We propose an algorithm to perform sparse spike estimation from Fourier measurements. Based on theoretical results on non-convex optimization techniques for off-the-grid sparse spike estimation, we present a simple projected descent algorithm coupled with an initialization procedure. Our algorithm permits to estimate the positions of large numbers of Diracs in 2d from random Fourier measurements. This opens the way for practical estimation of such signals for imaging applications as the algorithm scales well with respect to the dimensions of the problem. We present, along with the algorithm, theoretical qualitative insights explaining the success of our algorithm.