Particle localization with DPTV and sizing with IPI using a forward model and optimization

An inverse problem (IP) approach is proposed to simultaneously determine the three-dimensional position and size of bubbles or droplets in a two phase flow from a single camera image. The method is based on interferometric particle imaging (IPI) and defocusing particle tracking velocimetry. A forward model (FM) is introduced that integrates a scattering model based on geometrical optics and the Lorentz–Mie theory, along with a wave propagation model based on the Huygens–Fresnel principle to simulate particle images. Using bounding boxes from object detection methods as initialization, the InvP approach approximates the position and diameter of each particle in the image. The performance of the presented approach is evaluated on the grounds of the data achieved by Sax et al (2025 Phys. Rev. Appl. 24 044083): As key aspects it achieves sub-pixel accuracy in position determination, exceeds the diameter accuracy of current FFT-based benchmarks on real data and furthermore achieves sub-micrometre precision in diameter resolution, even for three-dimensionally distributed particles. The InvP approach achieves a decoupling of the diameter estimation from the out-of-plane position estimation, thus avoiding error propagation from one to the other, which significantly increases the sizing accuracy. The incorporated FM accounts for aliasing effects in the interference pattern, effectively increasing the measurable volume both closer to and further from the focal plane. This improvement qualifies the approach to measure closer to the focal plane, which in turn allows to obtain images with higher signal-to-noise ratio (SNR). The InvP approach is capable of handling significantly lower SNRs compared to commonly applied algorithms and noise levels at which detection algorithms typically fail, presenting significant potential for single optical access IPI in side- and backscatter regions where low SNR usually necessitates sophisticated data processing methods. Notably, the InvP approach is largely unaffected by particle image overlaps, addressing another major challenge in single-camera particle tracking and sizing at high source densities in a given field of view.