Deep convolutional neural networks (DCNNs) are routinely used for image segmentation of biomedical data sets to obtain quantitative measurements of cellular structures like tissues. These cellular structures often contain gaps in their boundaries, leading to poor segmentation performance when using DCNNs like the U-Net. The gaps can usually be corrected by post-hoc computer vision (CV) steps, which are specific to the data set and require a disproportionate amount of work. As DCNNs are Universal Function Approximators, it is conceivable that the corrections should be obsolete by selecting the appropriate architecture for the DCNN. In this article, we present a novel theoretical framework for the gap-filling problem in DCNNs that allows the selection of architecture to circumvent the CV steps. Combining information-theoretic measures of the data set with a fundamental property of DCNNs, the size of their receptive field, allows us to formulate statements about the solvability of the gap-filling problem independent of the specifics of model training. In particular, we obtain mathematical proof showing that the maximum proficiency of filling a gap by a DCNN is achieved if its receptive field is larger than the gap length. We then demonstrate the consequence of this result using numerical experiments on a synthetic and real data set and compare the gap-filling ability of the ubiquitous U-Net architecture with variable depths. Our code is available at https://github.com/ai-biology/dcnn-gap-filling.

中文翻译：

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关于利用深度卷积神经网络解决间隙填充问题的充分条件


深度卷积神经网络 (DCNN) 通常用于生物医学数据集的图像分割，以获得组织等细胞结构的定量测量。这些细胞结构的边界通常包含间隙，导致使用 U-Net 等 DCNN 时分割性能较差。这些差距通常可以通过事后计算机视觉 (CV) 步骤来纠正，这些步骤特定于数据集，并且需要不成比例的工作量。由于 DCNN 是通用函数逼近器，因此可以想象，通过为 DCNN 选择适当的架构，校正应该被废弃。在本文中，我们针对 DCNN 中的间隙填充问题提出了一种新颖的理论框架，该框架允许选择架构来绕过 CV 步骤。将数据集的信息论测量与 DCNN 的基本属性（感受野的大小）相结合，使我们能够独立于模型训练的具体情况来制定有关填补空白问题的可解决性的陈述。特别是，我们获得了数学证明，表明如果 DCNN 的感受野大于间隙长度，则可以实现填充间隙的最大效率。然后，我们使用合成数据集和真实数据集上的数值实验来证明这一结果的结果，并比较普遍存在的具有可变深度的 U-Net 架构的间隙填充能力。我们的代码可在 https://github.com/ai-biology/dcnn-gap-filling 获取。