Mathematical inversion of covariance-map images to determine three-dimensional product scattering distributions

Two-dimensional "crushed" velocity-map images of photoinduced or electron-induced reaction products can be Abel-inverted to reconstruct the full three-dimensional scattering distribution. Here, we develop the equivalent of the inverse Abel transform for covariance-map images, which are increasingly used in the field of velocity-map imaging to uncover correlations between the velocities of two or more scattered products. To formulate the inverse problem, we first derive the forward transformation, which maps the underlying cylindrically symmetric scattering distribution to its corresponding covariance-map measurement. We express the unknown scattering distribution as an expansion in an orthonormal basis and reformulate the inverse problem as a linear least-squares problem, which we solve via QR decomposition. This approach provides flexible control over the reconstruction resolution and supports natural regularization through both basis truncation and data smoothing. We validate the method on synthetic test problems, demonstrating accurate recovery of both local and extended structures, and show convergence using error metrics. We then apply our method to the experimental data on the dissociation products of CF3I2+ generated in electron-molecule collisions. For a simple binary fragmentation channel, the method recovers a localized point-source distribution that is consistent with physical understanding. For a more complex channel, we show that appropriate regularization obtains stable, interpretable results. Our results demonstrate that this method is robust, flexible, and well-suited for analyzing high-quality covariance-map scattering data.