Tilt-Then-Blur or Blur-Then-Tilt? Clarifying the Atmospheric Turbulence Model

Imaging at a long distance often requires advanced image restoration algorithms to compensate for the distortions caused by atmospheric turbulence. However, unlike many standard restoration problems such as deconvolution, the forward image formation model of the atmospheric turbulence does not have a simple expression. Thanks to the Zernike representation of the phase, one can show that the forward model is a combination of tilt (pixel shifting due to the linear phase terms) and blur (image smoothing due to the high order aberrations). Confusions then arise between the ordering of the two operators. Should the model be tilt-then-blur, or blur-then-tilt? Some papers in the literature say that the model is tilt-then-blur, whereas more papers say that it is blur-then-tilt. This paper clarifies the differences between the two and discusses why the tilt-then-blur is the correct model. Recommendations are given to the research community.