On the analysis and interpretation of inhomogeneous quadratic forms as receptive fields

In this paper we introduce some mathematical and numerical tools to analyze and interpret inhomogeneous quadratic forms. The resulting characterization is in some aspects similar to that given by experimental studies of cortical cells, making it particularly suitable for application to second-order approximations and theoretical models of physiological receptive fields. We first discuss two ways of analyzing a quadratic form by visualizing the coefficients of its quadratic and linear term directly and by considering the eigenvectors of its quadratic term. We then present an algorithm to compute the optimal excitatory and inhibitory stimuli, i.e. the stimuli that maximize and minimize the considered quadratic form, respectively, given a fixed energy constraint. The analysis of the optimal stimuli is completed by considering their invariances, which are the transformations to which the quadratic form is most insensitive. We introduce a test to determine which of these are statistically significant. Next we propose a way to measure the relative contribution of the quadratic and linear term to the total output of the quadratic form. Furthermore, we derive simpler versions of the above techniques in the special case of a quadratic form without linear term and discuss the analysis of such functions in previous theoretical and experimental studies. In the final part of the paper we show that for each quadratic form it is possible to build an equivalent two-layer neural network, which is compatible with (but more general than) related networks used in some recent papers and with the energy model of complex cells. We show that the neural network is unique only up to an arbitrary orthogonal transformation of the excitatory and inhibitory subunits in the first layer

Prototyping Information Visualization in 3D City Models: a Model-based Approach

When creating 3D city models, selecting relevant visualization techniques is a particularly difficult user interface design task. A first obstacle is that current geodata-oriented tools, e.g. ArcGIS, have limited 3D capabilities and limited sets of visualization techniques. Another important obstacle is the lack of unified description of information visualization techniques for 3D city models. If many techniques have been devised for different types of data or information (wind flows, air quality fields, historic or legal texts, etc.) they are generally described in articles, and not really formalized. In this paper we address the problem of visualizing information in (rich) 3D city models by presenting a model-based approach for the rapid prototyping of visualization techniques. We propose to represent visualization techniques as the composition of graph transformations. We show that these transformations can be specified with SPARQL construction operations over RDF graphs. These specifications can then be used in a prototype generator to produce 3D scenes that contain the 3D city model augmented with data represented using the desired technique.Comment: Proc. of 3DGeoInfo 2014 Conference, Dubai, November 201