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We present an SE(3) Group Convolutional Neural Network along with a series of networks with different group actions for segmentation of Diffusion Weighted Imaging data. These networks gradually incorporate group actions that are natural for this type of data, in the form of convolutions that provide equivariant transformations of the data. This knowledge provides a potentially important inductive bias and may alleviate the need for data augmentation strategies. We study the effects of these actions on the performances of the networks by training and validating them using the diffusion data from the Human Connectome project. Unlike previous works that use Fourier-based convolutions, we implement direct convolutions, which are more lightweight. We show how incorporating more actions - using the SE(3) group actions - generally improves the performances of our segmentation while limiting the number of parameters that must be learned.

Purpose Applying machine learning techniques to magnetic resonance diffusion-weighted imaging (DWI) data is challenging due to the size of individual data samples and the lack of labeled data. It is possible, though, to learn general patterns from a very limited amount of training data if we take advantage of the geometry of the DWI data. Therefore, we present a tissue classifier based on a Riemannian deep learning framework for single-shell DWI data. Approach The framework consists of three layers: a lifting layer that locally represents and convolves data on tangent spaces to produce a family of functions defined on the rotation groups of the tangent spaces, i.e., a (not necessarily continuous) function on a bundle of rotational functions on the manifold; a group convolution layer that convolves this function with rotation kernels to produce a family of local functions over each of the rotation groups; a projection layer using maximization to collapse this local data to form manifold based functions. Results Experiments show that our method achieves the performance of the same level as state-of-the-art while using way fewer parameters in the model (<10 % ). Meanwhile, we conducted a model sensitivity analysis for our method. We ran experiments using a proportion (69.2%, 53.3%, and 29.4%) of the original training set and analyzed how much data the model needs for the task. Results show that this does reduce the overall classification accuracy mildly, but it also boosts the accuracy for minority classes. Conclusions This work extended convolutional neural networks to Riemannian manifolds, and it shows the potential in understanding structural patterns in the brain, as well as in aiding manual data annotation., Purpose: Applying machine learning techniques to magnetic resonance diffusion-weighted imaging (DWI) data is challenging due to the size of individual data samples and the lack of labeled data. It is possible, though, to learn general patterns from a very limited amount of training data if we take advantage of the geometry of the DWI data. Therefore, we present a tissue classifier based on a Riemannian deep learning framework for single-shell DWI data.Approach: The framework consists of three layers: a lifting layer that locally represents and convolves data on tangent spaces to produce a family of functions defined on the rotation groups of the tangent spaces, i.e., a (not necessarily continuous) function on a bundle of rotational functions on the manifold; a group convolution layer that convolves this function with rotation kernels to produce a family of local functions over each of the rotation groups; a projection layer using maximization to collapse this local data to form manifold based functions.Results: Experiments show that our method achieves the performance of the same level as state-of-the-art while using way fewer parameters in the model (< 10 % ). Meanwhile, we conducted a model sensitivity analysis for our method. We ran experiments using a proportion (69.2%, 53.3%, and 29.4%) of the original training set and analyzed how much data the model needs for the task. Results show that this does reduce the overall classification accuracy mildly, but it also boosts the accuracy for minority classes.Conclusions: This work extended convolutional neural networks to Riemannian manifolds, and it shows the potential in understanding structural patterns in the brain, as well as in aiding manual data annotation.

We present a Group Convolutional Network for Segmentation of Diffusion Weighted Imaging data (DWI). The network incorporates group actions that are natural for this type of data, in the form of equivariant convolutions, i.e., roto-translation equivariant convolutions. The equivariance property provides an important inductive bias and may alleviate the need for data augmentation strategies. Instead of performing group equivariant convolutions via spectral (Fourier-based) approaches, as is common for equivariance, we implement direct and light-weight regular group convolutions. We study the effect of equivariance and weight sharing over on performances of the networks on DWI scans from the Human Connectome project. We show how that full equivariance improves segmentations, while limiting the number of learnable parameters.

Convolutional neural networks (CNN) have been shown to be efficient and effective in image analysis tasks due to their built-in locality and their weight-sharing property. Since convolutions are translation equivariant - a shift in the input results in the same shift in the output - the networks preserve translation symmetry. In fact, for the most common image analysis tasks, the space that the data live in - Euclidean space - is a principal homogeneous space to the translation group, it is, consequently, generic and convenient to apply CNNs to this type of data directly. For data modalities that are not measured in spaces like the Euclidean space, nonetheless, the translation equivariance is not immediately satisfied. For example, if we translate a signal on a 2D unit sphere, based on the path that is taken, the resulting signal at the destination will have different orientations. Hence, in this thesis, we focus on generalizing CNNs to more general group actions other than simply translation. We take inspiration from the classical path in literature for generalized CNNs. We first lift the data to groups, and then convolutions are performed on the groups via group actions, after which we project the functions back to the original space to perform tasks. In this sense, the data should be modeled in a way such that it is a function mapping from the homogeneous space of the group it is being lifted to. On the other hand, the group that the data are lifted to is not arbitrary. What kind of actions should be incorporated into the group? In this thesis, we explore the group actions in the most natural way - the actions should be associated with the possible motions that come with the data. In other words, the group action should encode whatever motions the data might have in reality such that the model can capture these motions and thus be resistant to variations in real-world data. In this thesis, we choose Diffusion Weighted Magnetic Resonance Imaging (DWI) as the d

We present a tissue classifier for Magnetic Resonance Diffusion Weighted Imaging (DWI) data trained from a single subject with a single b-value. The classifier is based on a Riemannian Deep Learning framework for extracting features with rotational invariance, where we extend a G-CNN learning architecture generically on a Riemannian manifold. We validate our framework using single-shell DWI data with a very limited amount of training data - only 1 scan. The proposed framework mainly consists of three layers: a lifting layer that locally represents and convolves data on tangent spaces to produce a family of functions defined on the rotation groups of the tangent spaces, i.e., a section of a bundle of rotational functions on the manifold; a group convolution layer that convolves this section with rotation kernels to produce a new section; and a projection layer using maximisation to collapse this local data to form new manifold based functions. We present an instantiation on the 2-dimensional sphere where the DWI orientation data is in general represented, and we use it for voxel classification. We show that this allows us to learn a classifier for cerebrospinal fluid (CSF) - subcortical - grey matter - white matter classification from only one scan.