# Back-Projection Diffusion: Solving the Wideband Inverse Scattering Problem with Diffusion Models

arXiv: https://arxiv.org/abs/2408.02866 · HTML: https://mlsanity.com/paper/2408.02866/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Inverse problems and image restoration, one of 14 directions in Diffusion models.
- Builds on 28 of its 88 references that are in the map.
- First posted on arXiv 5 Aug 2024.

## Placed in

- Diffusion models / [Inverse problems and image restoration](https://mlsanity.com/diffusion/inverse-problems-and-image-restoration/)

## Builds on (most cited first, 28 of 28)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [Deep Learning Techniques for Inverse Problems in Imaging](https://mlsanity.com/paper/2005.06001/) (2020)
- [Training behavior of deep neural network in frequency domain](https://mlsanity.com/paper/1807.01251/) (2018)
- [Deep Equilibrium Architectures for Inverse Problems in Imaging](https://mlsanity.com/paper/2102.07944/) (2021)
- [Invertible generative models for inverse problems: mitigating representation error and dataset bias](https://mlsanity.com/paper/1905.11672/) (2019)
- [Conditional score-based diffusion models for Bayesian inference in infinite dimensions](https://mlsanity.com/paper/2305.19147/) (2023)
- [SwitchNet: a neural network model for forward and inverse scattering problems](https://mlsanity.com/paper/1810.09675/) (2018)
- [User-defined Event Sampling and Uncertainty Quantification in Diffusion Models for Physical Dynamical Systems](https://mlsanity.com/paper/2306.07526/) (2023)
- [Prior Image-Constrained Reconstruction using Style-Based Generative Models](https://mlsanity.com/paper/2102.12525/) (2021)
- [Solving Inverse Wave Scattering with Deep Learning](https://mlsanity.com/paper/1911.13202/) (2019)
- [WISE: full-Waveform variational Inference via Subsurface Extensions](https://mlsanity.com/paper/2401.06230/) (2023)
- [Implicit Neural Representation for Mesh-Free Inverse Obstacle Scattering](https://mlsanity.com/paper/2206.02027/) (2022)
- [A Neural Network Warm-Start Approach for the Inverse Acoustic Obstacle Scattering Problem](https://mlsanity.com/paper/2212.08736/) (2022)
- [Wide-band butterfly network: stable and efficient inversion via multi-frequency neural networks](https://mlsanity.com/paper/2011.12413/) (2020)
- [Optimization for Amortized Inverse Problems](https://mlsanity.com/paper/2210.13983/) (2022)
- [Butterfly-Net2: Simplified Butterfly-Net and Fourier Transform Initialization](https://mlsanity.com/paper/1912.04154/) (2019)
- [Deep Injective Prior for Inverse Scattering](https://mlsanity.com/paper/2301.03092/) (2023)
- [Solving the Wide-band Inverse Scattering Problem via Equivariant Neural Networks](https://mlsanity.com/paper/2212.06068/) (2022)

## Built on by (most cited first, 3 of 3)

- [Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach](https://mlsanity.com/paper/2506.03979/) (2025)
- [Unifying and extending Diffusion Models through PDEs for solving Inverse Problems](https://mlsanity.com/paper/2504.07437/) (2025)
- [Instance-Wise Adaptive Sampling for Dataset Construction in Approximating Inverse Problem Solutions](https://mlsanity.com/paper/2509.04583/) (2025)

## Abstract (arXiv)

We present Wideband Back-Projection Diffusion, an end-to-end probabilistic framework for approximating the posterior distribution induced by the inverse scattering map from wideband scattering data. This framework produces highly accurate reconstructions, leveraging conditional diffusion models to draw samples, and also honors the symmetries of the underlying physics of wave-propagation. The procedure is factored into two steps: the first step, inspired by the filtered back-propagation formula, transforms data into a physics-based latent representation, while the second step learns a conditional score function conditioned on this latent representation. These two steps individually obey their associated symmetries and are amenable to compression by imposing the rank structure found in the filtered back-projection formula. Empirically, our framework has both low sample and computational complexity, with its number of parameters scaling only sub-linearly with the target resolution, and has stable training dynamics. It provides sharp reconstructions effortlessly and is capable of recovering even sub-Nyquist features in the multiple-scattering regime.