In this paper, we consider the two-dimensional Fredholm integral equation of the first kind. The kernel of this equation models the scattering of a laser beam by spheroids having the same fixed orientation. The unknown function under the integral describes the distribution of spheroids along two semi-axes. The input data are the diffraction pattern corresponding to the scattering of the laser beam by the particles. We show that the Tikhonov regularization method allows one to reconstruct two-dimensional distributions in the case when the diffraction pattern is modulated by white noise with relative amplitude up to 1%. In applications, this means novel possibility of obtaining two-dimensional particle size distributions, rather than one-dimensional ones, as in the classical version of the method. This significantly expands the capabilities of particle sizing via static laser diffraction technique. We show that the solution of the corresponding equation is unique in space and exists when the right-hand-side function is in a set dense in . We provide test experimental results in the framework of laser ektacytometry of red blood cells, which serves as the main application of the proposed approach presently.

中文翻译：

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基于激光衍射数据的二维粒度分布的数值重建

在本文中，我们考虑第一类二维 Fredholm 积分方程。该方程的核心模拟了具有相同固定方向的球体对激光束的散射。积分下的未知函数描述了椭球体沿两个半轴的分布。输入数据是对应于粒子对激光束的散射的衍射图案。我们表明，当衍射图案由相对幅度高达 1% 的白噪声调制时，Tikhonov 正则化方法允许人们重建二维分布。在应用中，这意味着获得二维粒度分布的新可能性，而不是像在该方法的经典版本中那样的一维粒度分布。这显着扩展了通过静态激光衍射技术进行粒度测量的能力。我们证明了相应方程的解在空间上是唯一的，并且当右侧函数在 中的稠密集合中时存在。我们在红细胞激光 ektacytometry 框架内提供测试实验结果，这是目前所提出方法的主要应用。