# CNN-Based Projected Gradient Descent for Consistent Image Reconstruction

arXiv: https://arxiv.org/abs/1709.01809 · HTML: https://mlsanity.com/paper/1709.01809/

## Key facts

- Cited by 77 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Sep 2017.

## Built on by (most cited first, 30 of 77)

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- [Survey of Vulnerabilities in Large Language Models Revealed by Adversarial Attacks](https://mlsanity.com/paper/2310.10844/) (2023)
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- [Direct Diffusion Bridge using Data Consistency for Inverse Problems](https://mlsanity.com/paper/2305.19809/) (2023)
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- [Solving Inverse Problems by Joint Posterior Maximization with Autoencoding Prior](https://mlsanity.com/paper/2103.01648/) (2021)
- [Online Deep Equilibrium Learning for Regularization by Denoising](https://mlsanity.com/paper/2205.13051/) (2022)
- [MambaMIR: An Arbitrary-Masked Mamba for Joint Medical Image Reconstruction and Uncertainty Estimation](https://mlsanity.com/paper/2402.18451/) (2024)
- [SGD-Net: Efficient Model-Based Deep Learning with Theoretical Guarantees](https://mlsanity.com/paper/2101.09379/) (2021)
- [Sparse-view Cone Beam CT Reconstruction using Data-consistent Supervised and Adversarial Learning from Scarce Training Data](https://mlsanity.com/paper/2201.09318/) (2022)
- [PBI-Attack: Prior-Guided Bimodal Interactive Black-Box Jailbreak Attack for Toxicity Maximization](https://mlsanity.com/paper/2412.05892/) (2024)
- [Physics-/Model-Based and Data-Driven Methods for Low-Dose Computed Tomography: A survey](https://mlsanity.com/paper/2203.15725/) (2022)
- [Differentiable Forward Projector for X-ray Computed Tomography](https://mlsanity.com/paper/2307.05801/) (2023)
- [Fast and High-Quality Blind Multi-Spectral Image Pansharpening](https://mlsanity.com/paper/2103.09943/) (2021)
- [Wasserstein Patch Prior for Image Superresolution](https://mlsanity.com/paper/2109.12880/) (2021)
- [Towards Constituting Mathematical Structures for Learning to Optimize](https://mlsanity.com/paper/2305.18577/) (2023)
- [DuDoTrans: Dual-Domain Transformer Provides More Attention for Sinogram Restoration in Sparse-View CT Reconstruction](https://mlsanity.com/paper/2111.10790/) (2021)
- [WPPNets and WPPFlows: The Power of Wasserstein Patch Priors for Superresolution](https://mlsanity.com/paper/2201.08157/) (2022)
- [Bayesian Inversion for Nonlinear Imaging Models using Deep Generative Priors](https://mlsanity.com/paper/2203.10078/) (2022)
- [Diffusion Prior Regularized Iterative Reconstruction for Low-dose CT](https://mlsanity.com/paper/2310.06949/) (2023)
- [Improving Generalizability in Limited-Angle CT Reconstruction with Sinogram Extrapolation](https://mlsanity.com/paper/2103.05255/) (2021)
- [To be or not to be stable, that is the question: understanding neural networks for inverse problems](https://mlsanity.com/paper/2211.13692/) (2022)
- [Parallel Diffusion Model-based Sparse-view Cone-beam Breast CT](https://mlsanity.com/paper/2303.12861/) (2023)
- [Deep Algorithm Unrolling for Biomedical Imaging](https://mlsanity.com/paper/2108.06637/) (2021)
- [QN-Mixer: A Quasi-Newton MLP-Mixer Model for Sparse-View CT Reconstruction](https://mlsanity.com/paper/2402.17951/) (2024)
- [Denoising Plane Wave Ultrasound Images Using Diffusion Probabilistic Models](https://mlsanity.com/paper/2408.10987/) (2024)
- [Two-and-a-half Order Score-based Model for Solving 3D Ill-posed Inverse Problems](https://mlsanity.com/paper/2308.08511/) (2023)
- [RN-SDEs: Limited-Angle CT Reconstruction with Residual Null-Space Diffusion Stochastic Differential Equations](https://mlsanity.com/paper/2409.13930/) (2024)

## Abstract (arXiv)

We present a new method for image reconstruction which replaces the projector in a projected gradient descent (PGD) with a convolutional neural network (CNN). CNNs trained as high-dimensional (image-to-image) regressors have recently been used to efficiently solve inverse problems in imaging. However, these approaches lack a feedback mechanism to enforce that the reconstructed image is consistent with the measurements. This is crucial for inverse problems, and more so in biomedical imaging, where the reconstructions are used for diagnosis. In our scheme, the gradient descent enforces measurement consistency, while the CNN recursively projects the solution closer to the space of desired reconstruction images. We provide a formal framework to ensure that the classical PGD converges to a local minimizer of a non-convex constrained least-squares problem. When the projector is replaced with a CNN, we propose a relaxed PGD, which always converges. Finally, we propose a simple scheme to train a CNN to act like a projector. Our experiments on sparse view Computed Tomography (CT) reconstruction for both noiseless and noisy measurements show an improvement over the total-variation (TV) method and a recent CNN-based technique.