  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.
