Deconstructing Approximate Offsets

We consider the offset-deconstruction problem: Given a polygonal shape Q with n vertices, can it be expressed, up to a tolerance \eps in Hausdorff distance, as the Minkowski sum of another polygonal shape P with a disk of fixed radius? If it does, we also seek a preferably simple-looking solution P; then, P's offset constitutes an accurate, vertex-reduced, and smoothened approximation of Q. We give an O(n log n)-time exact decision algorithm that handles any polygonal shape, assuming the real-RAM model of computation. A variant of the algorithm, which we have implemented using CGAL, is based on rational arithmetic and answers the same deconstruction problem up to an uncertainty parameter \delta; its running time additionally depends on \delta. If the input shape is found to be approximable, this algorithm also computes an approximate solution for the problem. It also allows us to solve parameter-optimization problems induced by the offset-deconstruction problem. For convex shapes, the complexity of the exact decision algorithm drops to O(n), which is also the time required to compute a solution P with at most one more vertex than a vertex-minimal one.Comment: 18 pages, 11 figures, previous version accepted at SoCG 2011, submitted to DC

SFF-Oriented Modeling and Process Planning of Functionally Graded Materials Using a Novel Equal Distance Offset Approach

This paper deals with the modeling and process planning of solid freeform fabrication (SFF) of 3D functionally graded materials (FGMs). A novel approach of representation and process planning of FGMs, termed as equal distance offset (EDO), is developed. In EDO, a neutral arbitrary 3D CAD model is adaptively sliced into a series of 2D layers. Within each layer, 2D material gradients are designed and represented via dividing the 2D shape into several sub-regions enclosed by iso-composition contours. If needed, the material composition gradient within each of sub-regions can be further determined by applying the equal distance offset algorithm to each sub-region. Using this approach, an arbitrary-shaped 3D FGM object with linear or non-linear composition gradients can be represented and fabricated via suitable SFF machines.Mechanical Engineerin

Optimal learning of joint alignments with a faulty oracle

We consider the following problem, which is useful in applications such as joint image and shape alignment. The goal is to recover n discrete variables gi ∈ {0, . . . , k − 1} (up to some global offset) given noisy observations of a set of their pairwise differences {(gi − gj) mod k}; specifically, with probability 1 k + for some > 0 one obtains the correct answer, and with the remaining probability one obtains a uniformly random incorrect answer. We consider a learning-based formulation where one can perform a query to observe a pairwise difference, and the goal is to perform as few queries as possible while obtaining the exact joint alignment. We provide an easy-to-implement, time efficient algorithm that performs O (n lg n k^2 ) queries, and recovers the joint alignment with high probability. We also show that our algorithm is optimal by proving a general lower bound that holds for all non-adaptive algorithms. Our work improves significantly recent work by Chen and Cand´es [CC16], who view the problem as a constrained principal components analysis problem that can be solved using the power method. Specifically, our approach is simpler both in the algorithm and the analysis, and provides additional insights into the problem structure.First author draf

Bayesian galaxy shape measurement for weak lensing surveys - II. Application to simulations

We extend the Bayesian model fitting shape measurement method presented in Miller et al. (2007) and use the method to estimate the shear from the Shear TEsting Programme simulations (STEP). The method uses a fast model fitting algorithm which uses realistic galaxy profiles and analytically marginalises over the position and amplitude of the model by doing the model fitting in Fourier space. This is used to find the full posterior probability in ellipticity so that the shear can be estimated in a fully Bayesian way. The Bayesian shear estimation allows measurement bias arising from the presence of random noise to be removed. In this paper we introduce an iterative algorithm that can be used to estimate the intrinsic ellipticity prior and show that this is accurate and stable. By using the method to estimate the shear from the STEP1 simulations we find the method to have a shear bias of m ~ 0.005 and a variation in shear offset with PSF type of sigma_c ~ 0.0002. These values are smaller than for any method presented in the STEP1 publication that behaves linearly with shear. Using the method to estimate the shear from the STEP2 simulations we find than the shear bias and offset are m ~ 0.002 and c ~ -0.0007 respectively. In addition we find that the bias and offset are stable to changes in magnitude and size of the galaxies. Such biases should yield any cosmological constraints from future weak lensing surveys robust to systematic effects in shape measurement

Analysis of approximations and aperture distortion for 3D migration of bistatic radar data with the two-step approach

The two-step approach is a fast algorithm for 3D migration originally introduced to process zero-offset seismic data. Its application to monostatic GPR (Ground Penetrating Radar) data is straightforward. A direct extension of the algorithm for the application to bistatic radar data is possible provided that the TX-RX azimuth is constant. As for the zero-offset case, the two-step operator is exactly equivalent to the one-step 3D operator for a constant velocity medium and is an approximation of the one-step 3D operator for a medium where the velocity varies vertically. Two methods are explored for handling a heterogeneous medium; both are suitable for the application of the two-step approach, and they are compared in terms of accuracy of the final 3D operator. The aperture of the two-step operator is discussed, and a solution is proposed to optimize its shape. The analysis is of interest for any NDT application where the medium is expected to be heterogeneous, or where the antenna is not in direct contact with the medium (e.g., NDT of artworks, humanitarian demining, radar with air-launched antennas)

Investigation on shape deviation of horizontal interior circular channels fabricated by laser powder bed fusion

The fabrication of horizontal interior circular channels poses some unique challenges to the laser powder bed fusion (L-PBF) process. The engineering challenge is to be able to print horizontal interior channels using L-PBF without using support structures, while the scientific challenge is to predict the shape deviation in the horizontal channel. This paper studies the geometric fidelity (roundness and shape deviation) of L-PBF printed horizontal interior circular channels (diameters 1−3 mm) by developing experiment-based regression models and a preliminary computational fluid dynamics (CFD) simulation model. The roundness error is found to be affected by the shape/size of the melt pool, thermal stresses, beam offset, and the slicing algorithm. It is recommended that to decrease the roundness error, in addition to choosing a proper beam offset, the width/depth of the melt pool should be minimized by minimizing the volumetric energy density (smaller laser power or higher scanning speed). Shape deviation in overhanging structures is determined by the thermo-mechanical driven molten flow in the melt pool. Hanging structures with irregular profiles (dross) are formed due to the sinking of the melt pool on an unconsolidated powder bed under the effect of gravity, surface tension, and poor thermal conductivity. (Partially) unmelted powder randomly adheres to the edges of the melt pool enlarging the hanging structure and roughening the profile. Small laser power or large scanning speed benefits reducing the roundness error and hang-diameter ratio. 0° or 45° rotational linear scanning strategy can be selected for minimizing the roundness error or the hang-diameter ratio, respectively

Extraction of the beam elastic shape from uncertain FBG strain measurement points

Aim of the present paper is the analysis of the strain along the beam that is equipped with Glass Fibers Reinforced Polymers (GFRP) with an embedded set of optical Fiber Bragg Grating sensors (FBG), in the context of a project to equip with these new structural elements an Italian train bridge. Different problems are attacked, and namely: (i)during the production process [1] it is difficult to locate precisely the FBG along the reinforcement bar, therefore the following question appears: How can we associate the strain measurements to the points along the bar? Is it possible to create a signal analysis procedure such that this correspondence is found?(ii)the beam can be inflected and besides the strain at some points, we would like to recover the elastic shape of the deformed beam that is equipped with the reinforcement bars. Which signal processing do we use to determine the shape of the deformed beam in its inflection plane?(iii)if the beam is spatially inflected, in two orthogonal planes, is it possible to recover the beam spatial elastic shape? Object of the paper is to answer to these questions

The Douglas-Peucker algorithm for line simplification: Re-evaluation through visualization

The primary aim of this paper is to illustrate the value of visualization in cartography and to indicate that tools for the generation and manipulation of realistic images are of limited value within this application. This paper demonstrates the value of visualization within one problem in cartography, namely the generalisation of lines. It reports on the evaluation of the Douglas-Peucker algorithm for line simplification. Visualization of the simplification process and of the results suggest that the mathematical measures of performance proposed by some other researchers are inappropriate, misleading and questionable