On a Link Between Kernel Mean Maps and Fraunhofer Diffraction, with an Application to Super-Resolution Beyond the Diffraction Limit
By Stefan Harmeling, Michael Hirsch, Bernhard Schölkopf. 2013
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.
Subfields: Mathematical Optimization, Signal Processing
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In Mathematical Optimization
In Signal Processing
- Kenji Fukumizu
- Bharath K. Sriperumbudur
- Krikamol Muandet
- Jan Peters
- Bernhard Schölkopf
- Stefan Harmeling
- Dominik Janzing
- David Lopez-Paz
Citations in the corpus
- Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, methodology): “Based on this interpretation, Harmeling et al. (2013) asserts that the Fraunhofer diffraction is in fact a special case of kernel mean embedding and that in theory an object p(ξ) with bounded support can be recovered completely from its diffraction-limited image, using an argument from the…”
- Computing functions of random variables via reproducing kernel Hilbert space representations (2015, positive): “Wecanmapconditionaldistributionsandper-form Bayesian updates (Fukumizu et al. 2008, 2013; Zhang etal.2011),andwecanconnectkernelmapstoFourieroptics, leading to a physical realization as Fraunhofer diffraction (Harmeling et al. 2013).”