Two-dimensional spatial linear filters are constrained by general uncertainty relations that limit their attainable information resolution for orientation, spatial frequency, and two-dimensional (2D) spatial position. The theoretical lower limit for the joint entropy, or uncertainty, of these variables is achieved by an optimal 2D filter family whose spatial weighting functions are generated by exponentiated bivariate second-order polynomials with complex coefficients, the elliptic generalization of the one-dimensional elementary functions proposed in Gabor’s famous theory of communication [ J. Inst. Electr. Eng.93, 429 ( 1946)]. The set includes filters with various orientation bandwidths, spatial-frequency bandwidths, and spatial dimensions, favoring the extraction of various kinds of information from an image. Each such filter occupies an irreducible quantal volume (corresponding to an independent datum) in a four-dimensional information hyperspace whose axes are interpretable as 2D visual space, orientation, and spatial frequency, and thus such a filter set could subserve an optimally efficient sampling of these variables. Evidence is presented that the 2D receptive-field profiles of simple cells in mammalian visual cortex are well described by members of this optimal 2D filter family, and thus such visual neurons could be said to optimize the general uncertainty relations for joint 2D-spatial–2D-spectral information resolution. The variety of their receptive-field dimensions and orientation and spatial-frequency bandwidths, and the correlations among these, reveal several underlying constraints, particularly in width/length aspect ratio and principal axis organization, suggesting a polar division of labor in occupying the quantal volumes of information hyperspace. Such an ensemble of 2D neural receptive fields in visual cortex could locally embed coarse polar mappings of the orientation–frequency plane piecewise within the global retinotopic mapping of visual space, thus efficiently representing 2D spatial visual information by localized 2D spectral signatures.

Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters.

http://experimentationlab.berkeley.edu/sites/default/files/AFMImages/butt_AFM.pdf

Force measurements with the atomic force microscope: Technique, interpretation and applications

The relative efficiency of any particular image-coding scheme should be defined only in relation to the class of images that the code is likely to encounter. To understand the representation of images by the mammalian visual system, it might therefore be useful to consider the statistics of images from the natural environment (i.e., images with trees, rocks, bushes, etc). In this study, various coding schemes are compared in relation to how they represent the information in such natural images. The coefficients of such codes are represented by arrays of mechanisms that respond to local regions of space, spatial frequency, and orientation (Gabor-like transforms). For many classes of image, such codes will not be an efficient means of representing information. However, the results obtained with six natural images suggest that the orientation and the spatial-frequency tuning of mammalian simple cells are well suited for coding the information in such images if the goal of the code is to convert higher-order redundancy (e.g., correlation between the intensities of neighboring pixels) into first-order redundancy (i.e., the response distribution of the coefficients). Such coding produces a relatively high signal-to-noise ratio and permits information to be transmitted with only a subset of the total number of cells. These results support Barlow's theory that the goal of natural vision is to represent the information in the natural environment with minimal redundancy.

/pdf/relations-between-the-statistics-of-natural-images-and-the-12zwc2ox1f.pdf

Relations between the statistics of natural images and the response properties of cortical cells.

Unsupervised texture segmentation using Gabor filters

Neuronal Synchrony: A Versatile Code for the Definition of Relations?

Membranes are the most common cellular structures in both plants and animals. They are now recognized as being involved in almost all aspects of cellular activity ranging from motility and food entrapment in simple unicellular organisms, to energy transduction, immunorecognition, nerve conduction and biosynthesis in plants and higher organisms. This functional diversity is reflected in the wide variety of lipids and particularly of proteins that compose different membranes. An understanding of the physical principles that govern the molecular organization of membranes is essential for an understanding of their physiological roles since structure and function are much more interdependent in membranes than in, say, simple chemical reactions in solution. We must recognize, however, that the word ‘understanding’ means different things in different disciplines, and nowhere is this more apparent than in this multidisciplinary area where biology, chemistry and physics meet.

/pdf/physical-principles-of-membrane-organization-2enzb9mlqb.pdf

Physical principles of membrane organization.

On the basis of measured receptive field profiles and spatial frequency tuning characteristics of simple cortical cells, it can be concluded that the representation of an image in the visual cortex must involve both spatial and spatial frequency variables. In a scheme due to Gabor, an image is represented in terms of localized symmetrical and antisymmetrical elementary signals. Both measured receptive fields and measured spatial frequency tuning curves conform closely to the functional form of Gabor elementary signals. It is argued that the visual cortex representation corresponds closely to the Gabor scheme owing to its advantages in treating the subsequent problem of pattern recognition.

Mathematical description of the responses of simple cortical cells.

Repulsion of interfaces due to boundary water

Chain ordering in liquid crystals. II. Structure of bilayer membranes.

In the liquid crystalline phases of many organic materials molecular end chains take part in the ordering process. For a single chain, this ordering is calculated by exact summation over all conformations of a chain in the molecular field due to the neighboring molecules. The nematic orientational ordering of a system where molecules consist of rigid parts and chains is then obtained within a self‐consistent molecular field approximation. The results are in very good agreement with the experimental phase diagrams for homologous series of liquid crystals, and for the first time provide the explanation for the well known even‐odd effect in isotropic‐nematic transition temperatures. In agreement with observations, the corresponding transition entropies exhibit both the increase with increasing end‐chain length and the even‐odd variations. The decrease of order along the chain is also calculated and compared with the available data.

Stjepan Marčelja

Papers

Physical principles of membrane organization.

Mathematical description of the responses of simple cortical cells.

Repulsion of interfaces due to boundary water

Chain ordering in liquid crystals. II. Structure of bilayer membranes.

Chain ordering in liquid crystals. I. Even‐odd effect