# Recovery of Continuous 3D Refractive Index Maps from Discrete Intensity-Only Measurements using Neural Fields

arXiv: https://arxiv.org/abs/2112.00002 · HTML: https://mlsanity.com/paper/2112.00002/

## Key facts

- Cited by 48 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 72 references that are in the map.
- First posted on arXiv 27 Nov 2021.

## Builds on (most cited first, 30 of 33)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [pixelNeRF: Neural Radiance Fields from One or Few Images](https://mlsanity.com/paper/2012.02190/) (2020)
- [NeRF in the Wild: Neural Radiance Fields for Unconstrained Photo Collections](https://mlsanity.com/paper/2008.02268/) (2020)
- [Nerfies: Deformable Neural Radiance Fields](https://mlsanity.com/paper/2011.12948/) (2020)
- [Scene Representation Networks: Continuous 3D-Structure-Aware Neural Scene Representations](https://mlsanity.com/paper/1906.01618/) (2019)
- [Neural Scene Flow Fields for Space-Time View Synthesis of Dynamic Scenes](https://mlsanity.com/paper/2011.13084/) (2020)
- [Neural Body: Implicit Neural Representations with Structured Latent Codes for Novel View Synthesis of Dynamic Humans](https://mlsanity.com/paper/2012.15838/) (2020)
- [NeRV: Neural Reflectance and Visibility Fields for Relighting and View Synthesis](https://mlsanity.com/paper/2012.03927/) (2020)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Learning Deep CNN Denoiser Prior for Image Restoration](https://mlsanity.com/paper/1704.03264/) (2017)
- [MoDL: Model Based Deep Learning Architecture for Inverse Problems](https://mlsanity.com/paper/1712.02862/) (2017)
- [Plug-and-Play ADMM for Image Restoration: Fixed Point Convergence and Applications](https://mlsanity.com/paper/1605.01710/) (2016)
- [Image reconstruction by domain transform manifold learning](https://mlsanity.com/paper/1704.08841/) (2017)
- [NeX: Real-time View Synthesis with Neural Basis Expansion](https://mlsanity.com/paper/2103.05606/) (2021)
- [How to represent part-whole hierarchies in a neural network](https://mlsanity.com/paper/2102.12627/) (2021)
- [A deep convolutional neural network using directional wavelets for low-dose X-ray CT reconstruction](https://mlsanity.com/paper/1610.09736/) (2016)
- [Plug-and-Play Priors for Bright Field Electron Tomography and Sparse Interpolation](https://mlsanity.com/paper/1512.07331/) (2015)
- [Plug and play methods for magnetic resonance imaging (long version)](https://mlsanity.com/paper/1903.08616/) (2019)
- [Dynamic CT Reconstruction from Limited Views with Implicit Neural Representations and Parametric Motion Fields](https://mlsanity.com/paper/2104.11745/) (2021)
- [TermiNeRF: Ray Termination Prediction for Efficient Neural Rendering](https://mlsanity.com/paper/2111.03643/) (2021)
- [Scalable Plug-and-Play ADMM with Convergence Guarantees](https://mlsanity.com/paper/2006.03224/) (2020)
- [Provable Convergence of Plug-and-Play Priors with MMSE denoisers](https://mlsanity.com/paper/2005.07685/) (2020)
- [Data-driven Seismic Waveform Inversion: A Study on the Robustness and Generalization](https://mlsanity.com/paper/1809.10262/) (2018)
- [Deep speckle correlation: a deep learning approach towards scalable imaging through scattering media](https://mlsanity.com/paper/1806.04139/) (2018)
- [SIMBA: Scalable Inversion in Optical Tomography using Deep Denoising Priors](https://mlsanity.com/paper/1911.13241/) (2019)
- [Efficient and accurate inversion of multiple scattering with deep learning](https://mlsanity.com/paper/1803.06594/) (2018)
- [High-resolution 3D refractive index microscopy of multiple-scattering samples from intensity images](https://mlsanity.com/paper/1909.00023/) (2019)

## Built on by (most cited first, 30 of 48)

- [NeRFPlayer: A Streamable Dynamic Scene Representation with Decomposed Neural Radiance Fields](https://mlsanity.com/paper/2210.15947/) (2022)
- [FINER: Flexible spectral-bias tuning in Implicit NEural Representation by Variable-periodic Activation Functions](https://mlsanity.com/paper/2312.02434/) (2023)
- [InverseBench: Benchmarking Plug-and-Play Diffusion Priors for Inverse Problems in Physical Sciences](https://mlsanity.com/paper/2503.11043/) (2025)
- [A survey of synthetic data augmentation methods in computer vision](https://mlsanity.com/paper/2403.10075/) (2024)
- [Self-Supervised Coordinate Projection Network for Sparse-View Computed Tomography](https://mlsanity.com/paper/2209.05483/) (2022)
- [DINER: Disorder-Invariant Implicit Neural Representation](https://mlsanity.com/paper/2211.07871/) (2022)
- [Disorder-invariant Implicit Neural Representation](https://mlsanity.com/paper/2304.00837/) (2023)
- [Equivariant plug-and-play image reconstruction](https://mlsanity.com/paper/2312.01831/) (2023)
- [IMJENSE: Scan-specific Implicit Representation for Joint Coil Sensitivity and Image Estimation in Parallel MRI](https://mlsanity.com/paper/2311.12892/) (2023)
- [Weakly Convex Regularisers for Inverse Problems: Convergence of Critical Points and Primal-Dual Optimisation](https://mlsanity.com/paper/2402.01052/) (2024)
- [RHINO: Regularizing the Hash-based Implicit Neural Representation](https://mlsanity.com/paper/2309.12642/) (2023)
- [Block Coordinate Plug-and-Play Methods for Blind Inverse Problems](https://mlsanity.com/paper/2305.12672/) (2023)
- [Self-Supervised Scalable Deep Compressed Sensing](https://mlsanity.com/paper/2308.13777/) (2023)
- [Single View Refractive Index Tomography with Neural Fields](https://mlsanity.com/paper/2309.04437/) (2023)
- [Continuous 3D Myocardial Motion Tracking via Echocardiography](https://mlsanity.com/paper/2310.02792/) (2023)
- [Highly Accelerated MRI via Implicit Neural Representation Guided Posterior Sampling of Diffusion Models](https://mlsanity.com/paper/2407.02744/) (2024)
- [Enhancing Implicit Neural Representations via Symmetric Power Transformation](https://mlsanity.com/paper/2412.09213/) (2024)
- [Continuous Representation Methods, Theories, and Applications: An Overview and Perspectives](https://mlsanity.com/paper/2505.15222/) (2025)
- [Ray-driven Spectral CT Reconstruction Based on Neural Base-Material Fields](https://mlsanity.com/paper/2404.06991/) (2024)
- [Accelerated deep self-supervised ptycho-laminography for three-dimensional nanoscale imaging of integrated circuits](https://mlsanity.com/paper/2304.04597/) (2023)
- [Local Conditional Neural Fields for Versatile and Generalizable Large-Scale Reconstructions in Computational Imaging](https://mlsanity.com/paper/2307.06207/) (2023)
- [Recovering a Molecule's 3D Dynamics from Liquid-phase Electron Microscopy Movies](https://mlsanity.com/paper/2308.11927/) (2023)
- [A Test-Time Learning Approach to Reparameterize the Geophysical Inverse Problem with a Convolutional Neural Network](https://mlsanity.com/paper/2312.04752/) (2023)
- [Multiplane Quantitative Phase Imaging Using a Wavelength-Multiplexed Diffractive Optical Processor](https://mlsanity.com/paper/2403.11035/) (2024)
- [Towards the Spectral bias Alleviation by Normalizations in Coordinate Networks](https://mlsanity.com/paper/2407.17834/) (2024)
- [RSR-NF: Neural Field Regularization by Static Restoration Priors for Dynamic Imaging](https://mlsanity.com/paper/2503.10015/) (2025)
- [Towards Understanding the Benefits of Neural Network Parameterizations in Geophysical Inversions: A Study With Neural Fields](https://mlsanity.com/paper/2503.17503/) (2025)
- [V2V3D: View-to-View Denoised 3D Reconstruction for Light-Field Microscopy](https://mlsanity.com/paper/2504.07853/) (2025)
- [Plug-and-Play Posterior Sampling for Blind Inverse Problems](https://mlsanity.com/paper/2505.22923/) (2025)
- [Dual-Cycle: Self-Supervised Dual-View Fluorescence Microscopy Image Reconstruction using CycleGAN](https://mlsanity.com/paper/2209.11729/) (2022)

## Abstract (arXiv)

Intensity diffraction tomography (IDT) refers to a class of optical microscopy techniques for imaging the 3D refractive index (RI) distribution of a sample from a set of 2D intensity-only measurements. The reconstruction of artifact-free RI maps is a fundamental challenge in IDT due to the loss of phase information and the missing cone problem. Neural fields (NF) has recently emerged as a new deep learning (DL) approach for learning continuous representations of physical fields. NF uses a coordinate-based neural network to represent the field by mapping the spatial coordinates to the corresponding physical quantities, in our case the complex-valued refractive index values. We present DeCAF as the first NF-based IDT method that can learn a high-quality continuous representation of a RI volume from its intensity-only and limited-angle measurements. The representation in DeCAF is learned directly from the measurements of the test sample by using the IDT forward model, without any ground-truth RI maps. We qualitatively and quantitatively evaluate DeCAF on the simulated and experimental biological samples. Our results show that DeCAF can generate high-contrast and artifact-free RI maps and lead to up to 2.1 times reduction in MSE over existing methods.