Approximating the Maximum Overlap of Polygons under Translation

Let $P$ and $Q$ be two simple polygons in the plane of total complexity $n$, each of which can be decomposed into at most $k$ convex parts. We present an $(1-\varepsilon)$-approximation algorithm, for finding the translation of $Q$, which maximizes its area of overlap with $P$. Our algorithm runs in $O(c n)$ time, where $c$ is a constant that depends only on $k$ and $\varepsilon$. This suggest that for polygons that are "close" to being convex, the problem can be solved (approximately), in near linear time

Probabilistic Matching of Planar Regions

We analyze a probabilistic algorithm for matching shapes modeled by planar regions under translations and rigid motions (rotation and translation). Given shapes $A$ and $B$, the algorithm computes a transformation $t$ such that with high probability the area of overlap of $t(A)$ and $B$ is close to maximal. In the case of polygons, we give a time bound that does not depend significantly on the number of vertices

Approximating Convex Shapes With Respect to Symmetric Difference Under Homotheties

The symmetric difference is a robust operator for measuring the error of approximating one shape by another. Given two convex shapes P and C, we study the problem of minimizing the volume of their symmetric difference under all possible scalings and translations of C. We prove that the problem can be solved by convex programming. We also present a combinatorial algorithm for convex polygons in the plane that runs in O((m+n) log^3(m+n)) expected time, where n and m denote the number of vertices of P and C, respectively

Photonics design tool for advanced CMOS nodes

Recently, the authors have demonstrated large-scale integrated systems with several million transistors and hundreds of photonic elements. Yielding such large-scale integrated systems requires a design-for-manufacture rigour that is embodied in the 10 000 to 50 000 design rules that these designs must comply within advanced complementary metal-oxide semiconductor manufacturing. Here, the authors present a photonic design automation tool which allows automatic generation of layouts without design-rule violations. This tool is written in SKILL, the native language of the mainstream electric design automation software, Cadence. This allows seamless integration of photonic and electronic design in a single environment. The tool leverages intuitive photonic layer definitions, allowing the designer to focus on the physical properties rather than on technology-dependent details. For the first time the authors present an algorithm for removal of design-rule violations from photonic layouts based on Manhattan discretisation, Boolean and sizing operations. This algorithm is not limited to the implementation in SKILL, and can in principle be implemented in any scripting language. Connectivity is achieved with software-defined waveguide ports and low-level procedures that enable auto-routing of waveguide connections.Comment: 5 pages, 10 figure

Approximating Smallest Containers for Packing Three-dimensional Convex Objects

We investigate the problem of computing a minimal-volume container for the non-overlapping packing of a given set of three-dimensional convex objects. Already the simplest versions of the problem are NP-hard so that we cannot expect to find exact polynomial time algorithms. We give constant ratio approximation algorithms for packing axis-parallel (rectangular) cuboids under translation into an axis-parallel (rectangular) cuboid as container, for cuboids under rigid motions into an axis-parallel cuboid or into an arbitrary convex container, and for packing convex polyhedra under rigid motions into an axis-parallel cuboid or arbitrary convex container. This work gives the first approximability results for the computation of minimal volume containers for the objects described

matching, interpolation, and approximation ; a survey

In this survey we consider geometric techniques which have been used to measure the similarity or distance between shapes, as well as to approximate shapes, or interpolate between shapes. Shape is a modality which plays a key role in many disciplines, ranging from computer vision to molecular biology. We focus on algorithmic techniques based on computational geometry that have been developed for shape matching, simplification, and morphing

Automatic 3D building model generation using deep learning methods based on cityjson and 2D floor plans

In the past decade, a lot of effort is put into applying digital innovations to building life cycles. 3D Models have been proven to be efficient for decision making, scenario simulation and 3D data analysis during this life cycle. Creating such digital representation of a building can be a labour-intensive task, depending on the desired scale and level of detail (LOD). This research aims at creating a new automatic deep learning based method for building model reconstruction. It combines exterior and interior data sources: 1) 3D BAG, 2) archived floor plan images. To reconstruct 3D building models from the two data sources, an innovative combination of methods is proposed. In order to obtain the information needed from the floor plan images (walls, openings and labels), deep learning techniques have been used. In addition, post-processing techniques are introduced to transform the data in the required format. In order to fuse the extracted 2D data and the 3D exterior, a data fusion process is introduced. From the literature review, no prior research on automatic integration of CityGML/JSON and floor plan images has been found. Therefore, this method is a first approach to this data integration

2D multi-objective placement algorithm for free-form components

This article presents a generic method to solve 2D multi-objective placement problem for free-form components. The proposed method is a relaxed placement technique combined with an hybrid algorithm based on a genetic algorithm and a separation algorithm. The genetic algorithm is used as a global optimizer and is in charge of efficiently exploring the search space. The separation algorithm is used to legalize solutions proposed by the global optimizer, so that placement constraints are satisfied. A test case illustrates the application of the proposed method. Extensions for solving the 3D problem are given at the end of the article.Comment: ASME 2009 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, San Diego : United States (2009