Sharp Computational Images From Diffuse Beams: Factorization of the Discrete Delta Function

Discrete delta functions define the limits of attainable spatial resolution for all imaging systems. Here we construct broad, multi-dimensional discrete functions that replicate closely the action of a Dirac delta function for convolution under aperiodic boundary conditions. These arrays spread the energy of a sharp probe beam to simultaneously sample multiple points across the volume of a large object, without losing image sharpness. Applying these point-spread functions in any computational imaging system can reveal the underlying structure of objects less intrusively and with equal or better signal-to-noise ratio. These multi-dimensional arrays are related to previously known, but relatively rarely employed, one-dimensional integer Huffman sequences. Practical probes can now be made that are larger than the object under measure. Such arrays can be applied to ghost imaging, which has demonstrated potential to greatly improve signal-to-noise ratios and reduce the total dose required for tomographic imaging. The discrete arrays built here parallel the self-adjoint or Hermitian functions of the continuum that underpin classical wave theory and quantum mechanics.