An interpolation/extrapolation approach to X-ray imaging of solar flares

Previous Article
IPI Home
This Issue
Next Article
Uniqueness for an ill-posed reaction-dispersion model. Application to organic pollution in stream-waters

May 2012, 6(2): 147-162. doi: 10.3934/ipi.2012.6.147

An interpolation/extrapolation approach to X-ray imaging of solar flares

Silvia Allavena ^1, , Michele Piana ^1, , Federico Benvenuto ^1, and Anna Maria Massone ^2,

1.	Dipartimento di Matematica, Università di Genova, via Dodecaneso 35 16146 Genova, Italy, Italy, Italy
2.	CNR - Consiglio Nazionale delle Ricerche, SPIN, Genova, via Dodecaneso 33 16146 Genova, Italy

Received December 2010 Revised October 2011 Published May 2012

Abstract
Related Papers

We describe an interpolation/extrapolation procedure that reconstructs X-ray maps of solar flares using as input data sparse samples of the Fourier transform of the radiation flux, named visibilities. The algorithm is based on two steps: in the first step the performance of an interpolation routine is optimized by representing the visibilities according to favorable coordinates in the frequency plane. In the second step two extrapolation schemes are introduced, respectively based on the projection and the thresholding of the Landweber iterative method. The procedure is validated against realistic synthetic visibilities and applied to experimental measurements provided by the NASA satellite Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI).

Keywords: iterative regularization methods, soft thresholding., visibilities, Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI), X-ray solar imaging.

Mathematics Subject Classification: Primary: 65J22, 68U10; Secondary: 65Z0.

Citation: Silvia Allavena, Michele Piana, Federico Benvenuto, Anna Maria Massone. An interpolation/extrapolation approach to X-ray imaging of solar flares. Inverse Problems & Imaging, 2012, 6 (2) : 147-162. doi: 10.3934/ipi.2012.6.147

References:

[1]	M. J. Aschwanden, E. Schmal and the RHESSI Team, Reconstruction of RHESSI solar flare images with a forward-fitting method,, Solar Phys., 210 (2002), 193. doi: 10.1023/A:1022469811115.
[2]	H. Bialy, Iterative behandlung linearer funktionalgleichungen,, Arch. Rat. Mech. Anal., 4 (1959), 166. doi: 10.1007/BF00281385.
[3]	F. L. Bookstein, Principal warps: Thin-plate splines and the decomposition of deformations,, IEEE Trans. Pattern Anal. Mach. Int., 11 (1989), 567. doi: 10.1109/34.24792.
[4]	S. C. Bong, L. Jeongwoo, D. E. Gary and H. S. Yun, Spatio-spectral maximum entropy method. I. Formulation and test,, Astrophys. J., 636 (2006), 1159. doi: 10.1086/498225.
[5]	T. F. Chan and L. A. Vese, Active contours without edges,, IEEE Trans. Im. Proc., 10 (1997), 266. doi: 10.1109/83.902291.
[6]	D. L. Donoho, Superresolution via sparsity constraints,, SIAM J. Math. Anal., 23 (1992), 1309. doi: 10.1137/0523074.
[7]	I. Daubechies, M. Defrise and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,, Comm. Pure Appl. Math., 57 (2004), 1413. doi: 10.1002/cpa.20042.
[8]	R. W. Gerchberg, Super-resolution through error energy reduction,, Optical Acta, 21 (1974), 709. doi: 10.1080/713818946.
[9]	G. J. Hurford, et al., The RHESSI imaging concept,, Solar Phys., 210 (2002), 61. doi: 10.1023/A:1022436213688.
[10]	E. P. Kontar, I. G. Hannah, N. L. S. Jeffrey and M. Battaglia, The sub-arcsecond hard X-ray structure of loop footpoints in a solar flare,, Astrophys. J., 717 (2010). doi: 10.1088/0004-637X/717/1/250.
[11]	R. Lagendijk, J. Biemond and D. Boekee, Regularized iterative image restoration with ringing reduction,, IEEE Trans. Acoust. Speech Signal Process., 36 (1988). doi: 10.1109/29.9032.
[12]	R. P. Lin, et al., The Reuven Ramaty high-energy solar spectroscopic imager,, Solar Phys., 210 (2002), 3.
[13]	A. M. Massone, A. G. Emslie, G. J. Hurford, M. Prato, E. P. Kontar and M. Piana, Hard X-ray imaging of solar flares using interpolated visibilities,, Astrophys. J., 703 (2009), 2004. doi: 10.1088/0004-637X/703/2/2004.
[14]	M. Piana and M. Bertero, Projected Landweber method and preconditioning,, Inverse Problems, 13 (1997), 441. doi: 10.1088/0266-5611/13/2/016.
[15]	Marco Prato, A. Gordon Emslie, Eduard P. Kontar, Anna Maria Massone and Michele Piana, The location of centroids in photon and electron maps of solar flares,, Astrophys. J., 706 ().
[16]	E. J. Schmahl, R. L. Pernak, G. J. Hurford, J. Lee and S. Bong, Analysis of RHESSI flares using a radio astronomical technique,, Solar Phys., 240 (2007), 242. doi: 10.1007/s11207-007-0263-1.
[17]	A. R. Thompson, J. M. Moran and G. W. Swenson, "Interferometry and Synthesis in Radioastronomy,", Wiley, (2004).
[18]	A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov and A. Yagola, "Numerical Methods for the Solution of Ill-posed Problems,", Translated from the 1990 Russian original by R. A. M. Hoksbergen and revised by the authors, 328 (1990).

show all references

References:

[1]	M. J. Aschwanden, E. Schmal and the RHESSI Team, Reconstruction of RHESSI solar flare images with a forward-fitting method,, Solar Phys., 210 (2002), 193. doi: 10.1023/A:1022469811115.
[2]	H. Bialy, Iterative behandlung linearer funktionalgleichungen,, Arch. Rat. Mech. Anal., 4 (1959), 166. doi: 10.1007/BF00281385.
[3]	F. L. Bookstein, Principal warps: Thin-plate splines and the decomposition of deformations,, IEEE Trans. Pattern Anal. Mach. Int., 11 (1989), 567. doi: 10.1109/34.24792.
[4]	S. C. Bong, L. Jeongwoo, D. E. Gary and H. S. Yun, Spatio-spectral maximum entropy method. I. Formulation and test,, Astrophys. J., 636 (2006), 1159. doi: 10.1086/498225.
[5]	T. F. Chan and L. A. Vese, Active contours without edges,, IEEE Trans. Im. Proc., 10 (1997), 266. doi: 10.1109/83.902291.
[6]	D. L. Donoho, Superresolution via sparsity constraints,, SIAM J. Math. Anal., 23 (1992), 1309. doi: 10.1137/0523074.
[7]	I. Daubechies, M. Defrise and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,, Comm. Pure Appl. Math., 57 (2004), 1413. doi: 10.1002/cpa.20042.
[8]	R. W. Gerchberg, Super-resolution through error energy reduction,, Optical Acta, 21 (1974), 709. doi: 10.1080/713818946.
[9]	G. J. Hurford, et al., The RHESSI imaging concept,, Solar Phys., 210 (2002), 61. doi: 10.1023/A:1022436213688.
[10]	E. P. Kontar, I. G. Hannah, N. L. S. Jeffrey and M. Battaglia, The sub-arcsecond hard X-ray structure of loop footpoints in a solar flare,, Astrophys. J., 717 (2010). doi: 10.1088/0004-637X/717/1/250.
[11]	R. Lagendijk, J. Biemond and D. Boekee, Regularized iterative image restoration with ringing reduction,, IEEE Trans. Acoust. Speech Signal Process., 36 (1988). doi: 10.1109/29.9032.
[12]	R. P. Lin, et al., The Reuven Ramaty high-energy solar spectroscopic imager,, Solar Phys., 210 (2002), 3.
[13]	A. M. Massone, A. G. Emslie, G. J. Hurford, M. Prato, E. P. Kontar and M. Piana, Hard X-ray imaging of solar flares using interpolated visibilities,, Astrophys. J., 703 (2009), 2004. doi: 10.1088/0004-637X/703/2/2004.
[14]	M. Piana and M. Bertero, Projected Landweber method and preconditioning,, Inverse Problems, 13 (1997), 441. doi: 10.1088/0266-5611/13/2/016.
[15]	Marco Prato, A. Gordon Emslie, Eduard P. Kontar, Anna Maria Massone and Michele Piana, The location of centroids in photon and electron maps of solar flares,, Astrophys. J., 706 ().
[16]	E. J. Schmahl, R. L. Pernak, G. J. Hurford, J. Lee and S. Bong, Analysis of RHESSI flares using a radio astronomical technique,, Solar Phys., 240 (2007), 242. doi: 10.1007/s11207-007-0263-1.
[17]	A. R. Thompson, J. M. Moran and G. W. Swenson, "Interferometry and Synthesis in Radioastronomy,", Wiley, (2004).
[18]	A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov and A. Yagola, "Numerical Methods for the Solution of Ill-posed Problems,", Translated from the 1990 Russian original by R. A. M. Hoksbergen and revised by the authors, 328 (1990).

[1]	Nuutti Hyvönen, Martti Kalke, Matti Lassas, Henri Setälä, Samuli Siltanen. Three-dimensional dental X-ray imaging by combination of panoramic and projection data. Inverse Problems & Imaging, 2010, 4 (2) : 257-271. doi: 10.3934/ipi.2010.4.257
[2]	Dan Jane, Gabriel P. Paternain. On the injectivity of the X-ray transform for Anosov thermostats. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 471-487. doi: 10.3934/dcds.2009.24.471
[3]	François Rouvière. X-ray transform on Damek-Ricci spaces. Inverse Problems & Imaging, 2010, 4 (4) : 713-720. doi: 10.3934/ipi.2010.4.713
[4]	Wenzhong Zhu, Huanlong Jiang, Erli Wang, Yani Hou, Lidong Xian, Joyati Debnath. X-ray image global enhancement algorithm in medical image classification. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1297-1309. doi: 10.3934/dcdss.2019089
[5]	Ingenuin Gasser. Modelling and simulation of a solar updraft tower. Kinetic & Related Models, 2009, 2 (1) : 191-204. doi: 10.3934/krm.2009.2.191
[6]	Arun K. Kulshreshth, Andreas Alpers, Gabor T. Herman, Erik Knudsen, Lajos Rodek, Henning F. Poulsen. A greedy method for reconstructing polycrystals from three-dimensional X-ray diffraction data. Inverse Problems & Imaging, 2009, 3 (1) : 69-85. doi: 10.3934/ipi.2009.3.69
[7]	Zhenhua Zhao, Yining Zhu, Jiansheng Yang, Ming Jiang. Mumford-Shah-TV functional with application in X-ray interior tomography. Inverse Problems & Imaging, 2018, 12 (2) : 331-348. doi: 10.3934/ipi.2018015
[8]	Jakob S. Jørgensen, Emil Y. Sidky, Per Christian Hansen, Xiaochuan Pan. Empirical average-case relation between undersampling and sparsity in X-ray CT. Inverse Problems & Imaging, 2015, 9 (2) : 431-446. doi: 10.3934/ipi.2015.9.431
[9]	Jacques Féjoz. On "Arnold's theorem" on the stability of the solar system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3555-3565. doi: 10.3934/dcds.2013.33.3555
[10]	Daniela Calvetti, Erkki Somersalo. Microlocal sequential regularization in imaging. Inverse Problems & Imaging, 2007, 1 (1) : 1-11. doi: 10.3934/ipi.2007.1.1
[11]	Nicolas Fournier. A new regularization possibility for the Boltzmann equation with soft potentials. Kinetic & Related Models, 2008, 1 (3) : 405-414. doi: 10.3934/krm.2008.1.405
[12]	Ming Li, Ruming Zhang. Near-field imaging of sound-soft obstacles in periodic waveguides. Inverse Problems & Imaging, 2017, 11 (6) : 1091-1105. doi: 10.3934/ipi.2017050
[13]	Ennio Fedrizzi. High frequency analysis of imaging with noise blending. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 979-998. doi: 10.3934/dcdsb.2014.19.979
[14]	Daomin Cao, Hang Li. High energy solutions of the Choquard equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3023-3032. doi: 10.3934/dcds.2018129
[15]	Robert M. Strain, Keya Zhu. Large-time decay of the soft potential relativistic Boltzmann equation in $\mathbb{R}^3_x$. Kinetic & Related Models, 2012, 5 (2) : 383-415. doi: 10.3934/krm.2012.5.383
[16]	Alina Toma, Bruno Sixou, Françoise Peyrin. Iterative choice of the optimal regularization parameter in TV image restoration. Inverse Problems & Imaging, 2015, 9 (4) : 1171-1191. doi: 10.3934/ipi.2015.9.1171
[17]	Xiangtuan Xiong, Jinmei Li, Jin Wen. Some novel linear regularization methods for a deblurring problem. Inverse Problems & Imaging, 2017, 11 (2) : 403-426. doi: 10.3934/ipi.2017019
[18]	Stefan Kindermann, Antonio Leitão. Convergence rates for Kaczmarz-type regularization methods. Inverse Problems & Imaging, 2014, 8 (1) : 149-172. doi: 10.3934/ipi.2014.8.149
[19]	Victoria Martín-Márquez, Simeon Reich, Shoham Sabach. Iterative methods for approximating fixed points of Bregman nonexpansive operators. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1043-1063. doi: 10.3934/dcdss.2013.6.1043
[20]	Yanxing Cui, Chuanlong Wang, Ruiping Wen. On the convergence of generalized parallel multisplitting iterative methods for semidefinite linear systems. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 863-873. doi: 10.3934/naco.2012.2.863

2017 Impact Factor: 1.465

American Institute of Mathematical Sciences