Abstract Astigmatic optical systems encode the depth location of spherical objects in the defocus blur of their images. This allows the simultaneous imaging of 3D positions of a large number of such objects, which can act as tracer particles in the study of fluid flows. The challenge lies in decoding the depth information, as defocused particle images might be overlapping or have low maximum intensity values. Current methods are not able to simultaneously detect and locate overlapping and low-intensity particle images. In addition, their cost of computation increases with particle image density. We show how semi-synthetic images of defocused particle images with proximate center point positions can be employed to train an end-to-end trainable particle image detector. This allows for the detection of low-intensity and overlapping particle images in a single pass of an image through a neural network. We present a thorough evaluation of the uncertainty of the method for the application of particles in fluid flow measurements. We achieve a similar error in the depth predictions to previous algorithms for non-overlapping particle images. In the case of neighboring particle images, the location error increases with decreasing particle image center distances and peaks when particle image centers share the same location. When dealing with actual measurement images, the location error increases by approximately a factor of two when particle images share the same center point locations. The trained model detects low-intensity particle images close to the visibility limit and covers 91.4% of the depth range of a human annotator. For the employed experimental arrangement, this increased the depth range along which particle images can be detected by 67% over a previously employed thresholding detection method (Franchini et al. in Adv Water Resour 124:1–8, 2019). Graphic abstract

中文翻译：

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切割、重叠和定位：一种用于散光光学设置中粒子 3D 定位的深度学习方法

摘要 像散光学系统编码球面物体在其图像散焦模糊中的深度位置。这允许同时对大量此类物体的 3D 位置进行成像，这些物体可以作为流体流动研究中的示踪粒子。挑战在于解码深度信息，因为散焦的粒子图像可能重叠或具有较低的最大强度值。当前的方法无法同时检测和定位重叠和低强度的粒子图像。此外，它们的计算成本随着粒子图像密度的增加而增加。我们展示了如何使用具有近似中心点位置的散焦粒子图像的半合成图像来训练端到端可训练粒子图像检测器。这允许在一次通过神经网络的图像中检测低强度和重叠的粒子图像。我们对粒子在流体流量测量中的应用方法的不确定性进行了全面评估。我们在深度预测中实现了与先前非重叠粒子图像算法类似的误差。在相邻粒子图像的情况下，当粒子图像中心共享相同位置时，位置误差随着粒子图像中心距离和峰值的减小而增加。在处理实际测量图像时，当粒子图像共享相同的中心点位置时，位置误差会增加大约两倍。训练后的模型检测接近可见度极限的低强度粒子图像，覆盖 91 个。人类注释者深度范围的 4%。对于所采用的实验安排，这将粒子图像的检测深度范围比以前采用的阈值检测方法增加了 67%（Franchini 等人，在 Adv Water Resour 124:1-8, 2019 中）。图形摘要