TY - JOUR
T1 - A unified method for super-resolution recovery and real exponential-sum separation
AU - CHUI, Charles Kam-Tai
AU - Mhaskar, H. N.
N1 - Funding Information:
This author is also associated with the Statistics Department of Stanford University, CA 94305, and his research is partially supported by U.S. ARO Grant # W911NF-15-1-0385, Hong Kong Research Council Grant # 12300917, and Hong Kong Baptist University Grant # HKBU-RC-ICRS/16-17/03.The research of this author is supported in part by ARO Grant W911NF-15-1-0385.
PY - 2019/3
Y1 - 2019/3
N2 - In this paper, motivated by diffraction of traveling light waves, a simple mathematical model is proposed, both for the multivariate super-resolution problem and the problem of blind-source separation of real-valued exponential sums. This model facilitates the development of a unified theory and a unified solution of both problems in this paper. Our consideration of the super-resolution problem is aimed at applications to fluorescence microscopy and observational astronomy, and the motivation for our consideration of the second problem is the current need of extracting multivariate exponential features in magnetic resonance spectroscopy (MRS) for the neurologist and radiologist as well as for providing a mathematical tool for isotope separation in Nuclear Chemistry. The unified method introduced in this paper can be easily realized by processing only finitely many data, sampled at locations that are not necessarily prescribed in advance, with computational scheme consisting only of matrix-vector multiplication, peak finding, and clustering.
AB - In this paper, motivated by diffraction of traveling light waves, a simple mathematical model is proposed, both for the multivariate super-resolution problem and the problem of blind-source separation of real-valued exponential sums. This model facilitates the development of a unified theory and a unified solution of both problems in this paper. Our consideration of the super-resolution problem is aimed at applications to fluorescence microscopy and observational astronomy, and the motivation for our consideration of the second problem is the current need of extracting multivariate exponential features in magnetic resonance spectroscopy (MRS) for the neurologist and radiologist as well as for providing a mathematical tool for isotope separation in Nuclear Chemistry. The unified method introduced in this paper can be easily realized by processing only finitely many data, sampled at locations that are not necessarily prescribed in advance, with computational scheme consisting only of matrix-vector multiplication, peak finding, and clustering.
KW - Backward solution of heat equation
KW - Gaussian kernels
KW - Multivariate super-resolution
KW - Separation of exponential sums
UR - http://www.scopus.com/inward/record.url?scp=85043232722&partnerID=8YFLogxK
U2 - 10.1016/j.acha.2017.12.007
DO - 10.1016/j.acha.2017.12.007
M3 - Letter
AN - SCOPUS:85043232722
SN - 1063-5203
VL - 46
SP - 431
EP - 451
JO - Applied and Computational Harmonic Analysis
JF - Applied and Computational Harmonic Analysis
IS - 2
ER -