Blind inverse problems with isolated spikes. (English) Zbl 1508.94005
Summary: Assume that an unknown integral operator living in some known subspace is observed indirectly, by evaluating its action on a discrete measure containing a few isolated Dirac masses at an unknown location. Is this information enough to recover the impulse response location and the operator with a sub-pixel accuracy? We study this question and bring to light key geometrical quantities for exact and stable recovery. We also propose an in-depth study of the presence of additive white Gaussian noise. We illustrate the well-foundedness of this theory on the challenging optical imaging problem of blind deconvolution and blind deblurring with non-stationary operators.
MSC:
94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |
15A29 | Inverse problems in linear algebra |
45Q05 | Inverse problems for integral equations |
78A46 | Inverse problems (including inverse scattering) in optics and electromagnetic theory |
62M15 | Inference from stochastic processes and spectral analysis |
60G15 | Gaussian processes |
62D05 | Sampling theory, sample surveys |
44A12 | Radon transform |