Small-brain neural networks rapidly solve inverse problems with vortex Fourier encoders

We introduce a vortex phase transform with a lenslet-array to accompany shallow, dense, ``small-brain'' neural networks for high-speed and low-light imaging. Our single-shot ptychographic approach exploits the coherent diffraction, compact representation, and edge enhancement of Fourier-tranformed spiral-phase gradients. With vortex spatial encoding, a small brain is trained to deconvolve images at rates 5-20 times faster than those achieved with random encoding schemes, where greater advantages are gained in the presence of noise. Once trained, the small brain reconstructs an object from intensity-only data, solving an inverse mapping without performing iterations on each image and without deep-learning schemes. With this hybrid, optical-digital, vortex Fourier encoded, small-brain scheme, we reconstruct MNIST Fashion objects illuminated with low-light flux (5 nJ/cm$^2$) at a rate of several thousand frames per second on a 15 W central processing unit, two orders of magnitude faster than convolutional neural networks.