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Inversion of the noisy Radon transform on SO(3) by Gabor frames and sparse recovery principles. (English) Zbl 1227.42033

In this interesting paper, the authors consider the Radon transform \(R\) on the rotation group SO(3) and compute an approximate solution of the linear inverse problem \(Rf=g\), where only a noisy version \(g^\delta\) of \(g\) with \(\|g-g^{\delta}\|\leq \delta\) is available. First, the authors introduce the analytical framework suited for inverting the Radon transform on SO(3), namely, the Gabor transform on SO(3), its admissibility, corresponding coorbit spaces, atomic decompositions and frames. Then they focus on the problem of stable approximating the inverse of the Radon transform on SO(3). Very efficient approximation techniques as well as thrifty strategies for the computation of the stiffness matrix entries are discussed. Finally, the performance of this approach is demonstrated by numerical tests for a crystallographic recovery problem.

MSC:

42C15 General harmonic expansions, frames
65R32 Numerical methods for inverse problems for integral equations
44A12 Radon transform
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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[1] Averbuch, A.; Braverman, E.; Coifman, R.; Israeli, M.; Sidi, A., Efficient computation of oscillatory integrals via adaptive multiscale local Fourier bases, Appl. Comput. Harmon. Anal., 9, 1, 19-53 (2000) · Zbl 0958.65148
[2] Averbuch, A.; Braverman, E.; Coifman, R.; Israeli, M., On efficient computation of multidimensional oscillatory integrals with local Fourier bases, Nonlinear Anal. Ser. A: Theory and Methods, 47, 5, 3491-3502 (2001) · Zbl 1042.65520
[3] Bernstein, S.; Schaeben, H., A one-dimensional Radon transform on SO(3) and its application to texture goniometry, Math. Methods Appl. Sci., 28, 126989 (2005)
[4] v.d. Boogaart, K. G.; Hielscher, R.; Prestin, J.; Schaeben, H., Kernel-based methods for inversion of the radon transform on SO(3) and their applications to texture analysis, J. Comput. Appl. Math., 199, 122-140 (2007) · Zbl 1102.65134
[5] Bunge, H. J., Texture Analysis in Material Science (1982), Butterworths: Butterworths London
[6] Cerejeiras, P.; Schaeben, H.; Sommen, F., The spherical X-ray transform, Math. Methods Appl. Sci., 25, 1493507 (2002) · Zbl 1044.86007
[7] Dahlke, S.; Steidl, G.; Teschke, G., Frames and coorbit theory on homogeneous spaces with a special guidance on the sphere, Special Issue: Analysis on the Sphere. Special Issue: Analysis on the Sphere, J. Fourier Anal. Appl., 13, 4, 387-403 (2007) · Zbl 1141.42019
[8] Dahlke, S.; Steidl, G.; Teschke, G., Weighted coorbit spaces and Banach frames on homogeneous spaces, J. Fourier Anal. Appl., 10, 5, 507-539 (2004) · Zbl 1098.42025
[9] Daubechies, I.; Fornasier, M.; Loris, I., Accelerated projected gradient methods for linear inverse problems with sparsity constraints, J. Fourier Anal. Appl., 14, 5-6, 764-792 (2008) · Zbl 1175.65062
[10] Daubechies, I.; Defrise, M.; DeMol, C., An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math., 57, 1413-1541 (2004) · Zbl 1077.65055
[11] Daubechies, I.; Teschke, G., Variational image restoration by means of wavelets: simultaneous decomposition, deblurring and denoising, Appl. Comput. Harmon. Anal., 19, 1, 1-16 (2005) · Zbl 1079.68104
[12] Daubechies, I.; Teschke, G.; Vese, L., Iteratively solving linear inverse problems with general convex constraints, Inverse Probl. Imaging, 1, 1, 29-46 (2007) · Zbl 1123.65044
[13] Daubechies, I.; Teschke, G.; Vese, L., On some iterative concepts for image restoration, Adv. Imaging Electron Phys., 150, 1-51 (2008)
[14] Delanghe, R.; Sommen, F.; Souček, V., Clifford Analysis and Spinor-Valued Functions: A Function Theory for the Dirac Operator, Math. Appl., vol. 53 (1992), Kluwer Acad. Publ.: Kluwer Acad. Publ. Dordrecht · Zbl 0747.53001
[15] Hielscher, R., The Radon transform on the rotation group inversion and application to texture analysis, Dissertation, Department of Geosciences, University of Technology Freiberg, 2007
[16] Hielscher, R.; Potts, D.; Prestin, J.; Schaeben, H.; Schmalz, M., The Radon transform on \(SO(3)\): a Fourier slice theorem and numerical inversion, Inverse Problems, 24 (2008) · Zbl 1157.65070
[17] Kocks, U. F.; Tomé, C. N.; Wenk, H. R.; Mecking, H., Texture and Anisotropy (1998), Cambridge University Press: Cambridge University Press Cambridge
[18] Matthies, S.; Vinel, G.; Helmig, K., Standard Distributions in Texture Analysis, vol. 1 (1987), Akademie-Verlag: Akademie-Verlag Berlin
[19] Matviyenko, G., Optimized local trigonometric bases, Appl. Comput. Harmon. Anal., 3, 301-323 (1996) · Zbl 0858.42022
[20] Meister, L.; Schaeben, H., A concise quaternion geometry of rotations, MMAS, 28, 101126 (2004)
[21] G. Nawratil, H. Pottmann, Subdivision schemes for the fair discretization of the spherical motion group, Technical Report, Vienna University of Technology, June 2007.; G. Nawratil, H. Pottmann, Subdivision schemes for the fair discretization of the spherical motion group, Technical Report, Vienna University of Technology, June 2007. · Zbl 1156.65019
[22] Nikolayev, D. I.; Schaeben, H., Characteristics of the ultra-hyperbolic differential equation governing pole density functions, Inverse Problems, 15, 160319 (1999)
[23] Ramlau, R.; Teschke, G.; Zhariy, M., A compressive Landweber iteration for solving ill-posed inverse problems, Inverse Problems, 24, 6, 065013 (2008) · Zbl 1166.65024
[24] Randle, V.; Engler, O., Introduction to Texture Analysis Macrotexture: Microtexture and Orientation Mapping (2000), Gordon and Breach: Gordon and Breach London
[25] Schaeben, H.; Boogaart, K. G.v.d., Spherical harmonics in texture analysis, Tectonophysics, 370, 253-268 (2003)
[26] Schaeben, H.; Hielscher, R.; Fundenberger, J.; Potts, D.; Prestin, J., Orientation density function-controlled pole probability density function measurements: automated adaptive control of texture goniometers, J. Appl. Crystallogr., 40, 3, 570-579 (2007)
[27] Teschke, G., Multi-frame representations in linear inverse problems with mixed multi-constraints, Appl. Comput. Harmon. Anal., 22, 43-60 (2007) · Zbl 1144.94002
[28] Teschke, G.; Borries, C., Accelerated projected steepest descent method for nonlinear inverse problems with sparsity constraints, Inverse Problems, 26, 025007 (2010) · Zbl 1192.65071
[29] Torrésani, B., Phase space decompositions: local Fourier analysis on spheres, Signal Process., 43, 341-346 (1995), preprint CPT-93/P.2878, Marseille, 1993 · Zbl 0901.94004
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