Bogoliubov Renormalization Group and Symmetry of Solution in Mathematical Physics

Evolution of the concept known in the theoretical physics as the Renormalization Group (RG) is presented. The corresponding symmetry, that has been first introduced in QFT in mid-fifties, is a continuous symmetry of a solution with respect to transformation involving parameters (e.g., of boundary condition) specifying some particular solution. After short detour into Wilson's discrete semi-group, we follow the expansion of QFT RG and argue that the underlying transformation, being considered as a reparameterisation one, is closely related to the self-similarity property. It can be treated as its generalization, the Functional Self-similarity (FS). Then, we review the essential progress during the last decade of the FS concept in application to boundary value problem formulated in terms of differential equations. A summary of a regular approach recently devised for discovering the RG = FS symmetries with the help of the modern Lie group analysis and some of its applications are given. As a main physical illustration, we give application of new approach to solution for a problem of self-focusing laser beam in a non-linear medium.Comment: Contribution to the proceedings of conference "RG 2000" (Taxco, Mexico, Jan. 1999). To be published in Physics Report

M\"obius Invariants of Shapes and Images

Identifying when different images are of the same object despite changes caused by imaging technologies, or processes such as growth, has many applications in fields such as computer vision and biological image analysis. One approach to this problem is to identify the group of possible transformations of the object and to find invariants to the action of that group, meaning that the object has the same values of the invariants despite the action of the group. In this paper we study the invariants of planar shapes and images under the M\"obius group $\mathrm{PSL}(2,\mathbb{C})$, which arises in the conformal camera model of vision and may also correspond to neurological aspects of vision, such as grouping of lines and circles. We survey properties of invariants that are important in applications, and the known M\"obius invariants, and then develop an algorithm by which shapes can be recognised that is M\"obius- and reparametrization-invariant, numerically stable, and robust to noise. We demonstrate the efficacy of this new invariant approach on sets of curves, and then develop a M\"obius-invariant signature of grey-scale images

Dynamic texture recognition using time-causal and time-recursive spatio-temporal receptive fields

This work presents a first evaluation of using spatio-temporal receptive fields from a recently proposed time-causal spatio-temporal scale-space framework as primitives for video analysis. We propose a new family of video descriptors based on regional statistics of spatio-temporal receptive field responses and evaluate this approach on the problem of dynamic texture recognition. Our approach generalises a previously used method, based on joint histograms of receptive field responses, from the spatial to the spatio-temporal domain and from object recognition to dynamic texture recognition. The time-recursive formulation enables computationally efficient time-causal recognition. The experimental evaluation demonstrates competitive performance compared to state-of-the-art. Especially, it is shown that binary versions of our dynamic texture descriptors achieve improved performance compared to a large range of similar methods using different primitives either handcrafted or learned from data. Further, our qualitative and quantitative investigation into parameter choices and the use of different sets of receptive fields highlights the robustness and flexibility of our approach. Together, these results support the descriptive power of this family of time-causal spatio-temporal receptive fields, validate our approach for dynamic texture recognition and point towards the possibility of designing a range of video analysis methods based on these new time-causal spatio-temporal primitives.Comment: 29 pages, 16 figure

Cheng Equation: A Revisit Through Symmetry Analysis

The symmetry analysis of the Cheng Equation is performed. The Cheng Equation is reduced to a first-order equation of either Abel's Equations, the analytic solution of which is given in terms of special functions. Moreover, for a particular symmetry the system is reduced to the Riccati Equation or to the linear nonhomogeneous equation of Euler type. Henceforth, the general solution of the Cheng Equation with the use of the Lie theory is discussed, as also the application of Lie symmetries in a generalized Cheng equation.Comment: 10 pages. Accepted for publication in Quaestiones Mathematicae journa