The photography transforms and their analytic inversion formulas

TL;DR

This work develops a rigorous theory for reconstructing light fields from focal stacks by introducing three generalized photography transforms that extend the forward operator to arbitrary $n$. It proves that these photography transforms are equivalent to a coupled Radon transform and derives Fourier slice and convolution properties along with dual operators and a representation of the normal operator. Analytic inversion formulas are established for the $n=1$ and $n=2$ cases using (coupled) Riesz potentials, yielding FBP-like and Hilbert/Laplacian-based reconstruction schemes, with extensions to higher dimensions under coupling assumptions. The results provide a solid theoretical foundation for analytic light-field reconstruction from focal stacks and highlight practical challenges related to data sufficiency and stable numerics, motivating future numerical algorithms.

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.

Figures (4)

• Figure 1: The light field $f_F$ in lens plane $u_1 u_2$, reference plane $x_1 x_2$, and focal plane $\bar{x}_1\bar{x}_2$. $f_F(u_1,u_2,x_1,x_2)$ is the radiance along the given ray passing through the points $(u_1,u_2)$ and $(x_1,x_2)$.
• Figure 2: Two-plane parameterization of the light field $f_F(x_1,x_2,u_1,u_2)$.
• Figure 3: Reparameterization of $f_{F^{\prime}}(\bar{x}_1,u_1)$ by $f_{F}(x_1,u_1)$. The figure shows the simplified 2-dimensional case involving only $u_1$, $x_1$, and $\bar{x}_1$. By similar triangles, we have $x_1 = \frac{F}{F^{\prime}} \bar{x}_1 + \left(1-\frac{F}{F^{\prime}}\right)u_1$. Since the ray from $u_1$ to $x_1$ and the ray from $u_1$ to $\bar{x}_1$ are the same, there holds $f_{F^{\prime}}(\bar{x}_1,u_1)=f_F\left(\frac{F}{F^{\prime}} \bar{x}_1 + \left(1-\frac{F}{F^{\prime}}\right)u_1,u_1\right)$.
• Figure :

Theorems & Definitions (32)

• Definition 2.1
• Remark 2.2
• Definition 3.1
• Remark 3.2
• Theorem 3.3
• proof
• Corollary 3.4
• proof
• Theorem 3.5
• proof