Notes on Curvilinear Caustics in Diffusive Plumes

This short revision examines two long-exposure images featuring luminous filaments/segmentsimmersed in a turbid, smoke-like medium and illuminated by a colored lighting. It sets upa small mathematical model and accompanying measurements that treat the bright pathsas projected integral curves of a stochastic advection field generating caustics in the artici-pating media. Using edge extraction, box-counting and Fourier analysis, it estimates a frac-tal co-dimension consistent with the advective stretching of passive scalar level sets andshow a scale-invariant power spectrum. For the two images considered, the filamentaryedge sets have estimated box-counting dimensions of D1 ≈1.382and D2 ≈1.386(in R2)and the radially averaged power spectra follow approximate power laws over one decadeof spatial frequency. It then connects these observations to the geometry of general caus-tics (singularities of the optical mapping), the transport PDE ∂tI+ u ·∇I= κ∆I− αI+ Sfor the in-scattered intensity Iand to the differential geometry of projected curves withcurvature-driven intermittency.