Learning Nonlinear Statistical Regularities in Natural Images by Modeling the Outer Product of Image Intensities

Abstract

It is well known that there exist nonlinear statistical regularities in natural images. Existing approaches for capturing such regularities always model the image intensities by assuming a parameterized distribution for the intensities and learn the parameters. In the letter, we propose to model the outer product of image intensities by assuming a gaussian distribution for it. A two-layer structure is presented, where the first layer is nonlinear and the second layer is linear. Trained on natural images, the first-layer bases resemble the receptive fields of simple cells in the primary visual cortex (V1), while the second-layer units exhibit some properties of the complex cells in V1, including phase invariance and masking effect. The model can be seen as an approximation of the covariance model proposed in Karklin and Lewicki (2009) but has more robust and efficient learning algorithms.

Modeling outer products of features for image classification

“…To be self-contained, we briefly review the outer product model [8] first. Then, we demonstrate the advantage of this model over sparse coding with a toy problem, and discuss how it can be used for image classification.…”

“…Based on the statistics of outer products of natural image patches, we presented a model to reproduce properties of complex cells in V1 [8], which are known to be selective to orientations but invariant to phase changes. Stemmed from the discovery that natural image patches can be characterized by their covariance in the space of linear filter responses [12], this model has a simpler form considering that the outer product is closely related to the empirical covariance matrix.…”

“…Following the proof sketch in [13], and noting that (3) is convex, it is straightforward to prove that this modified feature-sign algorithm converges to the global optimum of the problem, as each feature-sign step strictly lowers the objective function until no improvements can be made. A rigorous proof can be found in [8].…”

“…Recently we proposed to model the outer product of the input data in order to capture its higher-order statistics, which has led to higher-level representation of the input [8]. The .…”