# Learned imaging with constraints and uncertainty quantification

arXiv: https://arxiv.org/abs/1909.06473 · HTML: https://mlsanity.com/paper/1909.06473/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Sep 2019.

## Built on by (most cited first, 8 of 8)

- [Reliable amortized variational inference with physics-based latent distribution correction](https://mlsanity.com/paper/2207.11640/) (2022)
- [Preconditioned training of normalizing flows for variational inference in inverse problems](https://mlsanity.com/paper/2101.03709/) (2021)
- [Deep Bayesian inference for seismic imaging with tasks](https://mlsanity.com/paper/2110.04825/) (2021)
- [Learning by example: fast reliability-aware seismic imaging with normalizing flows](https://mlsanity.com/paper/2104.06255/) (2021)
- [Uncertainty Quantification for Deep Unrolling-Based Computational Imaging](https://mlsanity.com/paper/2207.00698/) (2022)
- [Learned multiphysics inversion with differentiable programming and machine learning](https://mlsanity.com/paper/2304.05592/) (2023)
- [Velocity continuation with Fourier neural operators for accelerated uncertainty quantification](https://mlsanity.com/paper/2203.14386/) (2022)
- [CQnet: convex-geometric interpretation and constraining neural-network trajectories](https://mlsanity.com/paper/2302.10895/) (2023)

## Abstract (arXiv)

We outline new approaches to incorporate ideas from deep learning into wave-based least-squares imaging. The aim, and main contribution of this work, is the combination of handcrafted constraints with deep convolutional neural networks, as a way to harness their remarkable ease of generating natural images. The mathematical basis underlying our method is the expectation-maximization framework, where data are divided in batches and coupled to additional "latent" unknowns. These unknowns are pairs of elements from the original unknown space (but now coupled to a specific data batch) and network inputs. In this setting, the neural network controls the similarity between these additional parameters, acting as a "center" variable. The resulting problem amounts to a maximum-likelihood estimation of the network parameters when the augmented data model is marginalized over the latent variables.