Tomographic reconstruction of a binary image from few projections is considered. A novel {\em heuristic} algorithm is proposed, the central element of which is a nonlinear transformation f(p)=log(p/(1-p)) of the probability p that a pixel of the sought image be 1-valued. It consists of backprojections based on f(p) and iterative corrections. Application of this algorithm to a series of artificial test cases leads to exact binary reconstructions, (i.e. recovery of the binary image for each single pixel) from the knowledge of projection data over a few directions. Images up to 10^6 pixels are reconstructed in a few seconds. A series of test cases is performed for comparison with previous methods, showing a better efficiency and reduced computation times.