scholarly journals On the number of clumps resulting from the overlap of randomly placed figures in a plane

1983 ◽  
Vol 20 (1) ◽  
pp. 126-135 ◽  
Author(s):  
A. M. Kellerer
Keyword(s):  
Poisson Process ◽  
Field Of View ◽  
Two Dimensional ◽  
Expected Number ◽  
Random Direction ◽  
Closed Disc ◽  
Area And Perimeter ◽  
Isotropic Random

When two-dimensional figures, called laminae, are randomly placed on a plane domains result that can either be aggregates or individual laminae. The intersection of the union, U, of these domains with a specified field of view, F, in the plane is considered. The separate elements of the intersection are called clumps; they may be laminae, aggregates or partial laminae and aggregates. A formula is derived for the expected number of clumps minus enclosed voids. For bounded laminae homeomorphic to a closed disc with isotropic random direction the formula contains only their mean area and mean perimeter, the area and perimeter of F, and the intensity of the Poisson process.

scholarly journals On the number of clumps resulting from the overlap of randomly placed figures in a plane

1983 ◽  
Vol 20 (01) ◽  
pp. 126-135 ◽  
Author(s):  
A. M. Kellerer
Keyword(s):  
Poisson Process ◽  
Field Of View ◽  
Two Dimensional ◽  
Expected Number ◽  
Random Direction ◽  
Closed Disc ◽  
Area And Perimeter ◽  
Isotropic Random

When two-dimensional figures, called laminae, are randomly placed on a plane domains result that can either be aggregates or individual laminae. The intersection of the union, U, of these domains with a specified field of view, F, in the plane is considered. The separate elements of the intersection are called clumps; they may be laminae, aggregates or partial laminae and aggregates. A formula is derived for the expected number of clumps minus enclosed voids. For bounded laminae homeomorphic to a closed disc with isotropic random direction the formula contains only their mean area and mean perimeter, the area and perimeter of F, and the intensity of the Poisson process.

scholarly journals Counting figures in planar random configurations

1985 ◽  
Vol 22 (01) ◽  
pp. 68-81 ◽  
Author(s):  
A. M. Kellerer
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Poisson Process ◽  
Recent Result ◽  
Field Of View ◽  
Fixed Field ◽  
Area And Perimeter ◽  
Standard Deviations

Random configurations are considered that are generated by a Poisson process of figures in the plane, and a recent result is used to derive formulae for the estimation of the number of figures, and their mean area and perimeter. The formulae require merely the determination of the area, the perimeter, and the Euler–Poincaré characteristic of the random configurations in a fixed field of view. There are no similar formulae for the standard deviations of the estimates; their magnitudes in typical cases are therefore assessed by Monte Carlo simulations.

scholarly journals Counting figures in planar random configurations

1985 ◽  
Vol 22 (1) ◽  
pp. 68-81 ◽  
Author(s):  
A. M. Kellerer
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Poisson Process ◽  
Recent Result ◽  
Field Of View ◽  
Fixed Field ◽  
Area And Perimeter ◽  
Standard Deviations

Random configurations are considered that are generated by a Poisson process of figures in the plane, and a recent result is used to derive formulae for the estimation of the number of figures, and their mean area and perimeter. The formulae require merely the determination of the area, the perimeter, and the Euler–Poincaré characteristic of the random configurations in a fixed field of view. There are no similar formulae for the standard deviations of the estimates; their magnitudes in typical cases are therefore assessed by Monte Carlo simulations.

scholarly journals Limit theorems for isotropic random walks on triangle buildings

2002 ◽  
Vol 73 (3) ◽  
pp. 301-334 ◽  
Author(s):  
Marc Lindlbauer ◽  
Michael Voit
Keyword(s):  
Limit Theorem ◽  
Random Walks ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Two Dimensional ◽  
Littlewood Polynomials ◽  
Large Numbers ◽  
Moment Functions ◽  
Vertex Sets ◽  
Isotropic Random

AbstractThe spherical functions of triangle buildings can be described in terms of certain two-dimensional orthogonal polynomials on Steiner's hypocycloid which are closely related to Hall-Littlewood polynomials. They lead to a one-parameter family of two-dimensional polynimial hypergroups. In this paper we investigate isotropic random walks on the vertex sets of triangle buildings in terms of their projections to these hypergroups. We present strong laws of large numbers, a central limit theorem, and a local limit theorem; all these results are well-known for homogeneous trees. Proofs are based on moment functions on hypergroups and on explicit expansions of the hypergroup characters in terms of certain two-dimensional Tchebychev polynimials.

FOV constrained guidance law for nonstationary nonmaneuvering target interception with any impact angle

2021 ◽  
pp. 095441002110468
Author(s):  
Nikhil Kumar Singh ◽  
Sikha Hota
Keyword(s):  
Kinematic Model ◽  
Angular Spectrum ◽  
Impact Angle ◽  
Field Of View ◽  
Two Dimensional ◽  
Proportional Navigation ◽  
Guidance Law ◽  
Real World Applications ◽  
Target Interception ◽  
Impact Angles

This paper presents the nonstationary nonmaneuvering target interception with all possible desired impact angles in a two-dimensional (2D) aerial engagement scenario, where the target can move in any direction. The paper also considers the field-of-view (FOV) constraint for designing the guidance law so that the target is always visible while following the missile trajectory in the entire engagement time, which makes it feasible for real world applications. The guidance law is based on the pure proportional navigation (PPN) to achieve any impact angle of the entire angular spectrum. The proposed guidance law is then simulated for intercepting a nonstationary nonmaneuvering target using a kinematic model of a missile to demonstrate the efficacy of the presented scheme. A comparison with the related work existing in the literature has also been added to establish the superiority of the present work.

Three-Dimensional Image Analysis of Aggregate Particles from Orthogonal Projections

1996 ◽  
Vol 1526 (1) ◽  
pp. 98-103 ◽  
Author(s):  
C.-Y. Kuo ◽  
J.D. Frost ◽  
J.S. Lai ◽  
L.B. Wang
Keyword(s):  
Image Analysis ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Uniform Size ◽  
Standard Size ◽  
Orthogonal Projections ◽  
Projected Image ◽  
Dimensional Image ◽  
Area And Perimeter ◽  
Aggregate Particles

Digital image analysis provides the capability for rapid measurement of particle characteristics. When an image is captured and digitized, numerous measurements can be made in near real time for each particle. Usually, image analysis techniques treat particles as two-dimensional objects since only the two-dimensional projection of the particles is captured. In this study, three-dimensional analysis of aggregate particles that was performed by attaching aggregates in sample trays with two perpendicular faces is described. After the initial projected image of the aggregates is captured and measured, the sample trays are rotated 90 degrees so that the aggregates are now perpendicular to their original orientation and the dimensions of the aggregates in the new projected image are captured and measured. The long, intermediate, and short particle dimensions ( dL, dI, and dS, respectively) provide direct measures of the flatness and elongation of the particles. Some other shape indexes can also be derived from the measurements of area and perimeter length. The proposed image analysis method was verified by comparing the results obtained with manual measurements of particle dimensions for uniform size [passing 12.7 mm (1/2 in.) sieve and retained on 9.5 mm (3/8 in.) sieve] aggregates. Three-dimensional image analysis was also performed on five aggregates of standard size No. 89 from different sources, and the results are summarized herein. The proposed method is expected to improve field quality control of aggregates used in hot mix asphalt.

The weak convergence of a class of estimators of the variance function of a two-dimensional Poisson process

1978 ◽  
Vol 15 (02) ◽  
pp. 433-439 ◽  
Author(s):  
A. M. Liebetrau
Keyword(s):  
Weak Convergence ◽  
Poisson Process ◽  
Gaussian Process ◽  
Higher Dimensions ◽  
Variance Function ◽  
Stationary Gaussian Process ◽  
Two Dimensional ◽  
Second Moment ◽  
Class Of Estimators

Results of a previous paper (Liebetrau (1977a)) are extended to higher dimensions. An estimator V∗(t 1, t 2) of the variance function V(t 1, t 2) of a two-dimensional process is defined, and its first- and second-moment structure is given assuming the process to be Poisson. Members of a class of estimators of the form where and for 0 < α i < 1, are shown to converge weakly to a non-stationary Gaussian process. Similar results hold when the t′i are taken to be constants, when V is replaced by a suitable estimator and when the dimensionality of the underlying Poisson process is greater than two.

Small Field-of-view single-shot EPI-DWI of the prostate: Evaluation of spatially-tailored two-dimensional radiofrequency excitation pulses

2016 ◽  
Vol 26 (2) ◽  
pp. 168-176 ◽  
Author(s):  
Ulrike I. Attenberger ◽  
Nils Rathmann ◽  
Metin Sertdemir ◽  
Philipp Riffel ◽  
Anja Weidner ◽  
...  
Keyword(s):  
Field Of View ◽  
Single Shot ◽  
Small Field ◽  
Two Dimensional ◽  
Small Field Of View
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