Sharp images from diffuse beams: factorisation of the discrete delta function

TL;DR

This paper introduces multi-dimensional discrete functions that mimic delta functions, enabling diffuse imaging with improved resolution and signal-to-noise ratio, applicable to ghost imaging and large-scale tomographic systems.

Contribution

It constructs broad, multi-dimensional arrays that replicate delta functions under aperiodic convolution, expanding imaging capabilities and connecting to Huffman sequences.

Findings

Arrays can sample multiple points simultaneously without losing sharpness.

Diffuse point-spread functions improve structure detection with better SNR.

Large arrays enable applications in ghost imaging and tomography.

Abstract

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 under aperiodic convolution. 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. A diffuse point-spread function applied in any 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 point-spread functions can now be made sufficiently large to span the size of the object under measure. Such large 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 continuum self-adjoint or Hermitian functions that underpin wave theory and quantum mechanics.