A dynamic scale-space paradigm
We present a novel mathematical, physical and logical framework for describing an input image of the dynamics of physical fields, in particular the optic field dynamics. Our framework is required to be invariant under a particular gauge group, i.e., a group or set of transformations consistent with the symmetries of that physical field dynamics enveloping renormalisation groups. It has to yield a most concise field description in terms of a complete and irreducible set of equivalences or invariants. Furthermore, it should be robust to noise, i.e., unresolvable perturbations (morphisms) of the physical field dynamics present below a specific dynamic scale, possibly not covered by the gauge group, do not affect Lyapunov or structural stability measures expressed in equivalences above that dynamic scale. The related dynamic scale symmetry encompasses then a gauge invariant similarity operator with which similarly prepared ensembles of physical field dynamics are probed and searched for partial equivalences coming about at higher scales. The framework of our dynamic scale-space paradigm is partly based on the initialisation of joint (non)local equivalences for the physical field dynamics external to, induced on and stored in a vision system and represented by an image, possibly at various scales. These equivalences are consistent with the scale-space paradigm considered and permit a faithful segmentation and interpretation of the dynamic scale-space at initial scale. Among the equivalences are differential invariants, integral invariants and topological invariants not affected by the considered gauge group. These equivalences form a quantisation of the external, induced and stored physical field dynamics, and are associated to a frame field, co-frame field, metric and/or connection invariant under the gauge group. Examples of these equivalences are the curvature and torsion two-forms of general relativity, the Burgers and Frank vector density fields of crystal theory (in both disciplines these equivalences measure the inhomogeneity of translational and (affine) rotation groups over space-time), and the winding numbers and other topological charges popping up in electromagnetism and chromodynamics. Besides based on a gauge invariant initialisation of equivalences the framework of our dynamic scale-space paradigm assumes that a robust, i.e. stable and reproducible, partially equivalent representation of the physical field dynamics is acquired by a multi-scale filtering technique adapted to those initial equivalences. Effectively, the hierarchy of nested structures of equivalences, by definition too invariant under the gauge group, is obtained by applying an exchange principle for a free energy of the physical field dynamics (represented through the equivalences) that in turn is linked to a statistical partition function. This principle is operationalised as a topological current of free energy between different regions of the physical field dynamics. It translates for each equivalence into a process governed by a system of integral and/or partial differential equations (PDES) with local and global initial-boundary conditions (IBC). The scaled physical field dynamics is concisely classified in terms of local and nonlocal equivalences, conserved densities or curvatures of the dynamic scale-space paradigm that in generally are not coinciding with all initial equivalences. Our dynamic scale-space paradigm distinguishes itself intrinsically from the standard ones that are mainly developed for scalar fields. A dynamic scale-space paradigm is also operationalised for non-scalar fields like curvature and torsion tensor fields and even more complex nonlocal and global topological fields supported by the physical field dynamics. The description of the dynamic scale-spaces are given in terms of again equivalences, and the paradigms, in terms, of symmetries, curvatures and conservation laws. The topological characteristics of the paradigm form then a representation of the logical framework. A simple example of a dynamic scale-space paradigm is presented for a time-sequence of two-dimensional satellite images in the visual spectrum. The segmentation of the sequence in fore- and background dynamics at various scales is demonstrated together with a detection of ridges, courses and inflection lines allowing a concise triangulation of the image. Furthermore, the segmentation procedure of a dynamic scale-space is made explicit allowing a true hierarchically description in terms of nested equivalences. How to unify all the existing scale-space paradigms using our frame work is illustrated. This unification comes about by a choice of gauge and renormalisation group, and setting up a suitable scale-space paradigm that might be user-defined. How to extend and to generalise the existing scale-space paradigm is elaborated on. This is illustrated by pointing out how to retain a pure topological or covariant scale-space paradigm from an initially segmented image that instead of a scalar field also can represent a density field coinciding with dislocation and disclination fields capturing the cutting and pasting procedures underlying the image formation.
gauge group equivalence frame field co-frame field metric connection curvature exchange principle free energy statistical partition function topological current conserved density