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Shape analysis of objects in images is a critical area of research, and several approaches, including those that utilize elastic Riemannian metrics, have been proposed. While elastic techniques for shape analysis of curves are pretty advanced, the corresponding results for higher-dimensional objects (surfaces and disks) are less developed. This pap...
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Mode shape analysis of the vibration of cracked nanobeam on elastic foundation is presented using the differential transform method. Euler-Bernoulli beam theory and Eringen’s theory of nonlocal elasticity are used to study the dynamic behaviour of nanobeam. The crack of the beam is modelled by a rotational spring where the beam is considered as two...
Citations
... The field is generated by minimizing an energy functional. By developing a numerical technique, Zhang and Srivastava [20] studied the properties of solid objects to map tensor fields back to the object space. ...
... For GVF, H (z), the curl(H ) = ∇ × H = 0 and Eq. (20) becomes, ...
... In Eqs. (20) and (22), 2 z +i |∇×H | 2 z and the SP z has a shape of a repelling or attracting focus in a clockwise or counter clockwise direction. ...
The objects’ features play significant role in the machine learning classification. The present paper proofs and validates that the shapes of vector field (VF) singular points (SPs) embedded into image objects may improve classification accuracy. For this purpose the present paper develops two VFs vu^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{{\hat{u}}}$$\end{document} and vϕ^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\hat{\phi }}$$\end{document} with real and complex SPs. The VFs are developed on the solution u^(x,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{u}}(x,y)$$\end{document} of a particular form of the Poisson equation. Further, we define the mappings between the SPs of ∇u^(x,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla {\hat{u}}(x,y)$$\end{document}, vu^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{{\hat{u}}}$$\end{document} and vϕ^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{{\hat{\phi }}}$$\end{document}. Next, we develop the local Polya’s model of a VF and prove that the shapes of the SPs are invariant according to scaling, translation and weak rotations. This property implies that embedding the shapes of the SPs into the image objects augments the set of objects features, which leads to the advantage of increasing the classification statistics. We validate the invariance and the advantage with sets of experiments classifying the public image datasets ISIC2020 and COIL100. For the purpose of classification, we designed a new convolution neural network optimized to classify SP shapes and image objects features. The paper ends with conclusions on the contributions, advantages and the bottlenecks of this study.
... The space of all Riemannian (Lorentzian, resp.) metrics plays an important role in many areas of pure and applied mathematics and in particular in mathematical physics: it is the natural configuration space for Einstein's equation in general relativity [30], it is the central object in Teichmüller theory [38,79], and it appears in the context of mathematical shape analysis [21,80]. The above are just a few examples of an extensive list. ...
We show for a certain class of operators A and holomorphic functions f that the functional calculus A↦f(A)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A\mapsto f(A)$$\end{document} is holomorphic. Using this result we are able to prove that fractional Laplacians (1+Δg)p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1+\Delta ^g)^p$$\end{document} depend real analytically on the metric g in suitable Sobolev topologies. As an application we obtain local well-posedness of the geodesic equation for fractional Sobolev metrics on the space of all Riemannian metrics.
