Blindly separating mixtures of multiple layers with spatial shifts

We address the problem of blindly separating mixtures of multiple layer images with unknown spatial shifts and mixing coefficients. Our proposed method can handle the over-determined, determined and under-determined cases where mixtures are more than, as many as and fewer than layers, respectively. The method is fast in over-determined and determined cases, with the same complexity as the fast Fourier transform (FFT), and can separate more layers from fewer mixtures in the under-determined case. It consists of two main steps. First, a novel sparse blind separation algorithm is applied, to estimate the spatial shifts, the mixing coefficients and the edge image of each layer. Second, all layers are reconstructed, by large scale linear programming in the under-determined case, or by least-squares solutions in other cases. The effectiveness of this technology is shown in the experiments on two simulated mixtures of four layers with spatial shifts, real mixture photos containing transparency and reflections, and real mixture images in a dissolve from a video.