Slow feature analysis and decorrelation filtering for separating correlated sources

We generalize the method of Slow Feature Analysis for vector-valued functions of multivariables and apply it to the problem of blind source separation, in particular image separation. For the linear case, exact mathematical analysis is given, which shows in particular that the sources are perfectly separated by SFA if and only if they and their first order derivatives are uncorrelated. When the sources are correlated, we apply the following technique called decorrelation filtering: use a linear filter to decorrelate the sources and their derivatives, then apply the separating matrix obtained on the filtered sources to the original sources. We show that if the filtered sources are perfectly separated by this matrix, then so are the original sources. We show how to numerically obtain such a decorrelation filter by solving a nonlinear optimization problem. This technique can also be applied to other linear separation methods, whose output signals are decorrelated, such as ICA.