Quadratic forms in natural images

Several studies have succeeded in correlating natural image statistics with receptive field properties of neurons in the primary visual cortex. If we determine the parameters of linear transformations that make their output values as independent as possible when input data are natural images, we obtain parameter values that correspond to simple cell characteristics. It was also proved that, by making output values as temporally coherent as possible, simple cell characteristics also emerge. However, complex cell properties have not been fully explained by previous studies of natural image statistics. In this study, we examine whether we could reproduce complex cell properties by determining the parameters of two-layer networks that make their outputs as independent and sparse as possible or as temporally coherent as possible. Input-output functions of two-layer networks correspond to quadratic forms and they form a class of functions that includes complex cell responses and many other functions. Therefore, we employed two-layer networks as a framework for discussing complex cell properties as in previous studies. By maximizing the independence and sparseness of output values of two-layer networks without considering the temporal structure of input images, squared responses of simple cells are obtained and complex cell properties are not reproduced. On the other hand, by maximizing the temporal coherence of output, we obtain complex cell properties among other kinds of input-output functions. In previous studies, the measure of temporal coherence was the squared difference between the responses to two consecutive input images. We obtain two-layer networks that minimize this measure and show that some of them exhibit properties of complex cells but not clearly. We propose the sparseness of difference between responses to two consecutive inputs as an alternative measure of temporal coherence. We formulate an algorithm to maximize the sparseness of difference and show that complex cell properties emerge more clearly.