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2009
AI
This chapter explores the fundamentals of measure theory and probability theory as they apply to computer graphics. By outlining the definitions and properties of measures and integrals, it sets a foundation for fair sampling techniques within complex spaces, essential for applications like Monte Carlo integration. The chapter also discusses practical techniques for evaluating integrals, the significance of weighted measures, and provides illustrative examples to enhance understanding.
Journal of Electronic Imaging, 2017
Abstract. The detection of the contour of a binary object is a common problem; however, the area of a region, and its moments, can be a significant parameter. In several metrology applications, the area of planar objects must be measured. The area is obtained by counting the pixels inside the contour or using a discrete version of Green’s formula. Unfortunately, we obtain the area enclosed by the polygonal line passing through the centers of the pixels along the contour. We present a modified version of Green’s theorem in the discrete plane, which allows for the computation of the exact area of a two-dimensional region in the class of polyominoes. Penalties are introduced and associated with each successive pair of Freeman displacements along the contour in an eight-connectivity system. The proposed equation is shown to be true and properties of the equation related to the topology of the regions are presented. The proposed approach is adapted for faster computation than the combina...
Canadian Journal of Mathematics, 1970
It has been pointed out repeatedly in the literature that the methods of integral geometry (a mathematical theory founded by Wilhelm Blaschke and considerably extended by several mathematicians) provide highly suitable means for the solution of problems concerning "geometrical probabilities" [2; 6; 12; 15]. The possibilities for the application of these integral geometric results to the evaluation of probabilities, satisfying certain conditions of invariance with respect to a group of transformations which acts on the probability space, are obviously not yet exhausted. In this article, such applications are presented. First, some concepts and notation are introduced (§1). In the next section we derive some integral geometric relations (§ 2). These results are generalizations of known systems of formulae and they are valid in the ^-dimensional Euclidean space. In § 3, we determine mean-value formulae for the fundamental characteristics of point-sets, generated by randomly placed convex bodies. In particular, the rotational mean values of the fundamental characteristics of a zonotope (and of a parallelotope) with given lengths of its edges are investigated (§ 4). Finally, in § 5 we deal with the probability laws for certain agglomerations of planes, distributed at random in the ^-dimensional Euclidean space. The results in this section are the ^-dimensional extensions of some formulae in the special case k = 3, which have been developed by Hadwiger and myself in an earlier paper [11]. In the present work, some elementary knowledge of integral geometry is assumed to be known; in particular, we do not explain the concept of integral geometric densities. Details of invariant measures of sets of geometrical figures may be found in the introductory texts [1; 6; 8; 15].
This article explains how to measure area using both simple and complicated measurement methods. Elementary school children obtain area by counting triangles; junior-high school students use Cartesian coordinates; and high-school students study Heron's formula. In this article additional methods such as the trapezoid formula and Amsler's linear planimeter are presented.
The Mathematical Gazette, 2016
2012
Let us say that an element of a given family of subsets of ^d can be reconstructed using n test sets if there exist T_1,...,T_n ⊂^d such that whenever A,B∈ and the Lebesgue measures of A ∩ T_i and B ∩ T_i agree for each i=1,...,n then A=B. Our goal will be to find the least such n. We prove that if consists of the translates of a fixed reasonably nice subset of ^d then this minimum is n=d. In order to obtain this result we reconstruct a translate of a fixed function using d test sets as well, and also prove that under rather mild conditions the measure function f_K,θ (r) = ^d-1 (K ∩{x ∈^d : <x,θ> = r}) of the sections of K is absolutely continuous for almost every direction θ. These proofs are based on techniques of harmonic analysis. We also show that if consists of the magnified copies rE+t (r> 1, t∈^d) of a fixed reasonably nice set E⊂^d, where d> 2, then d+1 test sets reconstruct an element of . This fails in : we prove that an interval, and even an interval of lengt...
Pattern Recognition, 2016
In this paper we consider the shape space as the set of smooth simple closed curves in R 2 (parameterized curves), modulo translations, rotations and scale changes. An algorithm to obtain the intrinsic average of a sample data (set of planar shape realizations), from the identification of the shape space with an infinite dimensional Grassmannian is proposed using a gradient descent type algorithm. A simulation study is carried out to check the performance of the algorithm.
Pattern Recognition Letters, 2016
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Highlights • We prove a new formula for computing image integrals over d-dimensional rectangles. • The time complexity is the same as the well known integral image approach. • The required temporary memory is significantly reduced from N d to N d−1. • The precission loss issue produced when computing large integral images is reduced. • Application: we dramatically reduce time and memory required in image registration.
HAL (Le Centre pour la Communication Scientifique Directe), 2007
In this paper we prove that the function giving the frequency of a class of patterns of digital planes with respect to the slopes of the plane is continuous and piecewise affine, moreover the regions of affinity are precised. It allows to prove some combinatorial properties of a class of patterns called (m, n)-cubes. This study has also some consequences on local estimators of area: they never converge to the exact area when the resolution tends to zero for almost all region of plane. Actually we can prove with the same technics that this result is true for the regions of hyperplanes for any dimension d ≥ 3.
Computers & Graphics, 2009
In this paper, we prove that the function giving the frequency of a class of patterns of digital planes with respect to the slopes of the plane is continuous and piecewise affine, moreover the regions of affinity are specified. It allows to prove some combinatorial properties of a class of patterns called ðm; nÞ-cubes. This study has also some consequences on local estimators of area: we prove that the local estimators restricted to regions of plane never converge to the exact area when the resolution tends to zero for almost all slopes of plane. All the results are generalized for the regions of hyperplanes in any dimension dX3.
International Journal of Science and Mathematics …, 2010
The issue of the area of irregular shapes is absent from the modern mathematical textbooks in elementary education in Greece. However, there exists a collection of books written for educational purposes by famous Greek scholars dating from the eighteenth century, which propose certain techniques concerning the estimation of the area of such shapes. We claim that, when students deal for an adequate period with a succession of carefully designed tasks of the same conceptual basis-in our case that of the area of irregular shapes-then they "reinvent" problem-solving techniques for the estimation of their area, given that they have not been taught anything about these shapes. These techniques, in some cases, are almost the same as the abovementioned historical ones. In other cases, they could be considered to be an adaptation or extension of these.
The study seeks to investigate the effect of a good knowledge of mathematical concept of area on the quantification and estimation of tiles in a building construction. To achieve the purpose of this study, one null hypothesis was formulated. An experimental design was adopted for this study. A Sample of ten (10) tillers who attained primary education was collected from a building construction site. This was divided into two groups of five (5) tillers each, the experimental group tillers taught with mathematics concept of area) and the control group (tillers taught area without mathematics concept of area). The instrument for data collection was workers ability on quantification and estimation test (WAQET), with reliability index of 0.87. The hypothesis was tested using the independent t-test analysis at p>O.O5 level of sigi4flcance, the result showed a mean score of (10. 09) of the experimental group which was higher than the mean score of (8.02) for the control group. The analysis reveals that tillers who had a good knowledge of mathematical concept of area were better in quantification and estimation of tiles.
2014
This report gives a thorough description of the determination of two-dimensional areas and the estimation of area uncertainty. Considering the formulas for area determination, the basis is regular shapes, e.g. triangles, squares, rectangles, rhombi and regular (closed) polygons. The given formulas are valid as such; however, they also indicate how they can be simplified. The report presents strict as well as approximate formulas for area uncertainty estimates. The approach is mainly from a statistical point of view. However, there are some geometric factors/ variations which have an influence on the size of the area. They are dealt with by mathematical means. In addition, less mathematical, more argument-based (philosophical) descriptions of other "areal defects", having an influence on the area uncertainty, are presented. Examples of these defects are given to illustrate their impact on uncertainty estimation. The report is mainly an English translation of an earlier Swedish report [1]. Therefore, some minor parts refer to Swedish conditionse.g. regarding coordinate systems, map projections etc.
2014
This tutorial gives an overview of some of the basic techniques of measure theory. It includes a study of Borel sets and their generators for Polish and for analytic spaces, the weak topology on the space of all finite positive measures including its metrics, as well as measurable selections. Integration is covered, and product measures are introduced, both for finite and for arbitrary factors, with an application to projective systems. Finally, the duals of the Lp-spaces are discussed, together with the Radon-Nikodym Theorem and the Riesz Representation Theorem. Case studies include applications to stochastic Kripke models, to bisimulations, and to quotients for transition kernels.
Colloids and Surfaces A: Physicochemical and Engineering Aspects, 2004
A new method of characterising the morphology of disordered systems is presented based on the evolution of a family of integral geometric measures during erosion and dilation operations. The method is used to determine the accuracy of model reconstructions of random systems. It is shown that the use of erosion/dilation operations on the original image leads to an accurate discrimination of morphology. We consider the morphology of an experimental system and use the method to optimally match a reconstructed model morphology.
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