Geometric Maxwell equations and the structure of diffusive scale spaces

Nicolas F. Rougon; Francoise J. Preteux

doi:10.1117/12.216348

11 August 1995 Geometric Maxwell equations and the structure of diffusive scale spaces

Nicolas F. Rougon, Francoise J. Preteux

Proceedings Volume 2568, Neural, Morphological, and Stochastic Methods in Image and Signal Processing; (1995) https://doi.org/10.1117/12.216348
Event: SPIE's 1995 International Symposium on Optical Science, Engineering, and Instrumentation, 1995, San Diego, CA, United States

Abstract

In (linear or nonlinear) diffusive scale-space representations, local variations of the luminance field with respect to infinitesimal scale transitions are described via a first-order parabolic partial differential equation modeling a generalized diffusion process. A geometric characterization of the scale-space structure is then classically derived by analyzing the properties of the deformation flow induced by scale transitions along specific geometric structures embedded on the photometric surface. In particular, studying the simultaneous deformation of the dual families of curves consisting of isophotes and stream lines of the luminance field yields a Euclidean-invariant geometric description of generalized diffusion processes. In this paper, the generalized diffusion equation is interpreted within the framework of the relativistic electromagnetic (EM) theory as a Lorentz gauge condition expressing the trace-invariance of an EM quadripotential with covariant scalar and contravariant vector components respectively related to luminance and geometric properties of the image. This gauge condition determines an EM quadrifield and quadricharge which satisfy Maxwell equations. Deriving the general expressions of these quadrivectors as functions of Euclidean characteristics of isophotes and stream lines leads to identifying Lorentz-invariants which synthetize under an extremely compact form intrinsic multiscale image properties. In addition, weak formulations of diffusive scale-spaces are consistently re-expressed in terms of Em energy density. The specific cases of linear scale-spaces, corresponding to purely electric fields, and of classical anisotropic diffusion models are studied in detail, providing a significant insight in the understanding of the deep structure of diffusive scale-spaces.

Citation Download Citation

Nicolas F. Rougon and Francoise J. Preteux "Geometric Maxwell equations and the structure of diffusive scale spaces", Proc. SPIE 2568, Neural, Morphological, and Stochastic Methods in Image and Signal Processing, (11 August 1995); https://doi.org/10.1117/12.216348

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