Total variation denoising in l1l^1 anisotropy

We aim at constructing solutions to the minimizing problem for the variant of Rudin-Osher-Fatemi denoising model with rectilinear anisotropy and to the gradient flow of its underlying anisotropic total variation functional. We consider a naturally defined class of functions piecewise constant on rectangles (PCR). This class forms a strictly dense subset of the space of functions of bounded variation with an anisotropic norm. The main result shows that if the given noisy image is a PCR function, then solutions to both considered problems also have this property. For PCR data the problem of finding the solution is reduced to a finite algorithm. We discuss some implications of this result, for instance we use it to prove that continuity is preserved by both considered problems.Comment: 34 pages, 9 figure

On three soft rectangle packing problems with guillotine constraints

We investigate how to partition a rectangular region of length $L_1$ and height $L_2$ into $n$ rectangles of given areas $(a_1, \dots, a_n)$ using two-stage guillotine cuts, so as to minimize either (i) the sum of the perimeters, (ii) the largest perimeter, or (iii) the maximum aspect ratio of the rectangles. These problems play an important role in the ongoing Vietnamese land-allocation reform, as well as in the optimization of matrix multiplication algorithms. We show that the first problem can be solved to optimality in $\mathcal{O}(n \log n)$, while the two others are NP-hard. We propose mixed integer programming (MIP) formulations and a binary search-based approach for solving the NP-hard problems. Experimental analyses are conducted to compare the solution approaches in terms of computational efficiency and solution quality, for different objectives

Motion of discrete interfaces in periodic media

We study the motion of discrete interfaces driven by ferromagnetic interactions in a two-dimensional periodic environment by coupling the minimizing movements approach by Almgren, Taylor and Wang and a discrete-to-continuous analysis. The case of a homogeneous environment has been recently treated by Braides, Gelli and Novaga, showing that the effective continuous motion is a flat motion related to the crystalline perimeter obtained by $\Gamma$-convergence from the ferromagnetic energies, with an additional discontinuous dependence on the curvature, giving in particular a pinning threshold. In this paper we give an example showing that in general the motion does not depend only on the $\Gamma$-limit, but also on geometrical features that are not detected in the static description. In particular we show how the pinning threshold is influenced by the microstructure and that the effective motion is described by a new homogenized velocity.Comment: arXiv admin note: substantial text overlap with arXiv:1407.694

A new transfer-matrix algorithm for exact enumerations: Self-avoiding polygons on the square lattice

We present a new and more efficient implementation of transfer-matrix methods for exact enumerations of lattice objects. The new method is illustrated by an application to the enumeration of self-avoiding polygons on the square lattice. A detailed comparison with the previous best algorithm shows significant improvement in the running time of the algorithm. The new algorithm is used to extend the enumeration of polygons to length 130 from the previous record of 110.Comment: 17 pages, 8 figures, IoP style file

The Unreasonable Success of Local Search: Geometric Optimization

What is the effectiveness of local search algorithms for geometric problems in the plane? We prove that local search with neighborhoods of magnitude $1/\epsilon^c$ is an approximation scheme for the following problems in the Euclidian plane: TSP with random inputs, Steiner tree with random inputs, facility location (with worst case inputs), and bicriteria $k$-median (also with worst case inputs). The randomness assumption is necessary for TSP

Self-adapting structuring and representation of space

The objective of this report is to propose a syntactic formalism for space representation. Beside the well known advantages of hierarchical data structure, the underlying approach has the additional strength of self-adapting to a spatial structure at hand. The formalism is called puzzletree because its generation results in a number of blocks which in a certain order -- like a puzzle - reconstruct the original space. The strength of the approach does not lie only in providing a compact representation of space (e.g. high compression), but also in attaining an ideal basis for further knowledge-based modeling and recognition of objects. The approach may be applied to any higher-dimensioned space (e.g. images, volumes). The report concentrates on the principles of puzzletrees by explaining the underlying heuristic for their generation with respect to 2D spaces, i.e. images, but also schemes their application to volume data. Furthermore, the paper outlines the use of puzzletrees to facilitate higher-level operations like image segmentation or object recognition. Finally, results are shown and a comparison to conventional region quadtrees is done

Pants Decomposition of the Punctured Plane

A pants decomposition of an orientable surface S is a collection of simple cycles that partition S into pants, i.e., surfaces of genus zero with three boundary cycles. Given a set P of n points in the plane, we consider the problem of computing a pants decomposition of the surface S which is the plane minus P, of minimum total length. We give a polynomial-time approximation scheme using Mitchell's guillotine rectilinear subdivisions. We give a quartic-time algorithm to compute the shortest pants decomposition of S when the cycles are restricted to be axis-aligned boxes, and a quadratic-time algorithm when all the points lie on a line; both exact algorithms use dynamic programming with Yao's speedup.Comment: 5 pages, 1 grayscale figur

Punctured polygons and polyominoes on the square lattice

We use the finite lattice method to count the number of punctured staircase and self-avoiding polygons with up to three holes on the square lattice. New or radically extended series have been derived for both the perimeter and area generating functions. We show that the critical point is unchanged by a finite number of punctures, and that the critical exponent increases by a fixed amount for each puncture. The increase is 1.5 per puncture when enumerating by perimeter and 1.0 when enumerating by area. A refined estimate of the connective constant for polygons by area is given. A similar set of results is obtained for finitely punctured polyominoes. The exponent increase is proved to be 1.0 per puncture for polyominoes.Comment: 36 pages, 11 figure