Scattering networks yield powerful and robust hierarchical image descriptors which do not require lengthy training and which work well with very few training data. However, they rely on sampling the scale dimension. Hence, they become sensitive to scale variations and are unable to generalize to unseen scales. In this work, we define an alternative feature representation based on the Riesz transform. We detail and analyze the mathematical foundations behind this representation. In particular, it inherits scale equivariance from the Riesz transform and completely avoids sampling of the scale dimension. Additionally, the number of features in the representation is reduced by a factor four compared to scattering networks. Nevertheless, our representation performs comparably well for texture classification with an interesting addition: scale equivariance. Our method yields superior performance when dealing with scales outside of those covered by the training dataset. The usefulness of the equivariance property is demonstrated on the digit classification task, where accuracy remains stable even for scales four times larger than the one chosen for training. As a second example, we consider classification of textures.

翻译：散射网络能够生成强大且鲁棒的层次化图像描述符，这类描述符无需长时间训练，且在训练数据极少的情况下依然表现良好。然而，这类网络依赖于对尺度维度的采样，因此对尺度变化敏感且无法泛化至未见过的尺度。本研究基于Riesz变换定义了一种替代性特征表示。我们详细阐述并分析了该表示的数学基础：该表示从Riesz变换中继承了尺度等变性，完全避免了尺度维度的采样，同时相较于散射网络，其表示中的特征数量减少至四分之一。尽管如此，我们的表示在纹理分类任务中仍能达到可比拟的性能，并具有一个显著优势——尺度等变性。当处理训练数据集未覆盖的尺度时，该方法展现出更优的表现。在数字分类任务中，即使测试尺度为训练尺度四倍，分类精度依然保持稳定，充分证明了等变特性的实用价值。此外，我们以纹理分类作为第二个应用案例进行验证。