Abstract:Anisotropic image analysis is ubiquitous in medical and scientific imaging, and while the literature on the subject is extensive, the robustness to numerical rotations of numerous methods remains to be studied. Indeed, the principal directions and angular profile of a rotated image are often expected to rotate accordingly. In this work, we propose a new spectral method for the anisotropic analysis of images (EquivAnIA) using two established directional filters, namely cake wavelets, and ridge filters. We show that it is robust to numerical rotations throughout extensive experiments on synthetic and real-world images containing geometric structures or textures, and we also apply it successfully for a task of angular image registration. The code is available at https://github.com/jscanvic/Anisotropic-Analysis
Abstract:The simplicity and effectiveness of the UNet architecture makes it ubiquitous in image restoration, image segmentation, and diffusion models. They are often assumed to be equivariant to translations, yet they traditionally consist of layers that are known to be prone to aliasing, which hinders their equivariance in practice. To overcome this limitation, we propose a new alias-free UNet designed from a careful selection of state-of-the-art translation-equivariant layers. We evaluate the proposed equivariant architecture against non-equivariant baselines on image restoration tasks and observe competitive performance with a significant increase in measured equivariance. Through extensive ablation studies, we also demonstrate that each change is crucial for its empirical equivariance. Our implementation is available at https://github.com/jscanvic/UNet-AF
Abstract:DeepInverse is an open-source PyTorch-based library for solving imaging inverse problems. The library covers all crucial steps in image reconstruction from the efficient implementation of forward operators (e.g., optics, MRI, tomography), to the definition and resolution of variational problems and the design and training of advanced neural network architectures. In this paper, we describe the main functionality of the library and discuss the main design choices.
Abstract:The design of convolutional neural architectures that are exactly equivariant to continuous translations is an active field of research. It promises to benefit scientific computing, notably by making existing imaging systems more physically accurate. Most efforts focus on the design of downsampling/pooling layers, upsampling layers and activation functions, but little attention is dedicated to normalization layers. In this work, we present a novel theoretical framework for understanding the equivariance of normalization layers to discrete shifts and continuous translations. We also determine necessary and sufficient conditions for normalization layers to be equivariant in terms of the dimensions they operate on. Using real feature maps from ResNet-18 and ImageNet, we test those theoretical results empirically and find that they are consistent with our predictions.




Abstract:Self-supervised methods have recently proved to be nearly as effective as supervised methods in various imaging inverse problems, paving the way for learning-based methods in scientific and medical imaging applications where ground truth data is hard or expensive to obtain. This is the case in magnetic resonance imaging and computed tomography. These methods critically rely on invariance to translations and/or rotations of the image distribution to learn from incomplete measurement data alone. However, existing approaches fail to obtain competitive performances in the problems of image super-resolution and deblurring, which play a key role in most imaging systems. In this work, we show that invariance to translations and rotations is insufficient to learn from measurements that only contain low-frequency information. Instead, we propose a new self-supervised approach that leverages the fact that many image distributions are approximately scale-invariant, and that can be applied to any inverse problem where high-frequency information is lost in the measurement process. We demonstrate throughout a series of experiments on real datasets that the proposed method outperforms other self-supervised approaches, and obtains performances on par with fully supervised learning.