# CoIL: Coordinate-based Internal Learning for Imaging Inverse Problems

arXiv: https://arxiv.org/abs/2102.05181 · HTML: https://mlsanity.com/paper/2102.05181/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Feb 2021.

## Built on by (most cited first, 30 of 33)

- [Neural Fields in Visual Computing and Beyond](https://mlsanity.com/paper/2111.11426/) (2021)
- [WIRE: Wavelet Implicit Neural Representations](https://mlsanity.com/paper/2301.05187/) (2023)
- [NeRP: Implicit Neural Representation Learning with Prior Embedding for Sparsely Sampled Image Reconstruction](https://mlsanity.com/paper/2108.10991/) (2021)
- [Beyond Periodicity: Towards a Unifying Framework for Activations in Coordinate-MLPs](https://mlsanity.com/paper/2111.15135/) (2021)
- [Implicit Neural Representation for Cooperative Low-light Image Enhancement](https://mlsanity.com/paper/2303.11722/) (2023)
- [Implicit Neural Representation in Medical Imaging: A Comparative Survey](https://mlsanity.com/paper/2307.16142/) (2023)
- [Unified Implicit Neural Stylization](https://mlsanity.com/paper/2204.01943/) (2022)
- [A survey of synthetic data augmentation methods in computer vision](https://mlsanity.com/paper/2403.10075/) (2024)
- [SplineCam: Exact Visualization and Characterization of Deep Network Geometry and Decision Boundaries](https://mlsanity.com/paper/2302.12828/) (2023)
- [Residual Multiplicative Filter Networks for Multiscale Reconstruction](https://mlsanity.com/paper/2206.00746/) (2022)
- [3DGR-CT: Sparse-View CT Reconstruction with a 3D Gaussian Representation](https://mlsanity.com/paper/2312.15676/) (2023)
- [Learning A Physical-aware Diffusion Model Based on Transformer for Underwater Image Enhancement](https://mlsanity.com/paper/2403.01497/) (2024)
- [Fast Context-Based Low-Light Image Enhancement via Neural Implicit Representations](https://mlsanity.com/paper/2407.12511/) (2024)
- [MINER: Multiscale Implicit Neural Representations](https://mlsanity.com/paper/2202.03532/) (2022)
- [Medical Image Registration via Neural Fields](https://mlsanity.com/paper/2206.03111/) (2022)
- [Neural Volumetric Reconstruction for Coherent Synthetic Aperture Sonar](https://mlsanity.com/paper/2306.09909/) (2023)
- [Implicit-Zoo: A Large-Scale Dataset of Neural Implicit Functions for 2D Images and 3D Scenes](https://mlsanity.com/paper/2406.17438/) (2024)
- [From Activation to Initialization: Scaling Insights for Optimizing Neural Fields](https://mlsanity.com/paper/2403.19205/) (2024)
- [Curvature-Aware Training for Coordinate Networks](https://mlsanity.com/paper/2305.08552/) (2023)
- [Implicit Neural Representations for Deconvolving SAS Images](https://mlsanity.com/paper/2112.08539/) (2021)
- [Preconditioners for the Stochastic Training of Neural Fields](https://mlsanity.com/paper/2402.08784/) (2024)
- [Generative modeling of living cells with SO(3)-equivariant implicit neural representations](https://mlsanity.com/paper/2304.08960/) (2023)
- [Neural Computed Tomography](https://mlsanity.com/paper/2201.06574/) (2022)
- [K-Space Transformer for Undersampled MRI Reconstruction](https://mlsanity.com/paper/2206.06947/) (2022)
- [Single-Shot Plug-and-Play Methods for Inverse Problems](https://mlsanity.com/paper/2311.13682/) (2023)
- [Estimating Neural Orientation Distribution Fields on High Resolution Diffusion MRI Scans](https://mlsanity.com/paper/2409.09387/) (2024)
- [Bias for Action: Video Implicit Neural Representations with Bias Modulation](https://mlsanity.com/paper/2501.09277/) (2025)
- [I-INR: Iterative Implicit Neural Representations](https://mlsanity.com/paper/2504.17364/) (2025)
- [Low-light phase retrieval with implicit generative priors](https://mlsanity.com/paper/2402.17745/) (2024)
- [FUTON: Fourier Tensor Network for Implicit Neural Representations](https://mlsanity.com/paper/2602.13414/) (2026)

## Abstract (arXiv)

We propose Coordinate-based Internal Learning (CoIL) as a new deep-learning (DL) methodology for the continuous representation of measurements. Unlike traditional DL methods that learn a mapping from the measurements to the desired image, CoIL trains a multilayer perceptron (MLP) to encode the complete measurement field by mapping the coordinates of the measurements to their responses. CoIL is a self-supervised method that requires no training examples besides the measurements of the test object itself. Once the MLP is trained, CoIL generates new measurements that can be used within a majority of image reconstruction methods. We validate CoIL on sparse-view computed tomography using several widely-used reconstruction methods, including purely model-based methods and those based on DL. Our results demonstrate the ability of CoIL to consistently improve the performance of all the considered methods by providing high-fidelity measurement fields.