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

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

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

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).
Citation: Silvia Allavena, Michele Piana, Federico Benvenuto, Anna Maria Massone. An interpolation/extrapolation approach to X-ray imaging of solar flares. Inverse Problems and 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), 193-211. doi: 10.1023/A:1022469811115.

[2]

H. Bialy, Iterative behandlung linearer funktionalgleichungen, Arch. Rat. Mech. Anal., 4 (1959), 166-176. 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), 567-585. 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), 1159-1165. doi: 10.1086/498225.

[5]

T. F. Chan and L. A. Vese, Active contours without edges, IEEE Trans. Im. Proc., 10 (1997), 266-277. doi: 10.1109/83.902291.

[6]

D. L. Donoho, Superresolution via sparsity constraints, SIAM J. Math. Anal., 23 (1992), 1309-1331. 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), 1413-1457. doi: 10.1002/cpa.20042.

[8]

R. W. Gerchberg, Super-resolution through error energy reduction, Optical Acta, 21 (1974), 709-720. doi: 10.1080/713818946.

[9]

G. J. Hurford, et al., The RHESSI imaging concept, Solar Phys., 210 (2002), 61-90. 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), 250. 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), 1874. doi: 10.1109/29.9032.

[12]

R. P. Lin, et al., The Reuven Ramaty high-energy solar spectroscopic imager, Solar Phys., 210 (2002), 3-32.

[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), 2004-2016. doi: 10.1088/0004-637X/703/2/2004.

[14]

M. Piana and M. Bertero, Projected Landweber method and preconditioning, Inverse Problems, 13 (1997), 441-463. 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), 242-252. 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, Mathematics and its Applications, 328, Kluwer Academic Publishers Group, Dordrecht, 1995.

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), 193-211. doi: 10.1023/A:1022469811115.

[2]

H. Bialy, Iterative behandlung linearer funktionalgleichungen, Arch. Rat. Mech. Anal., 4 (1959), 166-176. 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), 567-585. 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), 1159-1165. doi: 10.1086/498225.

[5]

T. F. Chan and L. A. Vese, Active contours without edges, IEEE Trans. Im. Proc., 10 (1997), 266-277. doi: 10.1109/83.902291.

[6]

D. L. Donoho, Superresolution via sparsity constraints, SIAM J. Math. Anal., 23 (1992), 1309-1331. 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), 1413-1457. doi: 10.1002/cpa.20042.

[8]

R. W. Gerchberg, Super-resolution through error energy reduction, Optical Acta, 21 (1974), 709-720. doi: 10.1080/713818946.

[9]

G. J. Hurford, et al., The RHESSI imaging concept, Solar Phys., 210 (2002), 61-90. 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), 250. 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), 1874. doi: 10.1109/29.9032.

[12]

R. P. Lin, et al., The Reuven Ramaty high-energy solar spectroscopic imager, Solar Phys., 210 (2002), 3-32.

[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), 2004-2016. doi: 10.1088/0004-637X/703/2/2004.

[14]

M. Piana and M. Bertero, Projected Landweber method and preconditioning, Inverse Problems, 13 (1997), 441-463. 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), 242-252. 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, Mathematics and its Applications, 328, Kluwer Academic Publishers Group, Dordrecht, 1995.

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