Multi-kernel unmixing and super-resolution ...
Document type :
Article dans une revue scientifique: Article original
Title :
Multi-kernel unmixing and super-resolution using the Modified Matrix Pencil method
Author(s) :
Journal title :
Journal of Fourier Analysis and Applications
Publisher :
Springer Verlag
Publication date :
2020-01-22
ISSN :
1069-5869
English keyword(s) :
Matrix Pencil
super-resolution
unmixing kernels
mixture models
sampling
approximation
signal recovery Mathematics
super-resolution
unmixing kernels
mixture models
sampling
approximation
signal recovery Mathematics
HAL domain(s) :
Mathématiques [math]
English abstract : [en]
Consider L groups of point sources or spike trains, with the l'th group represented by $x_l (t)$. For a function $g : R → R$, let $g_l (t) = g(t/µ_l)$ denote a point spread function with scale $µ_l > 0$, and with $µ_1 < · ...
Show more >Consider L groups of point sources or spike trains, with the l'th group represented by $x_l (t)$. For a function $g : R → R$, let $g_l (t) = g(t/µ_l)$ denote a point spread function with scale $µ_l > 0$, and with $µ_1 < · · · < µ_L$. With $y(t) = \sum_{l=1}^{L} (g_l * x_l)(t)$, our goal is to recover the source parameters given samples of y, or given the Fourier samples of y. This problem is a generalization of the usual super-resolution setup wherein $L = 1$; we call this the multi-kernel unmixing super-resolution problem. Assuming access to Fourier samples of y, we derive an algorithm for this problem for estimating the source parameters of each group, along with precise non-asymptotic guarantees. Our approach involves estimating the group parameters sequentially in the order of increasing scale parameters, i.e., from group 1 to L. In particular, the estimation process at stage $1 ≤ l ≤ L$ involves (i) carefully sampling the tail of the Fourier transform of y, (ii) a deflation step wherein we subtract the contribution of the groups processed thus far from the obtained Fourier samples, and (iii) applying Moitra's modified Matrix Pencil method on a deconvolved version of the samples in (ii).Show less >
Show more >Consider L groups of point sources or spike trains, with the l'th group represented by $x_l (t)$. For a function $g : R → R$, let $g_l (t) = g(t/µ_l)$ denote a point spread function with scale $µ_l > 0$, and with $µ_1 < · · · < µ_L$. With $y(t) = \sum_{l=1}^{L} (g_l * x_l)(t)$, our goal is to recover the source parameters given samples of y, or given the Fourier samples of y. This problem is a generalization of the usual super-resolution setup wherein $L = 1$; we call this the multi-kernel unmixing super-resolution problem. Assuming access to Fourier samples of y, we derive an algorithm for this problem for estimating the source parameters of each group, along with precise non-asymptotic guarantees. Our approach involves estimating the group parameters sequentially in the order of increasing scale parameters, i.e., from group 1 to L. In particular, the estimation process at stage $1 ≤ l ≤ L$ involves (i) carefully sampling the tail of the Fourier transform of y, (ii) a deflation step wherein we subtract the contribution of the groups processed thus far from the obtained Fourier samples, and (iii) applying Moitra's modified Matrix Pencil method on a deconvolved version of the samples in (ii).Show less >
Language :
Anglais
Peer reviewed article :
Oui
Audience :
Internationale
Popular science :
Non
Collections :
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