We establish a link between Fourier optics and a recent construction from the machine learning community termed the kernel mean map. Using the Fraunhofer approximation, it identifies the kernel with the squared Fourier transform of the aperture. This allows us to use results about the invertibility of the kernel mean map to provide a statement about the invertibility of Fraunhofer diffraction, showing that imaging processes with arbitrarily small apertures can in principle be invertible, i.e., do not lose information, provided the objects to be imaged satisfy a generic condition. A real world experiment shows that we can super-resolve beyond the Rayleigh limit.
Several papers consider a bounded support constraint to overcome the diffraction limit. Another possible constraint is sparsity: Donoho @cite_7 studied the problem of recovering a sparse signal for which only low frequencies of its Fourier transform are available. Recently, Candes and Fernandez-Granda @cite_19 also studied conditions under which sparse signals can be recovered. The results apply to signals which have a sparse representation. Sparsity has effectively also been practically used to break the diffraction limit using hardware, e.g. in stimulated emission depletion microscopy (STED) @cite_13 .
@cite_19 @cite_13 @cite_7
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This paper develops a mathematical theory of super-resolution. Broadly speaking, super-resolution is the problem of recovering the fine details of an object---the high end of its spectrum---from coarse scale information only---from samples at the low end of the spectrum. Suppose we have many point sources at unknown locations in @math and with unknown complex-valued amplitudes. We only observe Fourier samples of this object up until a frequency cut-off @math . We show that one can super-resolve these point sources with infinite precision---i.e. recover the exact locations and amplitudes---by solving a simple convex optimization problem, which can essentially be reformulated as a semidefinite program. This holds provided that the distance between sources is at least @math . This result extends to higher dimensions and other models. In one dimension for instance, it is possible to recover a piecewise smooth function by resolving the discontinuity points with infinite precision as well. We also show that the theory and methods are robust to noise. In particular, in the discrete setting we develop some theoretical results explaining how the accuracy of the super-resolved signal is expected to degrade when both the noise level and the super-resolution factor vary. We propose a new type of scanning fluorescence microscope capable of resolving 35 nm in the far field. We overcome the diffraction resolution limit by employing stimulated emission to inhibit the fluorescence process in the outer regions of the excitation point-spread function. In contrast to near-field scanning optical microscopy, this method can produce three-dimensional images of translucent specimens. Preface to the Second Edition. Preface to the First Edition. 1 Ray Optics. 2 Wave Optics. 3 Beam Optics. 4 Fourier Optics. 5 Electromagnetic Optics. 6 Ploarization Optics. 7 Photonic-Crystal Optics. 8 Guided-Wave Optics. 9 Fiber Optics. 10 Resonator Optics. 11 Statistical Optics. 12 Photon Optics. 13 Photon and Atoms. 14 Laser Amplifiers. 15 Lasers. 16 Semiconductor Optics. 17 Semiconductor Photon Sources. 18 Semiconductor Photon Detectors. 19 Acousto-Optics. 20 Electro-Optics. 21 Nonlinear Optics. 22 Ultrafast Optics. 23 Optical Interconnects and Switches. 24 Optical Fiber Communications. A Fourier Transform. B Linear Systems. C Modes of Linear Systems. Symbols and Units. Authors. Index.