Archive for February 2013

Neurobiology of Scale-Invariance

Image Credits: Paul Paradis


The scale-Invariance term indicates various properties like self-similarity or spatial scale-invariance, avalanche dynamics or temporal scale-invariance and complex networks or topological scale-invariance.  Generally speaking, scale-invariant systems have some properties that remain constant when looking at them either at different length or time scales. Constant quantities allow prediction of future behavior, no surprise that conserved quantities are fundamental in physics. This invariance is somewhat different though; still it can be used to extract useful information.

Visual perception is far more complex and powerful than our experience suggests.  Moreover, in attempting to understand vision and implement it in a computational device, the fact that a species’ senses developed in concert with the ecological niche in which that species evolved is a natural consideration; in this case, that means an evolutionary visual context consisting of natural objects, including mountains, rivers, trees, and other animals. Noting that neural representations of visual inputs are related to their statistical structure, natural structures display an inseparable size hierarchy indicative of scale invariance, and scale invariance also occurs near a critical point in wide range of physical systems including ferromagnetic; researchers at the Salk Institute for Biological Studies and the University of California-San Diego recently demonstrated what their paper describes as 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 traditional way images are represented in vision is by an array of pixels with gray levels.  However, we know that visual perception is based on a log scale of luminance. The challenge was to find a new representation that would make the log levels explicit.  The idea was in using bit planes, later generalized to power in any integer base.  Scientists started looking at the bit planes of natural images.  It became apparent that each layer looked like a 2D Ising model at different temperatures; that is, the high-order bits were cold and the low order bits were hot.  An Ising model is a mathematical model of ferromagnetism in statistical mechanics, consisting of discrete variables that represent magnetic dipole moments of atomic spins that can be in one of two states (+1 or −1). Taken together, these bit planes represent a 3D quasimagnet with interesting properties. Understanding retinal encoding and possibly obtaining further insight into how the neocortex represents scale invariance requires, in turn, an understanding of the statistical structure in natural image hierarchies. Moreover, the brain is not a passive image receptor, but rather actively generates sensory models derived from sensory experience. The Bolzmann machine is a device that spin glasses with arbitrary connectivity running at a finite temperature, generalizing Hopfield nets, which run at zero temperature can represent image statistical structure.  Application of this idea is a unique approach, in which certain aspects of the Boltzmann machine’s input representations are learned from natural images.

There is remarkably simple learning algorithm that finds the connection weights for a network that could represent the probability distribution for an ensemble of inputs.  When scientists applied Boltzmann machine learning to natural images as inputs, they found positive pairwise connections that fell off with distance on each layer, much like the 2D Ising model for a ferromagnet, and negative pairwise weights between the layers, representing antiferromagnetic interactions. The theory of second-order phase transitions is vital to understanding the significance of what the scientists had found.  There are 15 bit planes corresponding to pixels with 15 bit integers, each corresponding to a different temperature.

Scale invariance had been observed in natural images for decades based on the power law drop-off in power as a function of spatial scale.  At a phase transition, the spatial correlation length becomes infinite and there is a critical slowing. This suggests that the reason there is structure at every spatial scale in the natural world is because nature is, in some sense, sitting at a phase transition between order and disorder. In terms of the evolution and neurobiology of perceptual invariants, the biological systems that have evolved to survive in this world may take advantage of this structure, and in particular the organization of the visual system may reflect those statistics and most of the information in natural images is captures in 3 bit planes, which may be why photoreceptors are linear over a single order of magnitude. Adaptation mechanisms in the retina shift the linear region over 10 orders of magnitude in luminance.  Scientists have trained the Boltzmann machine on only the connections between pixels in the "visible" input layer. The next step is to use this as the input layer in a hierarchy of hidden layers, such as that found in human visual systems, which are around 12 layers deep.  There are great advances in computer power and algorithms that now allow Boltzmann machines to be trained in deep networks. In the longer term, this new input representation may benefit computer vision.


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