Hierarchical Modeling of Local Image Features through $L_p$-Nested Symmetric Distributions

We introduce a new family of distributions, called $L_p${\em -nested symmetric distributions}, whose densities access the data exclusively through a hierarchical cascade of $L_p$-norms. This class generalizes the family of spherically and $L_p$-spherically symmetric distributions which have recently been successfully used for natural image modeling. Similar to those distributions it allows for a nonlinear mechanism to reduce the dependencies between its variables. With suitable choices of the parameters and norms, this family also includes the Independent Subspace Analysis (ISA) model, which has been proposed as a means of deriving filters that mimic complex cells found in mammalian primary visual cortex. $L_p$-nested distributions are easy to estimate and allow us to explore the variety of models between ISA and the $L_p$-spherically symmetric models. Our main findings are that, without a preprocessing step of contrast gain control, the independent subspaces of ISA are in fact more dependent than the individual filter coefficients within a subspace and, with contrast gain control, where ISA finds more than one subspace, the filter responses were almost independent anyway.