On the Receptive Fields of Markov Random Fields: Predictions from a Probabilistic Model of Scene Statistics

Michael J. Black and Stefan Roth

Department of Computer Science, Brown University

A large literature on natural image statistics suggests the modeling of images in terms of a linear combination of Gabor-like filters. A variety of sparse image coding methods all arrive at similar filters and it has been often noted how these filters resemble the observed spatial receptive fields of V1 simple cells. Related recent work by Welling et al (2002) proposes a probabilistic model of image patches as a product of Student t distributions. As with standard sparse coding methods, this approach defines a model of image patches and not images. To model the probability of an entire image one must account for spatially overlapping receptive fields. Previous sparse coding methods fail to model the correlated image structure present in overlapping image regions.

Recently we proposed a "field of experts" (FOE) model (Roth and Black 2005) that defines a probability density over images by learning a Markov random field (MRF) model with a large neighborhood system. Mathematically the FOE model appears similar to previous methods yet, by capturing the correlated image structure in overlapping receptive fields, the resulting predictions regarding the nature of the underlying filters is radically changed. No longer do these filters have the characteristic appearance of Gabor-like filters. The question then arises: if the brain accounts for spatial correlation of natural images in a probabilistically sound way (e.g. as an MRF) then could cells in visual cortex "implement" these novel filters and, if so, could current analysis techniques uncover them?

Following Ringach et al (2002), we address this by simulating the response of cells in which images are filtered by our FOE filters and the filter response undergoes a standard threshold non-linearity. Using standard stimuli and methods we find that simulated cells with these novel filters exhibit orientation tuning consistent with observations of simple cells in V1. We mapped the receptive fields of these synthetic cells using reverse correlation and observed that, in most cases, the recovered receptive fields appeared similar to standard orientation-tuned simple cells (i.e. they are well fit by a Gabor). In a some cases we observe more complex receptive fields which are suggestive of more complex feature detectors. The results suggest that standard reverse correlation methods for recovering receptive fields my be insufficient to elucidate the underlying computational properties of V1 populations. In particular if the brain effectively implements an MRF, probing individual cells may not reveal the underlying model. Current experimental data may be insufficient to rule out our MRF hypothesis. The results also suggest that further analysis of V1 cells that do not exhibit classical orientation tuning may provide evidence for the processing suggested by the FOE model.


D. L. Ringach, M. J. Hawken and R. Shapley, Receptive field structure of neurons in monkey primary visual cortex revealed by stimulation with natural iamge sequences, Journal of Vision, Vol 2, pp. 12-24, 2002.

S. Roth and M. J. Black, Fields of experts: A framework for learning image priors with applications, under review, IEEE Conference on Computer Vision and Pattern Recognition, 2005.

M. Welling, G. Hinton, and S.Osindero. Learning sparse topographic representations with products of Student-t distributions. NIPS, pp. 1359-1366, 2002


This work was supported by NIH-NINDS R01 NS 50967-01 as part of the NSF/NIH Collaborative Research in Computational Neuroscience Program and by Intel Corporation.


Energy Based Models of Motor Cortical Population Activity
Wood, F.and Black, M.
to appear Society for Neuroscience, 2005. Online.