Bessel Equivariant Networks for Inversion of Transmission Effects in Multi-Mode Optical Fibres

We develop a new type of model for solving the task of inverting the transmission effects of multi-mode optical fibres through the construction of an $\mathrm{SO}^{+}(2,1)$-equivariant neural network. This model takes advantage of the of the azimuthal correlations known to exist in fibre speckle patterns and naturally accounts for the difference in spatial arrangement between input and speckle patterns. In addition, we use a second post-processing network to remove circular artifacts, fill gaps, and sharpen the images, which is required due to the nature of optical fibre transmission. This two stage approach allows for the inspection of the predicted images produced by the more robust physically motivated equivariant model, which could be useful in a safety-critical application, or by the output of both models, which produces high quality images. Further, this model can scale to previously unachievable resolutions of imaging with multi-mode optical fibres and is demonstrated on $256 \times 256$ pixel images. This is a result of improving the trainable parameter requirement from $\mathcal{O}(N^4)$ to $\mathcal{O}(m)$, where $N$ is pixel size and $m$ is number of fibre modes. Finally, this model generalises to new images, outside of the set of training data classes, better than previous models.