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Register a new userA Matlab Implementation of a Flat Norm Motivated Polygonal Edge Matching Method using a Decomposition of Boundary into Four 1-Dimensional Currents
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Added by
Simon Morgan
Publication date
2009
fields
Informatics Engineering
and research's language is
English
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We describe and provide code and examples for a polygonal edge matching method.
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Currents represent generalized surfaces studied in geometric measure theory. They range from relatively tame integral currents representing oriented compact manifolds with boundary and integer multiplicities, to arbitrary elements of the dual space of differential forms. The flat norm provides a natural distance in the space of currents, and works by decomposing a $d$-dimensional current into $d$- and (the boundary of) $(d+1)$-dimensional pieces in an optimal way. Given an integral current, can we expect its flat norm decomposition to be integral as well? This is not known in general, except in the case of $d$-currents that are boundaries of $(d+1)$-currents in $mathbb{R}^{d+1}$ (following results from a corresponding problem on the $L^1$ total variation ($L^1$TV) of functionals). On the other hand, for a discretized flat norm on a finite simplicial complex, the analogous statement holds even when the inputs are not boundaries. This simplicial version relies on the total unimodularity of the boundary matrix of the simplicial complex -- a result distinct from the $L^1$TV approach. We develop an analysis framework that extends the result in the simplicial setting to one for $d$-currents in $mathbb{R}^{d+1}$, provided a suitable triangulation result holds. In $mathbb{R}^2$, we use a triangulation result of Shewchuk (bounding both the size and location of small angles), and apply the framework to show that the discrete result implies the continuous result for $1$-currents in $mathbb{R}^2$.
The classification of topological insulators predicts the existence of high-dimensional topological phases that cannot occur in real materials, as these are limited to three or fewer spatial dimensions. We use electric circuits to experimentally implement a four-dimensional (4D) topological lattice. The lattice dimensionality is established by circuit connections, and not by mapping to a lower-dimensional system. On the lattices three-dimensional surface, we observe topological surface states that are associated with a nonzero second Chern number but vanishing first Chern numbers. The 4D lattice belongs to symmetry class AI, which refers to time-reversal-invariant and spinless systems with no special spatial symmetry. Class AI is topologically trivial in one to three spatial dimensions, so 4D is the lowest possible dimension for achieving a topological insulator in this class. This work paves the way to the use of electric circuits for exploring high-dimensional topological models.
The scaled boundary finite element method (SBFEM) is a semi-analytical computational scheme, which is based on the characteristics of the finite element method (FEM) and combines the advantages of the boundary element method (BEM). This paper integrates the scaled boundary finite element method (SBFEM) and the polygonal mesh technique into a new approach to solving the steady-state and transient seepage problems. The proposed method is implemented in Abaqus using a user-defined element (UEL). The detailed implementations of the procedure, defining the UEL element, updating the RHS and AMATRX, and solving the stiffness/mass matrix by the eigenvalue decomposition are presented. Several benchmark problems from seepage are solved to validate the proposed implementation. Results show that the polygonal element of PS-SBFEM has a higher accuracy rate than the standard FEM element in the same element size. For the transient problems, the results between PS-SBFEM and the FEM are in excellent agreement. Furthermore, the PS-SBFEM with quadtree meshes shows a good effect for solving complex geometric in the seepage problem. Hence, the proposed method is robust accurate for solving the steady-state and transient seepage problems. The developed UEL source code and the associated input files can be downloaded from https://github.com/yangyLab/PS-SBFEM.
For $s$ $>$ 0, we consider an algorithm that computes all $s$-well separated pairs in certain point sets in $mathbb{R}^{n}$, $n$ $>1$. For an integer $K$ $>1$, we also consider an algorithm that is a permutation of Dijkstras algorithm, that computes $K$-nearest neighbors using a certain power weighted shortest path metric in $mathbb{R}^{n}$, $n$ $>$ $1$. We describe each algorithm and their respective dependencies on the input data. We introduce a way to combine both algorithms into a fused algorithm. Several open problems are given for future research.
In this paper, we present a decomposition model for stereo matching to solve the problem of excessive growth in computational cost (time and memory cost) as the resolution increases. In order to reduce the huge cost of stereo matching at the original resolution, our model only runs dense matching at a very low resolution and uses sparse matching at different higher resolutions to recover the disparity of lost details scale-by-scale. After the decomposition of stereo matching, our model iteratively fuses the sparse and dense disparity maps from adjacent scales with an occlusion-aware mask. A refinement network is also applied to improving the fusion result. Compared with high-performance methods like PSMNet and GANet, our method achieves $10-100times$ speed increase while obtaining comparable disparity estimation results.
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