Bas Peters

Bas Peters

We introduce CQnet, a neural network with origins in the CQ algorithm for solving convex split-feasibility problems and forward-backward splitting. CQnet's trajectories are interpretable as particles that are tracking a changing constraint set via its point-to-set distance function while being elements of another constraint set at every layer. More than just a convex-geometric interpretation, CQnet accommodates learned and deterministic constraints that may be sample or data-specific and are satisfied by every layer and the output. Furthermore, the states in CQnet progress toward another constraint set at every layer. We provide proof of stability/nonexpansiveness with minimal assumptions. The combination of constraint handling and stability put forward CQnet as a candidate for various tasks where prior knowledge exists on the network states or output.

Keegan Lensink, Bas Peters, Eldad Haber

Convolutional Neural Networks (CNN) have recently seen tremendous success in various computer vision tasks. However, their application to problems with high dimensional input and output, such as high-resolution image and video segmentation or 3D medical imaging, has been limited by various factors. Primarily, in the training stage, it is necessary to store network activations for back propagation. In these settings, the memory requirements associated with storing activations can exceed what is feasible with current hardware, especially for problems in 3D. Motivated by the propagation of signals over physical networks, that are governed by the hyperbolic Telegraph equation, in this work we introduce a fully conservative hyperbolic network for problems with high dimensional input and output. We introduce a coarsening operation that allows completely reversible CNNs by using a learnable Discrete Wavelet Transform and its inverse to both coarsen and interpolate the network state and change the number of channels. We show that fully reversible networks are able to achieve results comparable to the state of the art in 4D time-lapse hyper spectral image segmentation and full 3D video segmentation, with a much lower memory footprint that is a constant independent of the network depth. We also extend the use of such networks to Variational Auto Encoders with high resolution input and output.

Bas Peters, Eldad Haber, Justin Granek

Neural-networks have seen a surge of interest for the interpretation of seismic images during the last few years. Network-based learning methods can provide fast and accurate automatic interpretation, provided there are sufficiently many training labels. We provide an introduction to the field aimed at geophysicists that are familiar with the framework of forward modeling and inversion. We explain the similarities and differences between deep networks to other geophysical inverse problems and show their utility in solving problems such as lithology interpolation between wells, horizon tracking and segmentation of seismic images. The benefits of our approach are demonstrated on field data from the Sea of Ireland and the North Sea.

Bas Peters, Justin Granek, Eldad Haber

Geologic interpretation of large seismic stacked or migrated seismic images can be a time-consuming task for seismic interpreters. Neural network based semantic segmentation provides fast and automatic interpretations, provided a sufficient number of example interpretations are available. Networks that map from image-to-image emerged recently as powerful tools for automatic segmentation, but standard implementations require fully interpreted examples. Generating training labels for large images manually is time consuming. We introduce a partial loss-function and labeling strategies such that networks can learn from partially interpreted seismic images. This strategy requires only a small number of annotated pixels per seismic image. Tests on seismic images and interpretation information from the Sea of Ireland show that we obtain high-quality predicted interpretations from a small number of large seismic images. The combination of a partial-loss function, a multi-resolution network that explicitly takes small and large-scale geological features into account, and new labeling strategies make neural networks a more practical tool for automatic seismic interpretation.

Bas Peters, Justin Granek, Eldad Haber

Detecting a specific horizon in seismic images is a valuable tool for geological interpretation. Because hand-picking the locations of the horizon is a time-consuming process, automated computational methods were developed starting three decades ago. Older techniques for such picking include interpolation of control points however, in recent years neural networks have been used for this task. Until now, most networks trained on small patches from larger images. This limits the networks ability to learn from large-scale geologic structures. Moreover, currently available networks and training strategies require label patches that have full and continuous annotations, which are also time-consuming to generate. We propose a projected loss-function for training convolutional networks with a multi-resolution structure, including variants of the U-net. Our networks learn from a small number of large seismic images without creating patches. The projected loss-function enables training on labels with just a few annotated pixels and has no issue with the other unknown label pixels. Training uses all data without reserving some for validation. Only the labels are split into training/testing. Contrary to other work on horizon tracking, we train the network to perform non-linear regression, and not classification. As such, we propose labels as the convolution of a Gaussian kernel and the known horizon locations that indicate uncertainty in the labels. The network output is the probability of the horizon location. We demonstrate the proposed computational ingredients on two different datasets, for horizon extrapolation and interpolation. We show that the predictions of our methodology are accurate even in areas far from known horizon locations because our learning strategy exploits all data in large seismic images.