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Projection-Based Correction for Enhancing Deep Inverse Networks

Jorge Bacca

TL;DR

This work tackles the mismatch between learned reconstructions and physical measurement constraints in ill-posed inverse problems described by $y = A x + n$. It introduces a non-iterative projection-based correction that enforces data consistency by projecting a network's output onto the forward-model subspace, leveraging a deep range-null-space interpretation where $\hat{f}(y) = A^ y + \mathcal{N}(y)$. A noise-free variant enforces $A x = y$, while a regularized version accommodates noise with covariance $\Sigma$, yielding a closed-form update that preserves computational efficiency. Across inpainting, deblurring, SPI, and multiple architectures, the method improves reconstruction quality—especially in early training and low-to-moderate noise settings—by reducing unconstrained null-space components and enhancing data fidelity. The approach offers a practical, non-iterative refinement that can be integrated with existing deep inverse models to boost reliability in safety- and physics-constrained imaging applications.

Abstract

Deep learning-based models have demonstrated remarkable success in solving illposed inverse problems; however, many fail to strictly adhere to the physical constraints imposed by the measurement process. In this work, we introduce a projection-based correction method to enhance the inference of deep inverse networks by ensuring consistency with the forward model. Specifically, given an initial estimate from a learned reconstruction network, we apply a projection step that constrains the solution to lie within the valid solution space of the inverse problem. We theoretically demonstrate that if the recovery model is a well-trained deep inverse network, the solution can be decomposed into range-space and null-space components, where the projection-based correction reduces to an identity transformation. Extensive simulations and experiments validate the proposed method, demonstrating improved reconstruction accuracy across diverse inverse problems and deep network architectures.

Projection-Based Correction for Enhancing Deep Inverse Networks

TL;DR

This work tackles the mismatch between learned reconstructions and physical measurement constraints in ill-posed inverse problems described by . It introduces a non-iterative projection-based correction that enforces data consistency by projecting a network's output onto the forward-model subspace, leveraging a deep range-null-space interpretation where . A noise-free variant enforces , while a regularized version accommodates noise with covariance , yielding a closed-form update that preserves computational efficiency. Across inpainting, deblurring, SPI, and multiple architectures, the method improves reconstruction quality—especially in early training and low-to-moderate noise settings—by reducing unconstrained null-space components and enhancing data fidelity. The approach offers a practical, non-iterative refinement that can be integrated with existing deep inverse models to boost reliability in safety- and physics-constrained imaging applications.

Abstract

Deep learning-based models have demonstrated remarkable success in solving illposed inverse problems; however, many fail to strictly adhere to the physical constraints imposed by the measurement process. In this work, we introduce a projection-based correction method to enhance the inference of deep inverse networks by ensuring consistency with the forward model. Specifically, given an initial estimate from a learned reconstruction network, we apply a projection step that constrains the solution to lie within the valid solution space of the inverse problem. We theoretically demonstrate that if the recovery model is a well-trained deep inverse network, the solution can be decomposed into range-space and null-space components, where the projection-based correction reduces to an identity transformation. Extensive simulations and experiments validate the proposed method, demonstrating improved reconstruction accuracy across diverse inverse problems and deep network architectures.

Paper Structure

This paper contains 11 sections, 4 theorems, 44 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Deep Inverse Networks
  4. Projection-Based Correction
  5. Regularized Projection-Based Correction for Noisy Measurements
  6. Simulations and Results
  7. Well-Trained Network Condition
  8. Performance of the Proposed Projection-Based Correction for Different Inverse Problems
  9. Evaluation on Out-of-Distribution Data and Varying Noise Levels
  10. Conclusion
  11. Compressive Spectral Imaging

Key Result

Proposition 1

Let $\hat{f}$ be a reconstruction network satisfying the conditions of a well-trained deep inverse network defined in Definition defintion_well. Then, the expected reconstruction error over the training dataset satisfies Thus, $\hat{f}$ is a minimizer of eq:training.

Figures (5)

  • Figure 1: Projection-Based Correction Method. he measurement $\boldsymbol{y}$ is processed through a trained network $\hat{f}$ to obtain an initial estimation $\hat{f}(\boldsymbol{y})$. This estimation is then refined using a fixed projection operator to enhance the reconstruction while ensuring consistency with the measurement model $\hat{\boldsymbol{x}}$.
  • Figure 2: Training and testing comparison of (Left) MSE reconstruction error and (Right) Null-space consistency over epochs. The solid line represents the network performance, while the dashed line corresponds to the proposed Projection-Based Correction.
  • Figure 3: Visual results of the evaluated method. The leftmost column shows the measurement and the ground truth (GT). The results at an early epoch (20 epochs) and the final epoch (100 epochs) are displayed to assess the effect of the Projection-Based Correction method.
  • Figure 4: Effect of $\lambda$ parameter in the Projected-Based Correction Method in the present of $0.1$ of Gaussian Noise.
  • Figure 5: Visual results of the evaluated method. The leftmost column shows the measurement and the ground truth (GT). The results at an early epoch (20 epochs) and the final epoch (100 epochs) are displayed to assess the effect of the Projection-Based Correction method.

Theorems & Definitions (13)

  • Definition 1: Well-Trained Deep Inverse Network
  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Proposition 2
  • proof
  • proof
  • ...and 3 more