Abel type integral equations in stereology. II. Computational methods of solution and the random spheres approximation
Abel type integral equations in stereology. II. Computational methods of solution and the random spheres approximation
复制标题
体视学中的阿贝尔型积分方程。
DOI：
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发表时间：
1975
期刊：
影响因子：
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通讯作者：
A. Jakeman
中科院分区：
文献类型：
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作者：
Robert S. Anderssen;A. Jakeman
如本文第一部分所述，第二部分主要讨论体视学中Abel型积分方程解的稳定计算方法的建立。然而，为了给我们将要提出的方法类型提供一个基本原理，首先有必要回顾一下迄今为止在立体学文献中提出的不同类型的方法。这在第2节中完成。通过回顾，我们得出结论，在已知的情况下，稳定方法的构建可以基于使用谱微分和积积分来评估合适的反演公式。以随机球体模型为例，我们将在第3节中考察这些方法的性质。第4节通过检验随机球体近似的可靠性（Moran（1972）提出的立体学中未解决的问题）来说明所提出方法的潜力。我们从这个检验中得出结论，随机球近似对于近似球的粒子是相当可靠的。
As mentioned in Part I of this paper, Part II is mainly concerned with the construction of stable computational methods for the solution of integral equations of Abel type which occur in stereology. However, in order to develop a rationale for the types of methods which we shall propose, it is first necessary to review the different types of methods which have been suggested to date in the stereological literature. This is done in section 2. From the review, we conclude that the construction of stable methods can be based on the use of spectral differentiation and product integration to evaluate appropriate inversion formulae, when they are known. Using the Random Spheres Model as exemplification, we examine the nature of such methods in section 3. The potential of the proposed methods is illustrated in section 4 by examining the reliability of the random spheres approximation—an unsolved problem in stereology proposed by Moran (1972). We conclude from this examination that the random spheres approximation is quite reliable for particles which approximate spheres.