Mathematical Methods for Signal and Image Analysis and Representation by Luc Florack & Remco Duits & Geurt Jongbloed & Marie-Colette van Lieshout & Laurie Davies

Author:Luc Florack & Remco Duits & Geurt Jongbloed & Marie-Colette van Lieshout & Laurie Davies
Language: eng
Format: epub
ISBN: 9781447123538
Publisher: Springer London

We then identify , the usual gradient of the scalar function S evaluated at x. The unit σ(x) is treated as a local basis vector for the one-dimensional fibre at x, and so the quantity must itself be a section, i.e. a local multiple of the basis vector σ(x). In this way we obtain a local gauge field A (a covector field) with amplitude A(x) at x:

All in all, the components of the covariant derivative of the signal relative to a local basis can be written as

with2

Effectively the whole procedure thus boils down to what is known as the “principle of minimal substitution” in physics, whereby one replaces ordinary differentiation of the signal function (i.e. the coefficient of the local basis section σ(x)), by covariant differentiation:3∂ μS(x)→(∂ μ+A μ(x))S(x).

It remains an outstanding problem how the visual system gauges itself after exposure to a given signal. In any case, prolonged exposure to a stationary scene (S(x)>0 independent of time) and deprived from the possibility to make saccadic eye movements “nullifies” one’s percept completely, which can be understood as an adaptation mechanism in which the input stimulus has somehow evolved towards a “covariantly constant” steady state, i.e. , inducing a gauge field

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