On a Hidden Property in Computational Imaging

TL;DR

This work identifies a hidden property in the latent spaces of computational imaging: the measurement data and target properties in FWI, CT, and EM inversion follow the same one-way wave equations with identical speeds, while their initial conditions are linearly related. It introduces HINT, a unified framework that jointly embeds two modalities using a shared FINOLA-based autoregressive process and a linear mapper between latent spaces. Empirically, HINT achieves competitive or superior reconstruction/inversion across OpenFWI, RSNA CT, and Kimberlina EM datasets, with a compact parameterization and robustness across resolutions. The findings offer a new theoretical perspective on latent representations in inverse problems and suggest practical benefits for cross-domain consistency and interpretability.

Abstract

Computational imaging plays a vital role in various scientific and medical applications, such as Full Waveform Inversion (FWI), Computed Tomography (CT), and Electromagnetic (EM) inversion. These methods address inverse problems by reconstructing physical properties (e.g., the acoustic velocity map in FWI) from measurement data (e.g., seismic waveform data in FWI), where both modalities are governed by complex mathematical equations. In this paper, we empirically demonstrate that despite their differing governing equations, three inverse problems (FWI, CT, and EM inversion) share a hidden property within their latent spaces. Specifically, using FWI as an example, we show that both modalities (the velocity map and seismic waveform data) follow the same set of one-way wave equations in the latent space, yet have distinct initial conditions that are linearly correlated. This suggests that after projection into the latent embedding space, the two modalities correspond to different solutions of the same equation, connected through their initial conditions. Our experiments confirm that this hidden property is consistent across all three imaging problems, providing a novel perspective for understanding these computational imaging tasks.

Figures (9)

• Figure 1: Illustration of the hidden property. Different imaging problems share a common hidden property in the latent space: the two modalities involved in each problem follow the same set of one-way wave equations in the latent space, with different but linearly correlated initial conditions. For instance, CT projection data $p(\mathbf{d}, \mathbf{s})$ and CT image $f(x,y)$, once projected into the latent space, become two distinct but linearly correlated initial conditions of the same wave equation $\frac{\partial \bm{\zeta}}{\partial x} = \bm{\Lambda} \frac{\partial \bm{\zeta}}{\partial y}$.
• Figure 2: Comparsion of Vanila FINOLA with the proposed HINT. The subfigure a) is the framework for Vanilla FINOLA, which reconstructs images within one modality. The illustration figures are from chen2023image. The subfigure b) is the overview of our framework. Each measurement $\bm{P}$ is firstly encoded into a single vector $\bm{v_P}$. The latent vector $\bm{v_\psi}$ is then obtained from a linear transformation $\bm{T}$. A shared multi-path FINOLA layer is applied to autoregress the feature map $\bm{z_P}$ and $\bm{z_\psi}$, respectively. Finally, two separate decoders composed of upsampling and $3 \times 3$ convolutional layers are used to reconstruct the measurement and to invert the target property.
• Figure 3: Comparing HINT with a two-separate-FINOLAs network, where each embedding has its own set of wave speeds, in terms of SSIM. Evaluated on OpenFWI.
• Figure 4: Comparing HINT with nonlinear converters, in terms of SSIM. Evaluated on OpenFWI.
• Figure 5: Illustration of results on OpenFWI, compared with InversionNet and Auto-Linear.