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Shearlet approximations to the inverse of a family of linear operators
Journal of Inequalities and Applications volume 2013, Article number: 11 (2013)
The Radon transform plays an important role in applied mathematics. It is a fundamental problem to reconstruct images from noisy observations of Radon data. Compared with traditional methods, Colona, Easley and etc. apply shearlets to deal with the inverse problem of the Radon transform and receive more effective reconstruction. This paper extends their work to a class of linear operators, which contains Radon, Bessel and Riesz fractional integration transforms as special examples.
1 Introduction and preliminary
The Radon transform is an important tool in medical imaging. Although can be recovered analytically from the Radon data , the solution is unstable and those data are corrupted by some noise in practice . In order to recover the object f stably and control the amplification of noise in the reconstruction, many methods of regularization were introduced including the Fourier method, singular value decomposition, etc. . However, those methods produced a blurred version of the original one.
where the recovered function f is compactly supported and twice continuously differentiable away from a smooth edge; W denotes a Wiener sheet; ε is a noisy level. Because curvelets have complicated structure, Colonna, Easley, etc. used shearlets to deal with the problem (1.1) in 2010 and received an effective reconstructive algorithm .
Note that the Bessel transform and the Riesz fractional integration transform arise in many scientific areas ranging from physical chemistry to extragalactic astronomy. Then this paper considers a more general problem,
where K stands for a linear operator mapping the Hilbert space to another Hilbert space Y and satisfies
with , ( is the conjugate operator of K). Here and in what follows, denotes the Fourier transform of f. The next section shows that Radon, Bessel and Riesz fractional integration transforms satisfy the condition (1.3).
The current paper is organized as follows. Section 2 presents three examples for (1.3) and several lemmas. An approximation result is proved in the last section, which contains Theorem 4.2 of  as a special case.
At the end of this section, we introduce some basic knowledge of shearlets, which will be used in our discussions. The Fourier transform of a function is defined by
The classical method extends that definition to functions.
There exist many different constructions for discrete shearlets. We introduce the construction  by taking two functions of one variable such that with their supports , and
Here, stands for infinitely many times differentiable functions on the Euclidean space . Then two shearlet functions , are defined by
To introduce discrete shearlets, we need two shear matrices
and two dilation matrices
Define discrete shearlets for , , and . Then there exists such that
forms a Parseval frame of , where . More precisely, for ,
holds in . It should be pointed out that are modified for and , as seen in .
2 Examples and lemmas
In this section, we provide three important examples of a linear operator K satisfying and present some lemmas which will be used in the next section. To introduce the first one, define a subspace of ,
and a Hilbert space
with the inner product .
Example 2.1 Let and be the Euclidean measure on the line . Then the classical Radon transform R: defined by
where . Moreover, for each .
Because is dense in , one receives the desired conclusion . Here, for . In fact, by , where is the Riesz fractional integration transform defined by
with some normalizing constant . We rewrite . Let , , where stands for the unit ball of and represents an indicator function on the set A. Then by Theorem 9 of reference [, p.59], where Mf is the Hardy-Littlewood maximal function of f.
On the other hand, the Holder inequality implies
with . Take , one gets and , since due to the assumption and , .
In order to introduce the next example, we use to denote the convolution of f and g. □
Example 2.2 The Bessel operator defined by with and satisfies
Proof It is known that for . Hence, . For , . Thus,
To introduce the Riesz fractional integration transform, we define
Then (). For and , the Riesz fractional integration transform is defined by
Lemma 2.1 Let be the Schwartz space and with being the non-negative integer set. Define . Then with ,
holds for each .
Example 2.3 The transform defined by (2.1) satisfies for and .
Proof As proved in Examples 2.1, 2.2, it is sufficient to show that for ,
One proves (2.2) firstly for . Take with and
Define . Then and . By Lemma 2.1,
Let be the inverse Fourier transform of the function and . Then and . Moreover, the classical approximation theorem  tells
for . On the other hand, due to Theorem 16 [, p.69]. Hence, . That is,
in sense. Note that ; with and . Then
thanks to the Lebesgue dominated convergence theorem, which means in sense. This with (2.3), (2.4) shows (2.2) for .
In order to show (2.2) for , one can find such that and () by Theorem 4.2.1 in , where . Since , the above proved fact says
The same arguments as (2.4) and (2.5) show that and . Hence,
This completes the proof of (2.2) for . □
Next, we prove a lemma which will be used in the next section. For convenience, here and in what follows, we define with , . Then the shearlet system (introduced in Section 1) can be represented as , where if , and if .
Lemma 2.2 Let K satisfy and be shearlets introduced in the first section. Define and . Then and for ,
Proof By the Plancherel formula and the assumption , one knows that . Moreover,
due to and .
Next, one shows . Note that , and . Then and
Because , one receives . This completes the proof of Lemma 2.2. □
At the end of this section, we introduce two theorems which are important for our discussions. As in , we use to denote all sets with boundary ∂B given by
in a polar coordinate system. Here, and . Define , where are compactly supported on . Let , and
Then with standing for the cardinality of , the following conclusion holds .
Theorem 2.3 For , and
Theorem 2.4 
Let and with . Then
where denotes the normal distribution with mean u and variance 1, while is the soft thresholding function.
3 Main theorem
In this section, we give an approximation result, which extends the result [, Theorem 4.2] from the Radon transform to a family of linear operators. To do that, we introduce a set of significant shearlet coefficients as follows. Let
and , . Define , where
Consider the model with . Lemma 2.2 tells that , where is Gaussian noise with zero mean and bounded variance . Let and with
where is the soft thresholding function. Then the following result holds.
Theorem 3.1 Let be the solution to with and be defined as above. Then
Here and in what follows, E stands for the expectation operator.
Proof Since is a Parseval frame, and , . Moreover, and
In order to estimate , one observes due to Theorem 2.3. Then . By ,
(Here and in what follows, denotes for some constant ).
Next, one considers for and . Note that (, ). Then . Since , and
On the other hand, for . Hence, for . Moreover, , since m can be chosen big enough. Therefore,
due to the choice of . The similar (even simpler) arguments show with . This with (3.2) and (3.3) leads to
Finally, one estimates . By the definition of , . Applying Theorem 2.4 with , one obtains that
Hence, . By for , one knows that
Note that . Then , and . Since is uniformly bounded, . This with the choice of shows that
It remains to estimate . Clearly,
By Theorem 2.3, . Hence,
On the other hand, . According to Theorem 2.3, and
Combining this with (3.7) and (3.8), one knows that . Furthermore,
thanks to . Now, it follows from (3.5), (3.6) and (3.9) that
This with (3.1) and (3.4) leads to the desired conclusion . The proof is completed. □
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This work is supported by the National Natural Science Foundation of China (No. 11271038) and the Natural Science Foundation of Beijing (No. 1082003).
The authors declare that they have no competing interests.
LH and YL finished this work together. Two authors read and approved the final manuscript.
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Hu, L., Liu, Y. Shearlet approximations to the inverse of a family of linear operators. J Inequal Appl 2013, 11 (2013). https://doi.org/10.1186/1029-242X-2013-11
- inverse problems
- Radon transform