Linking diffusive fields to virtual waves as their propagative duals

In nondestructive and biomedical imaging, spatial patterns inside a sample are imaged without destroying it. Therefore, propagating waves, including electromagnetic or ultrasonic signals or even diffuse heat, are generated or modified by these internal patterns and transmit this structural information to the sample surface. There, the signals can be detected, and an image of the internal structure can be reconstructed from the measured signals. The amount of information about the interior of the sample that can be obtained from the detected signals at the sample surface is significantly influenced by the propagation from the internal structure to the surface. In the real world, all signal propagation is more or less irreversible. The entropy generated during propagation corresponds to the loss of information. In an idealized model, such propagating waves, called virtual waves, are described by the wave equation. They remain valid solutions of this equation when the direction of time is reversed, thus exhibiting reversibility and showing no entropy production and therefore no loss of information during propagation. A Fredholm integral equation of the first kind relates the diffusion fields locally for each point in space to the virtual waves as their propagating duals. These virtual waves can be calculated from the measured diffusion field at each detection point by inverting this integral equation. In the past, this Fredholm integral was derived using so-called “thermal waves” for thermography. This was often confusing because thermal diffusion cannot be described by the wave equation. Here we derive this relationship for the first time in real space without having to use Fourier frequency space. We have used the locally computed virtual waves from the measured diffusive surface signals for image reconstruction using established time-of-flight methods from ultrasound or radar imaging. This improves the spatial resolution in thermography and compensates for the dispersion of quantum wave packets in atom probe tomography.