On a link between kernel mean maps and Fraunhofer diffraction, with an application to super-resolution beyond the diffraction limit

Imaging devices such as telescopes and microscopes collect incoming light using lenses or mirrors of finite size. This finite size imposes a finite aperture on the light that reaches the optical system, leading to effects of diffraction. In particular, diffraction ensures that the image of a point can never be a point. For instance, an imaging system using a lens with an F -number f/D (where f is the focal length, and D is the diameter of the circular aperture) has an impulse response function (Airy disk) whose radius is 1.22λf/D on the sensor, where λ is the wave length of the light (for simplicity, assumed to be monochromatic). Another way to express the same insight uses the transfer function. For a lens focused at infinity, the transfer function is constant within a circle of radius ν 1/(2λf/D), and zero outside [23, p. 136]. This means, in a nutshell, that if we try to image a sinusoidal pattern with spatial frequency larger than ν, diffraction will annihilate that pattern. Likewise, if we decompose a general object into spatial frequencies by Fourier analysis, all components larger than ν will vanish. This article has been accepted for publication at the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Portland, 2013.