Tin Barisin

• This thesis is motivated by two groups of scientific disciplines: engineering sciences and mathematics. On the one hand, engineering sciences such as civil engineering want to design sustainable and cost-effective materials with desirable mechanical properties. The material behaviour depends on physical properties and production parameters. Therefore, physical properties are measured experimentally from real samples. In our case, computed tomography (CT) is used to non-destructively gain insight into the materials’ microstructure. This results in large 3d images which yield information on geometric microstructure characteristics. On the other hand, mathematical sciences are interested in designing methods with suitable and guaranteed properties. For example, a natural assumption of human vision is to analyse images regardless of object position, orientation, or scale. This assumption is formalized through the concepts of equivariance and invariance. In Part I, we deal with oriented structures in materials such as concrete or fiber-reinforced composites. In image processing, knowledge of the local structure orientation can be used for various tasks, e.g. structure enhancement. The idea of using banks of directed filters parameterized in the orientation space is effective in 2d. However, this class of methods is prohibitive in 3d due to the high computational burden of filtering when using a fine discretization of the unit sphere. Hence, we introduce a method for 3d pixel-wise orientation estimation and directional filtering inspired by the idea of adaptive refinement in discretized settings. Furthermore, an operator for distinction between isotropic and anisotropic structures is defined based on our method. Finally, usefulness of the method is shown on 3d CT images in three different tasks on a fiber-reinforced polymer, concrete with cracks, and partially closed foams. Additionally, our method is extended to construct line granulometry and characterize fiber length and orientation distributions in fiber-reinforced polymers produced by either 3d printing or by injection moulding. In Part II, we investigate how to introduce scale invariance for neural networks by using the Riesz transform. In classical convolutional 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, the network 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, a scale equivariant operator. As a consequence, this network naturally generalizes to unseen or even arbitrary scales in a single forward pass. As an application example, we consider segmenting cracks in CT 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 CT images featuring a wide range of crack widths. As an alternative to deep learning models, the Riesz transform is utilized to construct a scale equivariant scattering network, which does not require a lengthy training procedure and works with very few training examples. Mathematical foundations behind this representation are laid out and analyzed. We show that this representation with 4 times less features than the original scattering networks from Mallat performs comparably well on texture classification and gives superior performance when dealing with scales outside the training set distribution.