Scale-Time Kernels and Models

Receptive field sensitivity profiles of visual front-end cells in the LGN and V1 area in intact animals can be measured with increasing accuracy, both in the spatial and temporal domain. This urges the need for mathematical models. Scale-space theory, as a theory of (multiscale) apertures as operators on observed data, is concerned with the mathematical modeling of front-end visual system behaviour. This paper compares recent measurements on the spatio-temporal respons of LGN cell and V1 simple cell receptive fields [1] with Koenderink’s results from axiomatic reasoning for a real-time measuring spatio-temporal differential operator [2]. In this model time must be logarithmically remapped to make the operation causal in the temporal domain. 1 Scale-Space Kernel Derivation from Entropy Maximization The Gaussian kernel as the fundamental linear scale-space kernel for an uncommitted observation is now well established. Many fundamental derivations have been proposed (see for an extensive and complete overview Weickert [3]). In this paper we present an alternative way to derive the Gaussian kernel as the scalespace kernel of an uncommitted observation. It is based on the notion that the ’uncommittedness’ is expressed in a statistical way using the entropy of the observed signal. The reasoning is due to Mads Nielsen, IT-University Copenhagen [4]: First of all, we want to do a measurement, i.e. we have a device which has some integration area with a finite width by necessity. The measurement should be done at all locations in the same way, i.e. with either a series of identical detectors, or the same detector measuring at all places: the measurement should be invariant for translation. We want the measurement to be linear in the signal to be measured (e.g. the intensity): invariance for translation along the intensity axis. These requirements lead automatically to the formulation that the observation must be a convolution: h(x) = ∫ ∞ −∞ L(α)g(x−α)dα. L(x) is the observed variable, e.g. the luminance, g(x) is the aperture function, h(x) the result of the measurement. M. Kerckhove (Ed.): Scale-Space 2001, LNCS 2106, pp. 255–263, 2001. c © Springer-Verlag and IEEE/CS 2001 256 Bart M. ter Haar Romeny, Luc M.J. Florack, and Mads Nielsen The aperture function g(x) should be a unity filter, i.e. normalized, which means that the integral over its weighting profile should be unity: ∫ ∞ −∞ g(x)dx = 1. The mean of the filter g(x) should be at the location where we measure, e.g. at x0, so the expected value (or first moment) should be x0 : ∫ ∞ −∞ xg(x)dx = x0. Because we may take any point for x0, we may take for our further calculations as well the point x0 = 0. The size of the aperture is an essential element. We want to be free in choice of this size, so at least we want to find a family of filters where this size is a free parameter. We can then monitor the world at all these sizes by ’looking through’ the complete set of kernels simultaneously. We call this ’size’ σ. It has the dimension of length, and is the yardstick of our measurement. We call it the inner scale. Every physical measurement has an inner scale. It can be μm or lightyears, we need for every dimension a yardstick: σ. If we weight distances r(x) with our kernel, so we get ∫ ∞ −∞ r(x)g(x)dx, we will use r(x) = x 2 since with this choice we separate the dimensions: two orthogonal vectors fullfill r(a + b) = r(a) + r(b). We call the weighted metric σ: ∫ ∞ −∞ x g(x)dx = σ. Finally we incorporate the request to be as uncommitted as possible. We want no filter with some preference at this first stage of the observation. We want, in statistical terms, the ’orderlessness’ or disorder of the measurement as large as possible. There should be no ordering, ranking, structuring or whatsoever. Physically the measure for disorder is expressed through the entropy H = ∫ ∞ −∞ g(x) ln g(x)dx where ln x is the natural logarithm. We look for the g(x) for which the entropy is maximal, given the constraints derived before: ∫ ∞