Hierarchical model of natural images and the origin of scale invariance

Abstract

The study of natural images and how our brain processes them has been an area of intense research in neuroscience, psychology, and computer science. We introduced a unique approach to studying natural images by decomposing images into a hierarchy of layers at different logarithmic intensity scales and mapping them to a quasi-2D magnet. The layers were in different phases: "cold" and ordered at large-intensity scales, "hot" and disordered at small-intensity scales, and going through a second-order phase transition at intermediate scales. There was a single "critical" layer in the hierarchy that exhibited long-range correlation similar to that found in the 2D Ising model of ferromagnetism at the critical temperature. We also determined the interactions between layers mapped from natural images and found mutual inhibition that generated locally "frustrated" antiferromagnetic states. Almost all information in natural images was concentrated in a few layers near the phase transition, which has biological implications and also points to the hierarchical origin of scale invariance in natural images.critical point | vision | generative models O ur visual system evolved to survive in nature with scenes of mountains, rivers, trees, and other animals (1). The neural representations of visual inputs are related to their statistical structure (1-3). Structures in nature come in a hierarchy of sizes that cannot be separated, a signature of scale invariance, which also occurs near a critical point in many physical systems. The classic example of a critical point is a uniaxial ferromagnetic system going through a second-order phase transition in a zero magnetic field by increasing temperature. At the critical point, the system loses its magnetization due to thermal fluctuations. There are large regions ("islands") that are magnetized in one direction but are surrounded by large regions ("seas") that are magnetized in the opposite direction. The seas themselves are embedded in bigger islands, ad infinitum. The total magnetization is zero, but the correlation length diverges, which is visualized by growth of the sizes of seas and islands with the system size. At the critical point, the system is free of a length scale because fluctuations occur at scales of all lengths. The infinite correlation length is thus intricately linked with scale invariance. The scale invariance in natural images was first characterized by the 1/f 2 spatial power spectrum of pixel intensities (2). Here, we study scaling properties of natural images at a deeper level by finding a hierarchy of statistical structures, in which the scale invariance emerges near a second-order phase transition.Images are preprocessed in the retina by a complex network with ∼55 distinct cell types in mammals (5). The cerebral cortex receives a spatiotemporal stream of spikes that contain all the information in the visual inputs that has been coded by the retina. Understanding the hierarchies of statistical structures in natural images is essential for better understanding how ...

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