Scale invariance of an algorithm refers to its ability to treat objects equally independently of their size. For neural networks, scale invariance is typically achieved by data augmentation. However, when presented with a scale far outside the range covered by the training set, neural networks may fail to generalize. Here, we introduce the Riesz network, a novel scale invariant neural network. Instead of standard 2d or 3d convolutions for combining spatial information, the Riesz network is based on the Riesz transform which is a scale equivariant operation. As a consequence, this network naturally generalizes to unseen or even arbitrary scales in a single forward pass. As an application example, we consider detecting and segmenting cracks in tomographic images of concrete. In this context, 'scale' refers to the crack thickness which may vary strongly even within the same sample. To prove its scale invariance, the Riesz network is trained on one fixed crack width. We then validate its performance in segmenting simulated and real tomographic images featuring a wide range of crack widths. An additional experiment is carried out on the MNIST Large Scale data set.

翻译：算法的尺度不变性是指其能够同等对待不同大小的对象。对于神经网络而言，尺度不变性通常通过数据增强实现。然而，当遇到训练集覆盖范围之外的尺度时，神经网络可能无法泛化。本文引入Riesz网络，一种新型尺度不变神经网络。Riesz网络不采用标准的二维或三维卷积组合空间信息，而是基于尺度等变运算——Riesz变换。因此，该网络能够在单次前向传播中自然地泛化到未见甚至任意尺度。作为应用示例，我们考虑混凝土断层图像中裂纹的检测与分割。在此背景下，“尺度”指裂纹宽度，即使在同一样本中也可能剧烈变化。为证明其尺度不变性，Riesz网络在固定裂纹宽度上进行训练，随后在模拟和真实断层图像上验证其分割性能，这些图像涵盖宽范围裂纹宽度。此外，还在MNIST大规模数据集上进行了补充实验。