Valero et al, Complex-Valued Independent Component Analysis of Natural Images

Natural image statistics is a useful tool for understanding the functions of the visual part of the brain. The research on Natural image statistics mainly focused on the “linear” model which tries to extract the independent sources from original natural images which correspond to Gabor-like receptive fields in the primary visual cortex.

Nowadays, researchers stepped into description of the statistics of the signals after simple “linear” model stage. It is pointed out that a learned squared outputs of the simple cells may lead to complex responses. For instances, while FFT is applied on natural images, there are phases term and magnitude term, which can be generally described by complex numbers. Similarly, in natural images, some image might contain phase information of some source images, and magnitude images from other source images. Furthermore, generally speaking, the information contains in the phase terms are of great importance. Therefore, naturally, Complex Independent Component Analysis comes into play.

Traditional complex ICA, according to the authors in the paper, assume a uniform distribution over the complex plain, which is the phase term, in natural image. This is not always the cases. In the experiment made in the paper, using the traditional complex ICA, the learned features in complex plain differ a lot from actual features, which is shown in the following figure. The blue curve shows the phases of complex ICA sources, and the black curve shows the learned phases of complex ICA sources.

http://cl.ly/190Q0n2O2l312V171p0X

Therefore, in this paper, the author proposed an extension to the traditional complex ICA that also models the phase information in natural images. This is done by assuming a von Mises distribution for the phase information of the output signal, instead of the uniform distribution that the standard cICA assumes. The extension allows for a better fit to the signal, as the phase distributions are often far from uniform. This assumed distribution is capable of capturing two peaks in the phase information. After learning, the learned feature fits better than the feature learned by traditional complex independent component analysis. The result is shown as follows,

http://cl.ly/2s420N1C2W281B3W1V15

In this paper, the author assumed a modified distribution over the phase information of the natural images, and thus had a better fit to the phase information. Therefore, this type of new distribution is good in terms of accuracy.