Hölder’s inequality and (3.3) we get We then have For we use Hölder’s inequality and (3.4) to get For the result follows by a direct estimate. The case remains, and here we first observe that is almost diagonal on . Furthermore, if we let we have Before we can put these two observations into use, we need that We then have
4. Curvelet-Type Frames
In this section we study a family of curvelet molecules which is a small perturbation of curvelets . The goal is first to show that if is close enough to , then is a frame for . Next to get a frame expansion, we show that is also a frame. The results are inspired by  where perturbations of frames were studied in Triebel-Lizorkin and Besov spaces.
Let be a system that is close to in the sense that there exists such that for where , and all functions satisfy the following.(i)For we need (ii)Furthermore we need
Then is a family of curvelet molecules with regularity and motivated by being a tight frame for , we formally define as It follows from (3.6) and Proposition 3.4 that is a bounded linear functional on ; in fact we have Furthermore, is a norming family for as it satisfies . This can be used to show that is a bounded operator on , and for small enough this will be the key to showing that is a frame for .
Theorem 4.1. There exists such that if satisfies (4.1) for some , then one has
Proof. That is a norming family gives the upper bound; thus we only need to establish the lower bound. For this we use that is also a norming family so we have It then follows that By choosing we get the lower bound.
As one might guess from Theorem 4.1, the boundedness of the matrix on is the key to showing that is also a frame for .
Proposition 4.2. There exists such that if satisfies (4.1) for some and furthermore is a frame for , then is bounded on .
Proof. The fact that is a frame for ensures that is a bounded operator on . We first show that is bounded on . This will follow from showing that choosing small enough and using the Neumann series. Assume for a moment that satisfies By using that is self-adjoint, we then have So to show (4.9) it suffices to prove (4.10). Note that By setting we have the decomposition Since is a family of curvelet molecules with regularity , we have from (3.6) that are almost diagonal on . Next, we use Remark 3.3, and by Proposition 3.4, Consequently, (4.9) holds, and for sufficiently small the operator is bounded on . Finally, let and . By using (2.9) we have that , and as is a frame for , we have that is self-adjoint which gives If we combine this with being a norming family (4.5), we get which proves that is bounded on .
That is a frame for now follows as a consequence of being bounded on . We state the following results without proofs as they follow directly in the same way as in the Besov space case. The proofs can be found in . First, we have the frame expansion.
Lemma 4.3. Assume that is a family of curvelet molecules with regularity and a frame for . If is bounded on , then for one has in the sense of .
Next, we have that is a frame for .
Theorem 4.4. Assume that is a family of curvelet molecules with regularity and a frame for . Then is a frame for if and only if is bounded on .
It follows from Proposition 4.2, Lemma 4.3, and Theorem 4.4 that if is a family of curvelet molecules which is close enough to curvelets, then the representation , has the same sparse representation properties as curvelets when measured in . As a final result we also have a frame expansion with .
Lemma 4.5. Assume that is a family of curvelet molecules with regularity and a frame for . If the transpose of is bounded on , then for one has in the sense of .
All that remains now is to construct a flexible family of curvelet molecules which is close enough to curvelets in the sense of (4.1).
5. Construction of Curvelet-Type Systems
In this section we construct a flexible curvelet-type system. We do this by showing that finite linear combinations of shifts and dilates of a function with sufficient smoothness and decay can be used to construct a system that satisfies (4.1). From the previous section, we then have that the representation , , has the same sparse representation properties as curvelets when measured in .
First we take , , which for fixed satisfies Next, for we define , where . It then follows that We recall that curvelets (2.6) are a family of curvelet molecules for any regularity . From the definition of a family of curvelet molecules (Definition 2.1), we have that for curvelet molecules can be expressed as where must satisfy (2.8) and (2.9). So to construct a family of curvelet molecules which satisfy (4.1), we need to construct a family of functions such that satisfy (4.2) and (4.3). We define as for and to construct we also need the following set of finite linear combinations: We have omitted the construction of for as it follow in a similar way.
Proposition 5.1. Let and . If , fulfills (5.1) and fulfills then for any there exist ( independent of ) and such that
Proof. Let and , be given. We construct the approximation of in direct space in three steps. First by a convolution operator , then by which is the integral in taken over a dyadic cube , and finally by a discretization over smaller dyadic cubes . From (5.2) we have Define , where . For , we use the mean value theorem to get Inserting this in (5.8) we have For , we split the integral over and . If , then , and we have Integrating over with gives So by choosing sufficiently large in (5.10)–(5.12), we get For the next step we fix and choose . Let denote the dyadic cube with sidelength , sides parallel with the axes and centered at the origin. We then approximate with defined as In which case we have and it follows that We first estimate the integral for which gives and . Hence we obtain For , we split the integral over and . If , then , and we get Similar for we have By choosing sufficiently large in (5.17)–(5.19), we obtain For the final step we fix , choose , and approximate by a discretization where is the center of the dyadic cube and is the set of dyadic cubes with sidelength which together give . Note that , . We introduce which gives By using the mean value theorem, we then get where is the line segment between and . If and , then we have For and , we have , and hence for , it follows that By choosing sufficiently large, we obtain by combining (5.23)–(5.25) that Finally by combining (5.13), (5.20) and (5.26), we get To approximate in frequency space we use three steps similar to the approximation in direct space. Note that still fulfills (5.27) if we choose even larger. First we use to approximate in which case we have By choosing such that and such that for , we get Next, we fix , choose and limit the Fourier integral of to from the approximation in direct space, This gives In the last step, we fix and approximate by . We introduce which gives By combining (5.29)–(5.32) for sufficiently large , we get It follows that by choosing large enough fulfills both (5.7) and (5.4). Furthermore, we have , .
In this paper we studied a flexible method for generation curvelet-type systems with the same sparse representation properties as curvelets when measured in . With Proposition 4.2, Lemma 4.3, and Theorem 4.4 we proved that a system of curvelet molecules which is close enough to curvelets has these sparse representation properties. Furthermore, with Proposition 5.1 we gave a constructive description of how such a system of curvelet molecules can be constructed from finite linear combinations of shifts and dilates for a single function with sufficient smoothness and decay.
Examples of functions with sufficient smoothness and decay are the exponential function and the rational functions with sufficiently large. An example with compact support can be constructed by using a spline with compact support. Furthermore as the system is constructed using finite linear combinations of splines, we get a system consisting of modulated compactly supported splines.
H. Triebel, Theory of Function Spaces, vol. 78 of Monographs in Mathematics, Birkhäuser, Basel, switzerland, 1983.