Linking diffusive fields to virtual waves as their propagative duals

In non-destructive and biomedical imaging, the spatial patterns inside a sample are imaged without destroying it. Propagating waves, including electromagnetic or ultrasonic signals, or even diffuse heat, are generated or modified by these patterns and transmit structural information to the sample surface. There, these signals can be detected and used to reconstruct an image of the internal structure. In reality, all signal propagation is more or less irreversible. In an idealized model, such propagating waves are described by the wave equation and are called virtual waves. These waves remain valid solutions of the wave equation when the direction of time is reversed. Thus, they exhibit reversibility, showing no entropy production or loss of information during propagation. A Fredholm integral equation of the first kind relates the diffusion fields locally at each point in space to the virtual waves as their propagating duals. We can calculate these virtual waves from the measured diffusion field at each detection point by inverting this integral equation. We used the locally computed virtual waves from the measured diffusive surface signals for image reconstruction. This improves the spatial resolution in thermography, for instance, or compensates for the dispersion of quantum wave packets in atom probe tomography.