A Stduy on Optimal Shape Using Coordinate Transformation : Examination on Affine Transformation

Abstract This paper presents a new method to determine an optimal shape using affine transformation which is used in the field of Computer Aided Design (CAD), linear programming, and etc. We use affine transformation as coordinate transformation. Affine transformation is a linear transformation, so that shapes transformed must be linearly. Shape optimization of a inclined beam for example, we can deal with in the following manner. We define a simple cantilever beam first in initial design domain, and calculate an optimal shape. Then we use affine transformation remaining with optimal shape calculated in simple design domain and get to an optimal shape of the inclined beam. To compare with an optimal shape obtained by our proposed method, we calculate an optimal shape directly by conventional method in the same design domain after coordinate transformation. We show that affine transformation plays a role as scaling to structural optimization by finite element method and that necessary and sufficient conditions between design variables and shape transformation matrix may exist to get an exact optimal shape. We treat some numerical examples by our proposed method. In numerical examples, we consider shape optimization of inclined cantilever beam for simplicity. We show that some stepwise linear optimal shapes could be expressed from an optimal shape of a simple cantilever beam by using affine transformation. Optimal shape calculated by our method can obtain easily and speedy. Through some numerical examples, we could examine effectiveness of our proposed method.

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The Review of Algorithms to Correct Barrel Distortion Image

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.628.403 ◽

2012 ◽

Vol 628 ◽

pp. 403-409 ◽

Cited By ~ 1

Author(s):

Min Qi ◽

Xiao Xi Zhang ◽

Da Jian Li ◽

Yang Yu Fan

Keyword(s):

Coordinate Transformation ◽

Affine Transformation ◽

Basic Theory ◽

Degree Polynomial ◽

Research Directions ◽

Advantages And Disadvantages ◽

Polynomial Transformation ◽

Polar Coordinate ◽

The Common ◽

Barrel Distortion

Aiming at solving the problem of correcting barrel distortion of image, this paper concluded the common correction algorithms into three types which include Affine Transformation, Two Degree Polynomial Transformation and Polar Coordinate Transformation, and introduced the basic theory of each of their representative algorithm. Then, analyzed and compared the advantages and disadvantages of them according to the correction experiments. At last, pointed out the research directions and difficulties of this field. In conclusion, Polar Coordinate Transformation is the most appropriate method to correct barrel distortion image, and the improved algorithm based on Polar Coordinate Transformation is more flexible to work out current difficulties of this aspect.

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Digital Image Processing and Coordinate Transformation for Vic-3D System

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.333-335.1002 ◽

2013 ◽

Vol 333-335 ◽

pp. 1002-1006

Author(s):

Sha Jia Song ◽

Neng He

Keyword(s):

Image Processing ◽

Correlation Analysis ◽

Coordinate Transformation ◽

Affine Transformation ◽

Template Matching ◽

Least Square Method ◽

Least Square ◽

3D Measurement ◽

Matching Method ◽

3D Measurement System

This paper discusses the problem about the processing of the images collected by Vic-3D measurement system and the coordinate transformation between the pixel coordinate and plane coordinate. Correlation analysis on image information is carried out by using Matlab. Template matching method is used to get the pixel coordinates of the marked part on the images. Based on affine transformation and least square method, I transform the pixel coordinates of the marked part on the images into the plane coordinates.

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A Study on Optimal Shape Using Coordinate Transformation (Examination on eigenvalue problems)

The Proceedings of The Computational Mechanics Conference ◽

10.1299/jsmecmd.2000.13.287 ◽

2000 ◽

Vol 2000.13 (0) ◽

pp. 287-288

Keyword(s):

Coordinate Transformation ◽

Eigenvalue Problems ◽

Optimal Shape

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Research on PIV Algorithm for Matching of Particle Images Based on Affine Transformation

10.21203/rs.3.rs-453692/v1 ◽

2021 ◽

Author(s):

Jia Li ◽

Da-Long TAN ◽

Fei Zhao ◽

Xiang-Ji Yue

Keyword(s):

Coordinate System ◽

Coordinate Transformation ◽

Affine Transformation ◽

Particle Image ◽

Polar Coordinate System ◽

Polar Coordinate ◽

Affine Line ◽

Image Pairs ◽

Rigid Body Transformation ◽

Particle Images

Abstract For the problems of distortion and rotation in the matching of particle images of turbulent motion, according to the nature of affine transformation, using log-polar coordinate transformation, the matching is achieved by performing correlation calculations on the image line by line, and developed a matching algorithm (Turbulent Particle Image Matching, abbreviation: TPIM) for particle image pairs with affine transformation and rigid body transformation: by moving the interpretation window, the algorithm is no longer restricted by displacements of particles; by setting the affine lines according to the angle of the image in the log-polar coordinate system and using the affine line as the matching unit, the decoupling of different transformation factors is realized; according to the characteristic of non-uniform sampling in log-polar coordinate transformation, based on the principle of not losing image information, by reasonably setting the image mask and the rate of sampling, establishing the image pyramid and the relative coordinate system, the algorithm complexity is reduced to about 15% of the original. The experimental results of various types of particle images show that the matching accuracy of the TPIM algorithm can reach more than 99%.

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The Standard Metric

10.23943/princeton/9780691166902.003.0026 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Affine Transformation ◽

Convex Sets ◽

Finite Dimension ◽

Affine Subspace ◽

Affine Transformations ◽

Real Vector ◽

Euclidean Metric ◽

Affine Hyperplane ◽

Tits Building ◽

Convex Closure

This chapter presents some results about groups generated by reflections and the standard metric on a Bruhat-Tits building. It begins with definitions relating to an affine subspace, an affine hyperplane, an affine span, an affine map, and an affine transformation. It then considers a notation stating that the convex closure of a subset a of X is the intersection of all convex sets containing a and another notation that denotes by AGL(X) the group of all affine transformations of X and by Trans(X) the set of all translations of X. It also describes Euclidean spaces and assumes that the real vector space X is of finite dimension n and that d is a Euclidean metric on X. Finally, it discusses Euclidean representations and the standard metric.

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