Learning real and complex overcomplete representations from the statistics of natural images

We show how an overcomplete dictionary may be adapted to the statistics of natural images so as to provide a sparse representation of image content. When the degree of overcompleteness is low, the basis functions that emerge resemble those of Gabor wavelet transforms. As the degree of overcompleteness is increased, new families of basis functions emerge, including multiscale blobs, ridge-like functions, and gratings. When the basis functions and coefficients are allowed to be complex, they provide a description of image content in terms of local amplitude (contrast) and phase (position) of features. These complex, overcomplete transforms may be adapted to the statistics of natural movies by imposing both sparseness and temporal smoothness on the amplitudes. The basis functions that emerge form Hilbert pairs such that shifting the phase of the coefficient shifts the phase of the corresponding basis function. This type of representation is advantageous because it makes explicit the structural and dynamic content of images, which in turn allows later stages of processing to discover higher-order properties indicative of image content. We demonstrate this point by showing that it is possible to learn the higher-order structure of dynamic phase - i.e., motion - from the statistics of natural image sequences.