The photography transforms and their analytic inversion formulas

TL;DR

This paper introduces new photography transforms modeled as integral equations, extending Radon transform theory to higher dimensions, and derives their properties and inversion formulas for light field reconstruction.

Contribution

It rigorously models forward problems with three new transforms, proving their equivalence to Radon transforms and deriving novel inversion formulas.

Findings

Proposed three types of photography transforms with integral geometry properties.

Established the equivalence of these transforms to Radon transforms.

Derived analytic inversion formulas for the transforms.

Abstract

The light field reconstruction from the focal stack can be mathematically formulated as an ill-posed integral equation inversion problem. Although the previous research about this problem has made progress both in practice and theory, its forward problem and inversion in a general form still need to be studied. In this paper, to model the forward problem rigorously, we propose three types of photography transforms with different integral geometry characteristics that extend the forward operator to the arbitrary $n$-dimensional case. We prove that these photography transforms are equivalent to the Radon transform with the coupling relation between variables. We also obtain some properties of the photography transforms, including the Fourier slice theorem, the convolution theorem, and the convolution property of the dual operator, which are very similar to those of the classic Radon transform. Furthermore, the representation of the normal operator and the analytic inversion formula for the photography transforms are derived and they are quite different from those of the classic Radon transform.