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，其无需冗长训练过程，且在极少训练数据下仍表现优异。然而，由于依赖尺度维度的采样，该类方法对尺度变化敏感，难以泛化至未见尺度。本研究提出一种基于Riesz变换的替代特征表示方法。我们详细阐述并分析了该表示的数学基础，特别是其从Riesz变换继承的尺度等变性，彻底避免了尺度维度的采样。此外，与散射网络相比，该表示的特征数量减少了四分之三。尽管如此，我们的表示在纹理分类任务中仍展现出可媲美的性能，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，该方法表现出更优性能。在数字分类任务中，即使测试尺度达到训练尺度四倍大小时，准确率仍保持稳定，从而验证了等变性的实用性。第二个案例中，我们考察了纹理分类应用。