The goal of background reconstruction is to recover the background image of a scene from a sequence of frames showing this scene cluttered by various moving objects. This task is fundamental in image analysis, and is generally the first step before more advanced processing, but difficult because there is no formal definition of what should be considered as background or foreground and the results may be severely impacted by various challenges such as illumination changes, intermittent object motions, highly cluttered scenes, etc. We propose in this paper a new iterative algorithm for background reconstruction, where the current estimate of the background is used to guess which image pixels are background pixels and a new background estimation is performed using those pixels only. We then show that the proposed algorithm, which uses stochastic gradient descent for improved regularization, is more accurate than the state of the art on the challenging SBMnet dataset, especially for short videos with low frame rates, and is also fast, reaching an average of 52 fps on this dataset when parameterized for maximal accuracy using acceleration with a graphics processing unit (GPU) and a Python implementation.