- Open Access
Shearlet approximations to the inverse of a family of linear operators
© Hu and Liu; licensee Springer 2013
- Received: 14 August 2012
- Accepted: 18 December 2012
- Published: 7 January 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.
- inverse problems
- Radon transform
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 .
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.
The classical method extends that definition to functions.
holds in . It should be pointed out that are modified for and , as seen in .
with the inner product .
where . Moreover, for each .
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.
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. □
holds for each .
Example 2.3 The transform defined by (2.1) satisfies for and .
thanks to the Lebesgue dominated convergence theorem, which means in sense. This with (2.3), (2.4) shows (2.2) for .
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 .
due to and .
Because , one receives . This completes the proof of Lemma 2.2. □
Then with standing for the cardinality of , the following conclusion holds .
Theorem 2.4 
where denotes the normal distribution with mean u and variance 1, while is the soft thresholding function.
where is the soft thresholding function. Then the following result holds.
Here and in what follows, E stands for the expectation operator.
(Here and in what follows, denotes for some constant ).
This with (3.1) and (3.4) leads to the desired conclusion . The proof is completed. □
This work is supported by the National Natural Science Foundation of China (No. 11271038) and the Natural Science Foundation of Beijing (No. 1082003).
- Natterer F: The Mathematics of Computerized Tomography. Wiley, New York; 1986.MATHGoogle Scholar
- Natterer F, Wübbeling F SIAM Monographs on Mathematical Modeling and Computation. In Mathematical Methods in Image Reconstruction. SIAM, Philadelphia; 2001.View ArticleGoogle Scholar
- Candés EJ, Donoho DL: Continuous curvelet transform: II. Discretization and frames. Appl. Comput. Harmon. Anal. 2005, 19: 198–222. 10.1016/j.acha.2005.02.004MATHMathSciNetView ArticleGoogle Scholar
- Candés EJ, Donoho DL:New tight frames of curvelets and optimal representations of objects with singularities. Commun. Pure Appl. Math. 2004, 57: 219–266. 10.1002/cpa.10116MATHView ArticleGoogle Scholar
- Candés EJ, Donoho DL: Recovering edges in ill-posed inverse problems: optimality of curvelet frames. Ann. Stat. 2002, 30: 784–842.MATHView ArticleGoogle Scholar
- Easley G, Labate D, Lim WQ: Sparse directional image representations using the discrete shearlet transform. Appl. Comput. Harmon. Anal. 2008, 25: 25–46. 10.1016/j.acha.2007.09.003MATHMathSciNetView ArticleGoogle Scholar
- Guo K, Labate D: Optimally sparse multidimensional representation using shearlets. SIAM J. Math. Anal. 2007, 39: 298–318. 10.1137/060649781MATHMathSciNetView ArticleGoogle Scholar
- Colonna F, Easley G, Guo K, Labate D: Radon transform inversion using the shearlet representation. Appl. Comput. Harmon. Anal. 2010, 29: 232–250. 10.1016/j.acha.2009.10.005MATHMathSciNetView ArticleGoogle Scholar
- Lee NY, Lucier BJ: Wavelet methods for inverting the Radon transform with noisy data. IEEE Trans. Image Process. 2001, 10: 1079–1094.MathSciNetGoogle Scholar
- Strichartz RS: Radon inversion-variations on a theme. Am. Math. Mon. 1982, 89: 377–384. 10.2307/2321649MATHMathSciNetView ArticleGoogle Scholar
- Samko SG: Hypersingular Integrals and Their Applications. Taylor & Francis, London; 2002.MATHGoogle Scholar
- Zhou MQ: Harmonic Analysis. Beijing Normal University Press, Beijing; 1999.Google Scholar
- Lu SZ, Wang KY: Real Analysis. Beijing Normal University Press, Beijing; 2005.Google Scholar
- Donoho DL, Johnstone IM: Ideal spatial adaptation via wavelet shrinkage. Biometrika 1994, 81: 425–455. 10.1093/biomet/81.3.425MATHMathSciNetView ArticleGoogle Scholar
- Johnstone, IM: Gaussian Estimation: Sequence and Wavelet Models. http://www-stat.stanford.edu/imj/GE12–27–11 (1999)Google Scholar
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