Learning Equivariant Representations

State-of-the-art deep learning systems often require large amounts of data and computation. For this reason, leveraging known or unknown structure of the data is paramount. Convolutional neural networks (CNNs) are successful examples of this principle, their defining characteristic being the shift-equivariance. By sliding a filter over the input, when the input shifts, the response shifts by the same amount, exploiting the structure of natural images where semantic content is independent of absolute pixel positions. This property is essential to the success of CNNs in audio, image and video recognition tasks. In this thesis, we extend equivariance to other kinds of transformations, such as rotation and scaling. We propose equivariant models for different transformations defined by groups of symmetries. The main contributions are (i) polar transformer networks, achieving equivariance to the group of similarities on the plane, (ii) equivariant multi-view networks, achieving equivariance to the group of symmetries of the icosahedron, (iii) spherical CNNs, achieving equivariance to the continuous 3D rotation group, (iv) cross-domain image embeddings, achieving equivariance to 3D rotations for 2D inputs, and (v) spin-weighted spherical CNNs, generalizing the spherical CNNs and achieving equivariance to 3D rotations for spherical vector fields. Applications include image classification, 3D shape classification and retrieval, panoramic image classification and segmentation, shape alignment and pose estimation. What these models have in common is that they leverage symmetries in the data to reduce sample and model complexity and improve generalization performance. The advantages are more significant on (but not limited to) challenging tasks where data is limited or input perturbations such as arbitrary rotations are present

A Unified Single-stage Learning Model for Estimating Fiber Orientation Distribution Functions on Heterogeneous Multi-shell Diffusion-weighted MRI

Diffusion-weighted (DW) MRI measures the direction and scale of the local diffusion process in every voxel through its spectrum in q-space, typically acquired in one or more shells. Recent developments in micro-structure imaging and multi-tissue decomposition have sparked renewed attention to the radial b-value dependence of the signal. Applications in tissue classification and micro-architecture estimation, therefore, require a signal representation that extends over the radial as well as angular domain. Multiple approaches have been proposed that can model the non-linear relationship between the DW-MRI signal and biological microstructure. In the past few years, many deep learning-based methods have been developed towards faster inference speed and higher inter-scan consistency compared with traditional model-based methods (e.g., multi-shell multi-tissue constrained spherical deconvolution). However, a multi-stage learning strategy is typically required since the learning process relied on various middle representations, such as simple harmonic oscillator reconstruction (SHORE) representation. In this work, we present a unified dynamic network with a single-stage spherical convolutional neural network, which allows efficient fiber orientation distribution function (fODF) estimation through heterogeneous multi-shell diffusion MRI sequences. We study the Human Connectome Project (HCP) young adults with test-retest scans. From the experimental results, the proposed single-stage method outperforms prior multi-stage approaches in repeated fODF estimation with shell dropoff and single-shell DW-MRI sequences

Scalable and equivariant spherical CNNs by discrete-continuous (DISCO) convolutions

No existing spherical convolutional neural network (CNN) framework is both computationally scalable and rotationally equivariant. Continuous approaches capture rotational equivariance but are often prohibitively computationally demanding. Discrete approaches offer more favorable computational performance but at the cost of equivariance. We develop a hybrid discrete-continuous (DISCO) group convolution that is simultaneously equivariant and computationally scalable to high-resolution. While our framework can be applied to any compact group, we specialize to the sphere. Our DISCO spherical convolutions exhibit SO(3) rotational equivariance, where SO(n) is the special orthogonal group representing rotations in n-dimensions. When restricting rotations of the convolution to the quotient space SO(3)/SO(2) for further computational enhancements, we recover a form of asymptotic SO(3) rotational equivariance. Through a sparse tensor implementation we achieve linear scaling in number of pixels on the sphere for both computational cost and memory usage. For 4k spherical images we realize a saving of 109 in computational cost and 104 in memory usage when compared to the most efficient alternative equivariant spherical convolution. We apply the DISCO spherical CNN framework to a number of benchmark dense-prediction problems on the sphere, such as semantic segmentation and depth estimation, on all of which we achieve the state-of-the-art performance