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The two-dimensional Gabor function adapted to natural image statistics: A model of simple-cell receptive fields and sparse structure in images

The two-dimensional Gabor function is adapted to natural image statistics, leading to a tractable probabilistic generative model that can be used to model simple-cell receptive-field profiles, or generate basis functions for sparse coding applications. Learning is found to be most pronounced in three Gabor-function parameters representing the size and spatial frequency of the two-dimensional Gabor function, and characterized by a non-uniform probability distribution with heavy tails. All three parameters are found to be strongly correlated: resulting in a basis of multiscale Gabor functions with similar aspect ratios, and size-dependent spatial frequencies. A key finding is that the distribution of receptive-field sizes is scale-invariant over a wide range of values, so there is no characteristic receptive-field size selected by natural image statistics. The Gabor-function aspect ratio is found to be approximately conserved by the learning rules and is therefore not well-determined by natural image statistics. This allows for three distinct solutions: a basis of Gabor functions with sharp orientation resolution at the expense of spatial-frequency resolution; a basis of Gabor functions with sharp spatial-frequency resolution at the expense of orientation resolution; or a basis with unit aspect ratio. Arbitrary mixtures of all three cases are also possible. Two parameters controlling the shape of the marginal distributions in a probabilistic generative model fully account for all three solutions. The best-performing probabilistic generative model for sparse coding applications is found to be a Gaussian copula with Pareto marginal probability density functions.

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