This paper is concerned with inverse problems of the form
where is a linear operator between Hilbert spaces, and denotes the data distortion (noise). We allow unbounded operators and assume that is dense. Moreover, we assume that the unknown object is an element of a closed subspace space on which is bounded. We are particularly interested in problems, where (1.1) is ill-posed in which case the solution of (1.1) (if existent) is either not unique or the solution operator is not continuous (hence, solution process is unstable with respect to data perturbations). In order to stabilize the inversion of (1.1) one has to apply regularization methods, cf. [11, 23]. The basic idea of regularization is to include a-priori information about the unknown object into the solution process.
In this paper, we use sparsity based regularization, where the a-priori assumption on the unknown object is sparsity of with respect to a frame of , cf. [23, 14, 7, 17, 21, 3]. That is, we regularize the recovery of from measurements (1.1) by enforcing sparsity of with respect to a suitably chosen frame of . Sparse regularization is well investigated and has been applied to many different imaging problems, and by now there are many algorithms available that implement sparse regularization. However, when dealing with frames, there are at least two fundamentally different concepts implementing sparsity, namely the synthesis and the analysis variant. The reason for this lies in the fact that expansions of with respect to frames are not unique (which is in contrast to orthonormal basis expansions). In the synthesis variant, it is assumed that the unknown is a sparse linear combination of frame elements, whereas, in the analysis variant, it is required that a the inner products with respect to a given frame are sparse. The difference between these approaches has been pointed out clearly in .
Sparse regularization is widely used in inverse problems as it provides good regularization results and is able to preserve or emphasize features (e.g., edges) in the reconstruction. However, this often comes at the price of speed, since most of the algorithms implementing sparse regularization are based on variational formulations that are solved by iterative schemes.
In the present paper, we investigate a third variant of sparse regularization that is based on an operator adapted biorthogonal frame decomposition (BFD) of the unknown object, cf. [9, 4, 6] and which allows to define a direct (non-iterative) sparse regularization method. In the noise-free case (), explicit reproducing formulas for the unknown object can be derived from the BFD, where the frame coefficients of are calculated directly from the data . In the presence of noise (), regularized versions of those formulas are obtained by applying component-wise soft-thresholding to the calculated coefficients, where the soft-thresholding operator is defined as follows:
Definition 1.1 (Soft-thresholding).
Let be some index set.
For let .
For we define the component wise soft-thresholding by
Here and below we define for and . The advantage of the BFD-variant of sparse regularization lies in the fact that it admits a non-iterative (direct) and fast implementation of sparse regularization that can be easily implemented for several inverse problems.
We point out, that the three variants of sparse regularization (mentioned above) are equivalent if orthonormal bases are used instead of frames, but they are fundamentally different in the redundant case.
As the main theoretical results in this paper we show that the thrid variant of sparse regularization, which we call BFD-thresholding, defines a convergent regularization method and we derive linear convergence rates for sparse solutions. For the basis case, same results follow from existing results of -regularization [7, 14, 15]. In the redundant case, the results follow from  for the synthesis approach and from  for the analysis approach. In case of BFD-thresholding, we are not aware of any results concerning convergence analysis or convergence rates.
This paper is organized as follows. In Section 2 we define the biorthogonal frame decompositions of operators and give several examples of biorthogonal frame expansions for various operators using wavelet, curvelet, and shearlet frames. In section 3 we review the convergence theory of -regularization and the convergence rates. In section 4, we show that BFD-thresholding is a convergent regularization method and derive its convergence rates.
2 Biorthogonal frame decomposition
In this section, we introduce the concept of operator adapted biorthogonal frame decompositions (BFD) and discuss some classical examples of such BFDs in the case of the classical 2D Radon transform and the forward operator of photoacoustic tomography with a flat observation surface.
2.1 Formal definition
We define the operator adapted biorthogonal frame decomposition as a generalization of the wavelet vaguelette decomposition and the biorthogonal curvelet or shearlet decompositions to general frames, cf. [9, 4, 6].
Definition 2.1 (Biorthogonal frame decomposition (BFD)).
Let and let be a family of positive numbers. For a linear operator , we call a biorthogonal frame decomposition (BFD) for , if the following conditions hold:
is a frame of ,
is a frame of ,
The BFD generalizes the singular value decomposition (SVD) and the wavelet-vaguelette decomposition (WVD) (cf.
The BFD generalizes the singular value decomposition (SVD) and the wavelet-vaguelette decomposition (WVD) (cf.) as it allows the systems and to be non-orthogonal and redundant. Note that by 2 and 3, then frame satisfies , where we have made use of the identity . For typical inverse problems this yields a notable smoothness assumption on the involved elements of the frame .
Although the SVD has proven itself to be a useful tool for analyzing and solving inverse problems it also can have the following drawbacks: First, ONBs that are provided by the SVD (though optimally adapted to the operator in consideration), in many cases, don’t provide sparse representations of signals of interest and, hence, are not suitable for the use in sparse regularization. In particular, frames that provide sparse representation of signals (such as wavelets or wavelet-like systems) are often not part of an SVD. Second, SVD is often very hard to compute or not known analytically for many practical applications.
To overcome some of those difficulties wavelet-vaguelette decompositions were introduced. Nevertheless, this concept builds upon expansions of signals with respect to orthogonal wavelet-systems, which may not provide an optimal sparse representation of signals of interest, e.g., signals with sharp edges. Thus, by allowing general frames, the BCD offers great flexibility in the choice of a suitable function system for sparse regularization while retaining the advantages.
For a frame of , the synthesis operator is defined as and the corresponding analysis operator is defined as its adjoint, .
In what follows, the synthesis operator of a frame will be always denoted with the corresponding upper case letter, e.g., if is a frame, then denotes the corresponding synthesis and analysis operator.
In order to simplify the notation, we will also refer to the as BFD instead of using the full notation .
If a BFD exists for an operator , it immediately gives rise to a reproducing formula
where is the dual frame to (cf. ) and the corresponding synthesis operator. Moreover, denotes the operator performing pointwise division with , i.e.
Hence, from given (clean) data , one can calculate the frame coefficient of and obtain a reconstruction via (2.1). The key to the practical use of this reproducing formulas is the efficient implementation of the analysis and synthesis operators and , respectively. For particular cases, we will provide efficient and easy to implement algorithms for evaluation of and .
Note that, the reproducing formula (2.1) (similarly to the SVD) reveals the ill-posedness of the operator equation through the decay of the quasi-singular values . A regularized version of the reproducing formula (2.1) can be obtained by incorporating soft-thresholding of the frame coefficients. In section 4, we present a complete analysis of this approach as we are not aware of any results for the general BFD-thresholding in the context of regularization theory.
We now provide several examples of BFDs, including the wavelet vaguelette decomposition and the biorthogonal curvelet decomposition  for the Radon transform as well as a BFD for the forward operator of photoacoustic tomography with flat observation surface.
2.2 Radon transform
Definition 2.4 (Radon transform).
The Radon transform is defined by
It is well known that the Radon transform is bounded on , see . Let
be the Fourier transform with respect to the first component and consider the Riesz potential
for . The following hold
The commutation relation .
The filtered backprojection formula .
Isometry property .
Using these ingredients, one can obtain a BFD for the Radon transform as follows:
BFD for the Radon transform Let be either an orthonormal basis of wavelets with compact support, a (band-limited) curvelet or a shearlet tight frame with where is the scale index. Then is a BFD with
These results have been obtained in  for wavelet bases, in  for curvelet systems and in  for the shearlet frame. All cases are shown in similar manner and basically follow from 1, 2 and the fact that that for any of the considered systems. The limited data case has been studied in .
Equation (2.5) implies
This gives an efficient numerical algorithm for the evaluation of provided that is associated with an efficient algorithm. This is in particular the case for the wavelet, shearlet and curvelet frames as above.
We would like to note that in order to define a BFD for the case of curvelets or shearlets one needs to consider the Radon transform on subspaces of consisting of functions that are defined on unbounded domains (since band-limited curvelets or shearlets have non-compact support). However, since the Radon transform is an unbounded operator on , the reproducing formula (2.1) will not hold any more in general. The reproducing formulas with are at least available for the case that the object can be represented as a finite linear combination of curvelets or shearlets (cf.  and ).
Another possibility would be to consider projections of curvelet or shearlet frames onto the space , which would yield a frame for this space (cf. ) and then define the BFD in the same way as above. Because the Radon transform is continuous on , the reproducing formula (2.1) will hold for general linear combinations.
Computing BFD coefficients for the Radon transform Let be a wavelet, shearlet of curvelet frame and define by (2.5).
Compute via wavelet, curvelet or shearlet transform.
Apply rescaling .
Output: Coefficients .
2.3 Inversion of the wave equation
We consider a planar geometry, which has been considered in our previous work . Let denote the space of compactly supported functions that are supported in the half space . For consider the initial value problem
Definition 2.8 (Forward operator for the wave equation).
We define by , for , where is the solution of (2.8), and extending it by continuity to .
The isometry property implies that any frame gives a BFD by setting and . This, in particular, yields a wavelet vaguelette decomposition and a biorthogonal curvelet decomposition for the wave equation.
BFD for the wave equation Let be either a wavelet frame, a curvelet frame or a shearlet frame with where is the scale index. Then is a BFD with
As noted in  this result directly follows from the isometry property and the associated inversion formula .
The isometry property also gives an efficient numerical algorithm for computing analysis coefficients with respect to the frame in the case that is associated with an efficient algorithm.
Computing BFD coefficients for the wave equation Let be the curvelet frame and define by (2.9).
Compute via wavelet, curvelet or shearlet transform.
Output: Coefficients .
The algorithm described above can be used for any problem where the forward operator is an isometry. In the case of the wavelet transform this simple procedure has been previously used in .
3 Sparse -regularization
There are two fundamentally different and well studied instances of sparse frame based regularization, namely -analysis regularization and -synthesis regularization. They are defined by
respectively, with weights .
We call sparse if the set is finite.
If is sparse, we write where is the multi-valued signum function defined by and for . We will use the notation
Any element in the set is called -minimizing solution of . Note that -minimizing solutions exists whenever there is any solution with , as follows from [23, Theorem 3.25].
Below we recall well-posedness and convergence results for both variants. These results hold under the following quite week assumptions:
For certain sparse elements we will state linear error estimates which have been derived in[14, 16]. See [7, 17, 21, 3, 23] for some further works on sparse -regularization, and [8, 18, 22] for wavelet regularization methods.
3.1 -Analysis regularization
Let us define the -analysis Tikhonov functional by
Then we have .
Proposition 3.2 (Convergence of analysis regularization).
Existence: The functional has at least one minimizer
Stability: There exists a subsequence of and a minimizer such that . If the minimizer of is unique, then .
Convergence: Assume for with and suppose with . Consider a parameter choice such that . There is an -minimizing solution of and a subsequence with . If the -minimizing solution is unique, then .
See [14, Propositions 5, 6 and 7]. ∎
In order to derive convergence rates, one has to make additional assumptions on the exact solution to be recovered. Besides the sparsity this requires a certain interplay between and the forward operator .
Assumption 5 is a so-called source condition and the main restrictive assumption. It requires that there exists an element that satisfies the smoothness assumption . Because , the space is finite dimensional. Therefore, Condition 6 requires injectivity on a certain finite dimensional subspace.
Proposition 3.3 (Convergence rates for analysis regularization).
See [16, Theorem III.8]. ∎
3.2 Synthesis regularization
Let use denote the -synthesis Tikhonov functional by
Then it holds . Synthesis regularization can be seen as analysis regularization for the coefficient inverse problem and the analysis operator . Using Proposition 3.2 we therefore have the following result.
Proposition 3.4 (Convergence of synthesis regularization).
Existence: The functional has at least one minimizer.
Stability: There exists a subsequence of and such that . If the minimizer of is unique, then .
Convergence: Assume for with and with . Consider a parameter choice with . There exist an -minimizing solution of and a subsequence with . If the -minimizing solution is unique, then .
Follows from Proposition 3.2 with and in place of . ∎
We have linear convergence rates under the following additional assumptions on the element to be recovered.
Proposition 3.5 (Convergence rates for synthesis regularization).
Follows from Proposition 3.3. ∎
Because is bounded, the above convergence results can be transferred to convergence in the signal space . In particular, we have stability and convergence under the assumptions of made in Proposition 3.4, and linear convergence rates .
3.3 Sparse regularization using an SVD
In the special case that is part of an SVD, then analysis and synthesis regularization are equivalent and can be computed explicitly by soft-thresholding of the expansion coefficients.
Theorem 3.6 (Equivalence in the SVD case).
Because is an orthonormal basis of , we have which implies that . Now let be any minimizer of the -analysis Tikhonov functional . Let denote the orthogonal projection on . We have
The latter sum is minimized by componentwise soft-thresholding. This shows and concludes the proof. ∎
are all non-equivalent. Further, in this case (3.1), (3.2) have to be computed by iterative minimization algorithms. This requires repeated application of the forward and adjoint problem and therefore is time consuming. In the following section, we study BFD thresholding which is the analog of (3.6) for redundant systems. Despite the non-equivalence to -regularization, we are able to derive the same type of convergence results and linear convergence rates as for the analysis and synthesis variants of -regularization.
4 Regularization via BFD thresholding
Throughout this section we fix the following assumptions
In this section we show the well-posedness, convergence and convergence rates for BFD soft-thresholding.
4.1 BFD soft-thresholding
Any BFD gives an explicit inversion formula where is defined by (2.2). For ill-posed problems, and therefore the above reproducing formula is unstable when applied to noisy data instead of . Below we stabilize the inversion by including the soft-thresholding operation.
Definition 4.1 (BFD soft-thresholding).
Let be a BFD for . We define the nonlinear BFD soft-thresholding estimator by
As the main result in this paper we show that BFD soft-thresholding yields the same theoretical results as - regularization. Assuming efficient implementations for and , the BFD estimator has the advantage that it can be calculated non-iteratively and is therefore much faster than and .
Consider the -Tikhonov functional for the multiplication operator ,
The proof strategy used in this paper is based on the following Lemma.
Lemma 4.2 (-minimization for multiplication operators).
has a unique minimizer.
Note that the continuity of (see item 2 in the above lemma) is not obvious as is the composition of soft thresholding with the unbounded operator in . The characterization in item 2 as minimizer of the -Tikhonov functional and the existing stability results for -Tikhonov yields an elegant way to obtain the continuity of . Verifying the continuity directly would also be possible but seems harder task. A similar comment applied to the proof of Theorem 4.3 where we use the convergence of .
4.2 Convergence analysis
In this section, we show that is well-posed and convergent.
Theorem 4.3 (Well-posedness and convergence).
4.3 Convergence rates
Next we derive linear convergence rates for sparse solutions. Let use denote by the support of . To derive the convergence rates, we assume the following for the exact solution to be recovered.
is satisfied for some element that can be chosen such that for . Moreover, it follows that is injective on with . Because , we have .
Theorem 4.4 (Convergence rates).
As in the proof of Theorem 4.3 one obtains
According to the considerations below 4,5 the conditions 4–6 are satisfies for the operator in place of and with . The convergence rates result in Proposition 3.3, estimate (4.5), and the identity shown in Lemma 4.2 imply . Together with (4.4) this implies and concludes the proof. ∎
To overcome the inherent ill-posedness of inverse problems, regularization methods incorporate available prior information about the unknowns to be reconstructed. In this context, a useful prior is sparsity with respect to a certain frame. There are at least two different regularization strategies implementing sparsity with respect to a frame, namely -analysis regularization and -synthesis regularization. In this paper, we analyzed BFD-thresholding as a third variant of sparse regularization. One advantage of BFD-thresholding compared to other sparse regularization methods is its non-iterative nature leading to fast algorithms. Besides having a BFD, actually computing the BFD soft-thresholding estimator (4.1) requires the dual frame . While in the general situation, the BFD and the dual frame have to computed numerically, we have shown that for many practical examples (see Section 2) they are known explicitly and efficient algorithms are available for its numerical evaluation.
The BFD-approach presented in this paper is well studied in the context of statistical estimating using certain multi-scale systems. However, its analysis in the context of regularization theory has not been given so far. In this paper we closed this gap and presented a complete convergence analysis of BFD-thresholding as regularization method.
The work of M.H has been supported by the Austrian Science Fund (FWF), project P 30747-N32.
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