# Learning The Invisible: A Hybrid Deep Learning-Shearlet Framework for Limited Angle Computed Tomography

arXiv: https://arxiv.org/abs/1811.04602 · HTML: https://mlsanity.com/paper/1811.04602/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Nov 2018.

## Built on by (most cited first, 29 of 29)

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- [Limitations of Deep Learning for Inverse Problems on Digital Hardware](https://mlsanity.com/paper/2202.13490/) (2022)
- [Data Models for Dataset Drift Controls in Machine Learning With Optical Images](https://mlsanity.com/paper/2211.02578/) (2022)
- [Limited-Angle Tomography Reconstruction via Deep End-To-End Learning on Synthetic Data](https://mlsanity.com/paper/2309.06948/) (2023)
- [Deep Microlocal Reconstruction for Limited-Angle Tomography](https://mlsanity.com/paper/2108.05732/) (2021)
- [AAPM DL-Sparse-View CT Challenge Submission Report: Designing an Iterative Network for Fanbeam-CT with Unknown Geometry](https://mlsanity.com/paper/2106.00280/) (2021)
- [Efficient Hierarchical Bayesian Inference for Spatio-temporal Regression Models in Neuroimaging](https://mlsanity.com/paper/2111.01692/) (2021)
- [Training Adaptive Reconstruction Networks for Blind Inverse Problems](https://mlsanity.com/paper/2202.11342/) (2022)
- [Physics-assisted Generative Adversarial Network for X-Ray Tomography](https://mlsanity.com/paper/2204.03703/) (2022)
- [LRIP-Net: Low-Resolution Image Prior based Network for Limited-Angle CT Reconstruction](https://mlsanity.com/paper/2208.00207/) (2022)
- [On Learning the Invisible in Photoacoustic Tomography with Flat Directionally Sensitive Detector](https://mlsanity.com/paper/2204.10001/) (2022)
- [Space-Variant Total Variation boosted by learning techniques in few-view tomographic imaging](https://mlsanity.com/paper/2404.16900/) (2024)
- [WARPd: A linearly convergent first-order method for inverse problems with approximate sharpness conditions](https://mlsanity.com/paper/2110.12437/) (2021)
- [Advantage of Machine Learning over Maximum Likelihood in Limited-Angle Low-Photon X-Ray Tomography](https://mlsanity.com/paper/2111.08011/) (2021)
- [Transgressing the boundaries: towards a rigorous understanding of deep learning and its (non-)robustness](https://mlsanity.com/paper/2307.02454/) (2023)
- [Learned reconstruction methods for inverse problems: sample error estimates](https://mlsanity.com/paper/2312.14078/) (2023)
- [Mathematical Algorithm Design for Deep Learning under Societal and Judicial Constraints: The Algorithmic Transparency Requirement](https://mlsanity.com/paper/2401.10310/) (2024)
- [Iterative CT Reconstruction via Latent Variable Optimization of Shallow Diffusion Models](https://mlsanity.com/paper/2408.03156/) (2024)
- [Continuous approximation by convolutional neural networks with a sigmoidal function](https://mlsanity.com/paper/2209.13332/) (2022)
- [Stretched sinograms for limited-angle tomographic reconstruction with neural networks](https://mlsanity.com/paper/2306.10201/) (2023)
- [Meta-Prior: Meta learning for Adaptive Inverse Problem Solvers](https://mlsanity.com/paper/2311.18710/) (2023)
- [Multi-Modal Learning-based Reconstruction of High-Resolution Spatial Wind Speed Fields](https://mlsanity.com/paper/2312.08933/) (2023)
- [Robustness of data-driven approaches in limited angle tomography](https://mlsanity.com/paper/2403.11350/) (2024)
- [An incremental algorithm for non-convex AI-enhanced medical image processing](https://mlsanity.com/paper/2505.08324/) (2025)
- [The method of the approximate inverse for limited-angle CT](https://mlsanity.com/paper/2510.04369/) (2025)
- [Data-Consistent Learning of Inverse Problems](https://mlsanity.com/paper/2601.12831/) (2026)
- [Exact and Calibrated Diffusion Reconstruction for Digital Breast Tomosynthesis](https://mlsanity.com/paper/2607.12937/) (2026)

## Abstract (arXiv)

The high complexity of various inverse problems poses a significant challenge to model-based reconstruction schemes, which in such situations often reach their limits. At the same time, we witness an exceptional success of data-based methodologies such as deep learning. However, in the context of inverse problems, deep neural networks mostly act as black box routines, used for instance for a somewhat unspecified removal of artifacts in classical image reconstructions. In this paper, we will focus on the severely ill-posed inverse problem of limited angle computed tomography, in which entire boundary sections are not captured in the measurements. We will develop a hybrid reconstruction framework that fuses model-based sparse regularization with data-driven deep learning. Our method is reliable in the sense that we only learn the part that can provably not be handled by model-based methods, while applying the theoretically controllable sparse regularization technique to the remaining parts. Such a decomposition into visible and invisible segments is achieved by means of the shearlet transform that allows to resolve wavefront sets in the phase space. Furthermore, this split enables us to assign the clear task of inferring unknown shearlet coefficients to the neural network and thereby offering an interpretation of its performance in the context of limited angle computed tomography. Our numerical experiments show that our algorithm significantly surpasses both pure model- and more data-based reconstruction methods.