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Positive definiteness of Diffusion Kurtosis Imaging
Inverse obstacle scattering with limited-aperture data
1. | Department of Mathematics, Graduate School of Engineering, Gunma University, Japan |
2. | Department of Mathematics and Statistics, University of Helsinki, Finland |
References:
[1] |
M. Bruhl and M. Hanke, Numerical implementation of two noniterative methods for locating inclusions by impedance tomography, Inverse Problems, 16 (2000), 1029-1042.
doi: 10.1088/0266-5611/16/4/310. |
[2] |
F. Cakoni, D. Colton and P. Monk, "The Linear Sampling Method in Inverse Electromagnetic Scattering," CBMS-NSF Regional Conference Series in Applied Mathematics, 80, SIAM, Philadelphia, PA, 2011. |
[3] |
D. Colton, J. Coyle and P. Monk, Recent developments in inverse acoustic scattering theory, SIAM Review, 42 (2000), 369-414.
doi: 10.1137/S0036144500367337. |
[4] |
D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393.
doi: 10.1088/0266-5611/12/4/003. |
[5] |
D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory," Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983. |
[6] |
D. Colton and R. Kress, Eigenvalues of the far field operator for the Helmholtz equation in an absorbing medium, SIAM J. Appl. Math., 55 (1995), 1724-1735.
doi: 10.1137/S0036139993256114. |
[7] |
D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," 2nd edition, Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, 1998. |
[8] |
D. Colton and R. Kress, Using fundamental solutions in inverse scattering, Inverse Problems, 22 (2006), R49-R66.
doi: 10.1088/0266-5611/22/3/R01. |
[9] |
D. Colton and P. Monk, A linear sampling method for the detection of leukemia using microwaves, SIAM J. Appl. Math., 58 (1998), 926-941.
doi: 10.1137/S0036139996308005. |
[10] |
F. Hettlich, On the uniqueness of the inverse conductive scattering problem for the Helmholtz equation, Inverse Problems, 10 (1994), 129-144.
doi: 10.1088/0266-5611/10/1/010. |
[11] |
M. Ikehata, Reconstruction of an obstacle from the scattering amplitude at a fixed frequency, Inverse Problems, 14 (1998), 949-954.
doi: 10.1088/0266-5611/14/4/012. |
[12] |
M. Ikehata, Reconstruction of a source domain from the Cauchy data, Inverse Problems, 15 (1999), 637-645.
doi: 10.1088/0266-5611/15/2/019. |
[13] |
M. Ikehata, Reconstruction of obstacle from boundary measurements, Wave Motion, 30 (1999), 205-223.
doi: 10.1016/S0165-2125(99)00006-2. |
[14] |
M. Ikehata, How to draw a picture of an unknown inclusion from boundary measurements. Two mathematical inversion algorithms, J. Inv. Ill-Posed Problems, 7 (1999), 255-271.
doi: 10.1515/jiip.1999.7.3.255. |
[15] |
M. Ikehata, Reconstruction of the support function for inclusion from boundary measurements, J. Inv. Ill-Posed Problems, 8 (2000), 367-378. |
[16] |
M. Ikehata, Complex geometrical optics solutions and inverse crack problems, Inverse Problems, 19 (2003), 1385-1405.
doi: 10.1088/0266-5611/19/6/009. |
[17] |
M. Ikehata, Inverse scattering problems and the enclosure method, Inverse Problems, 20 (2004), 533-551.
doi: 10.1088/0266-5611/20/2/014. |
[18] |
M. Ikehata, The Herglotz wave function, the Vekua transform and the enclosure method, Hiroshima Math. J., 35 (2005), 485-506. |
[19] |
M. Ikehata, The probe and enclosure methods for inverse obstacle scattering problems. The past and present, New developments of functional equations in Mathematical Analysis, RIMS Kôkyûroku, 1702 (2010), 1-22. |
[20] |
M. Ikehata and T. Ohe, A numerical method for finding the convex hull of polygonal cavities using the enclosure method, Inverse Problems, 18 (2002), 111-124.
doi: 10.1088/0266-5611/18/1/308. |
[21] |
M. Ikehata and S. Siltanen, Numerical method for finding the convex hull of an inclusion in conductivity from boundary measurements, Inverse Problems, 16 (2000), 1043-1052.
doi: 10.1088/0266-5611/16/4/311. |
[22] |
V. Isakov, "Inverse Problems for Partial Differential Equations," 2nd edition, Applied Mathematical Sciences, 127, Springer, New York, 2006. |
[23] |
A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), 1489-1512.
doi: 10.1088/0266-5611/14/6/009. |
[24] |
A. Kirsch, New characterizations of solutions in inverse scattering theory, Appl. Anal., 76 (2000), 319-350.
doi: 10.1080/00036810008840888. |
[25] |
A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems," Oxford Lecture Series in Mathematics and its Applications, 36, Oxford University Press, Oxford, 2008. |
[26] |
R. Kress, On the numerical solution of a hypersingular integral equation in scattering theory, Journal of Computational and Applied Mathematics, 61 (1995), 345-360.
doi: 10.1016/0377-0427(94)00073-7. |
[27] |
S. Kusiak and J. Sylvester, The scattering support, Comm. Pure Appl. Math., 56 (2003), 1525-1548.
doi: 10.1002/cpa.3038. |
[28] |
S. Kusiak and J. Sylvester, The convex scattering support in a background medium, SIAM J. Math. Anal., 36 (2005), 1142-1158.
doi: 10.1137/S0036141003433577. |
[29] |
C. Liu, The Helmholtz equation on Lipshitz domains, preprint, 1995. |
[30] |
D. R. Luke and R. Potthast, The no response test-a sampling method for inverse scattering problems, SIAM J. Appl. Math., 63 (2003), 1292-1312.
doi: 10.1137/S0036139902406887. |
[31] |
F. W. J. Olver, "Asymptotics and Special Functions," Computer Science and Applied Mathematics, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. |
[32] |
R. Potthast, A point source method for inverse acoustic and electromagnetic obstacle scattering problems, IMA J. Appl. Math., 61 (1998), 119-140.
doi: 10.1093/imamat/61.2.119. |
[33] |
R. Potthast, Stability estimates and reconstructions in inverse scattering using singular sources, J. Comp. Appl. Math., 114 (2000), 247-274.
doi: 10.1016/S0377-0427(99)00201-0. |
[34] |
R. Potthast, On the convergence of the no reponse test, SIAM J. Math. Anal., 38 (2007), 1808-1824.
doi: 10.1137/S0036141004441003. |
[35] |
R. Potthast, J. Sylvester and S. Kusiak, A 'range test' for determining scatterers with unknown physical properties, Inverse Problems, 19 (2003), 533-547.
doi: 10.1088/0266-5611/19/3/304. |
[36] |
I. Vekua, On the solution of the equation $\Delta u+\lambda^2 u=0$, Bull. Acad. Sci. Georgian SSR, 3 (1942), 307-314. |
[37] |
I. Vekua, Modification of an integral transformation and some of its properties, Bull. Acad. Sci. Georgian SSR, 6 (1945), 177-183. |
show all references
References:
[1] |
M. Bruhl and M. Hanke, Numerical implementation of two noniterative methods for locating inclusions by impedance tomography, Inverse Problems, 16 (2000), 1029-1042.
doi: 10.1088/0266-5611/16/4/310. |
[2] |
F. Cakoni, D. Colton and P. Monk, "The Linear Sampling Method in Inverse Electromagnetic Scattering," CBMS-NSF Regional Conference Series in Applied Mathematics, 80, SIAM, Philadelphia, PA, 2011. |
[3] |
D. Colton, J. Coyle and P. Monk, Recent developments in inverse acoustic scattering theory, SIAM Review, 42 (2000), 369-414.
doi: 10.1137/S0036144500367337. |
[4] |
D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393.
doi: 10.1088/0266-5611/12/4/003. |
[5] |
D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory," Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983. |
[6] |
D. Colton and R. Kress, Eigenvalues of the far field operator for the Helmholtz equation in an absorbing medium, SIAM J. Appl. Math., 55 (1995), 1724-1735.
doi: 10.1137/S0036139993256114. |
[7] |
D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," 2nd edition, Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, 1998. |
[8] |
D. Colton and R. Kress, Using fundamental solutions in inverse scattering, Inverse Problems, 22 (2006), R49-R66.
doi: 10.1088/0266-5611/22/3/R01. |
[9] |
D. Colton and P. Monk, A linear sampling method for the detection of leukemia using microwaves, SIAM J. Appl. Math., 58 (1998), 926-941.
doi: 10.1137/S0036139996308005. |
[10] |
F. Hettlich, On the uniqueness of the inverse conductive scattering problem for the Helmholtz equation, Inverse Problems, 10 (1994), 129-144.
doi: 10.1088/0266-5611/10/1/010. |
[11] |
M. Ikehata, Reconstruction of an obstacle from the scattering amplitude at a fixed frequency, Inverse Problems, 14 (1998), 949-954.
doi: 10.1088/0266-5611/14/4/012. |
[12] |
M. Ikehata, Reconstruction of a source domain from the Cauchy data, Inverse Problems, 15 (1999), 637-645.
doi: 10.1088/0266-5611/15/2/019. |
[13] |
M. Ikehata, Reconstruction of obstacle from boundary measurements, Wave Motion, 30 (1999), 205-223.
doi: 10.1016/S0165-2125(99)00006-2. |
[14] |
M. Ikehata, How to draw a picture of an unknown inclusion from boundary measurements. Two mathematical inversion algorithms, J. Inv. Ill-Posed Problems, 7 (1999), 255-271.
doi: 10.1515/jiip.1999.7.3.255. |
[15] |
M. Ikehata, Reconstruction of the support function for inclusion from boundary measurements, J. Inv. Ill-Posed Problems, 8 (2000), 367-378. |
[16] |
M. Ikehata, Complex geometrical optics solutions and inverse crack problems, Inverse Problems, 19 (2003), 1385-1405.
doi: 10.1088/0266-5611/19/6/009. |
[17] |
M. Ikehata, Inverse scattering problems and the enclosure method, Inverse Problems, 20 (2004), 533-551.
doi: 10.1088/0266-5611/20/2/014. |
[18] |
M. Ikehata, The Herglotz wave function, the Vekua transform and the enclosure method, Hiroshima Math. J., 35 (2005), 485-506. |
[19] |
M. Ikehata, The probe and enclosure methods for inverse obstacle scattering problems. The past and present, New developments of functional equations in Mathematical Analysis, RIMS Kôkyûroku, 1702 (2010), 1-22. |
[20] |
M. Ikehata and T. Ohe, A numerical method for finding the convex hull of polygonal cavities using the enclosure method, Inverse Problems, 18 (2002), 111-124.
doi: 10.1088/0266-5611/18/1/308. |
[21] |
M. Ikehata and S. Siltanen, Numerical method for finding the convex hull of an inclusion in conductivity from boundary measurements, Inverse Problems, 16 (2000), 1043-1052.
doi: 10.1088/0266-5611/16/4/311. |
[22] |
V. Isakov, "Inverse Problems for Partial Differential Equations," 2nd edition, Applied Mathematical Sciences, 127, Springer, New York, 2006. |
[23] |
A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), 1489-1512.
doi: 10.1088/0266-5611/14/6/009. |
[24] |
A. Kirsch, New characterizations of solutions in inverse scattering theory, Appl. Anal., 76 (2000), 319-350.
doi: 10.1080/00036810008840888. |
[25] |
A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems," Oxford Lecture Series in Mathematics and its Applications, 36, Oxford University Press, Oxford, 2008. |
[26] |
R. Kress, On the numerical solution of a hypersingular integral equation in scattering theory, Journal of Computational and Applied Mathematics, 61 (1995), 345-360.
doi: 10.1016/0377-0427(94)00073-7. |
[27] |
S. Kusiak and J. Sylvester, The scattering support, Comm. Pure Appl. Math., 56 (2003), 1525-1548.
doi: 10.1002/cpa.3038. |
[28] |
S. Kusiak and J. Sylvester, The convex scattering support in a background medium, SIAM J. Math. Anal., 36 (2005), 1142-1158.
doi: 10.1137/S0036141003433577. |
[29] |
C. Liu, The Helmholtz equation on Lipshitz domains, preprint, 1995. |
[30] |
D. R. Luke and R. Potthast, The no response test-a sampling method for inverse scattering problems, SIAM J. Appl. Math., 63 (2003), 1292-1312.
doi: 10.1137/S0036139902406887. |
[31] |
F. W. J. Olver, "Asymptotics and Special Functions," Computer Science and Applied Mathematics, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. |
[32] |
R. Potthast, A point source method for inverse acoustic and electromagnetic obstacle scattering problems, IMA J. Appl. Math., 61 (1998), 119-140.
doi: 10.1093/imamat/61.2.119. |
[33] |
R. Potthast, Stability estimates and reconstructions in inverse scattering using singular sources, J. Comp. Appl. Math., 114 (2000), 247-274.
doi: 10.1016/S0377-0427(99)00201-0. |
[34] |
R. Potthast, On the convergence of the no reponse test, SIAM J. Math. Anal., 38 (2007), 1808-1824.
doi: 10.1137/S0036141004441003. |
[35] |
R. Potthast, J. Sylvester and S. Kusiak, A 'range test' for determining scatterers with unknown physical properties, Inverse Problems, 19 (2003), 533-547.
doi: 10.1088/0266-5611/19/3/304. |
[36] |
I. Vekua, On the solution of the equation $\Delta u+\lambda^2 u=0$, Bull. Acad. Sci. Georgian SSR, 3 (1942), 307-314. |
[37] |
I. Vekua, Modification of an integral transformation and some of its properties, Bull. Acad. Sci. Georgian SSR, 6 (1945), 177-183. |
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