[Paper Review] Joint Reconstruction and Calibration using Regularization by Denoising

Mingyang Xie, Yu Sun+3 more|arXiv (Cornell University)|Nov 26, 2020

Medical Imaging Techniques and Applications46 references4 citations

TL;DR

This paper proposes Calibrated RED (Cal-RED), a novel method that jointly reconstructs images and calibrates unknown parameters of the measurement operator in imaging inverse problems by extending the Regularization by Denoising (RED) framework. By iteratively updating both the image and the operator parameters using deep denoising priors, Cal-RED achieves state-of-the-art performance in CT reconstruction, reducing angular error from 5° to 0.65° and improving SNR from 10.16 dB (uncalibrated) to 21.13 dB, with only a 0.95 dB gap to the oracle RED method.

ABSTRACT

Regularization by denoising (RED) is a broadly applicable framework for solving inverse problems by using priors specified as denoisers. While RED has been shown to provide state-of-the-art performance in a number of applications, existing RED algorithms require exact knowledge of the measurement operator characterizing the imaging system, limiting their applicability in problems where the measurement operator has parametric uncertainties. We propose a new method, called Calibrated RED (Cal-RED), that enables joint calibration of the measurement operator along with reconstruction of the unknown image. Cal-RED extends the traditional RED methodology to imaging problems that require the calibration of the measurement operator. We validate Cal-RED on the problem of image reconstruction in computerized tomography (CT) under perturbed projection angles. Our results corroborate the effectiveness of Cal-RED for joint calibration and reconstruction using pre-trained deep denoisers as image priors.

Motivation & Objective

• To address the limitation of existing RED methods that require exact knowledge of the measurement operator in inverse problems with parametric uncertainties.
• To develop a joint reconstruction and calibration framework that leverages powerful deep denoisers as priors without requiring explicit regularization functions.
• To enable automatic calibration of imaging system parameters—such as projection angles in CT—alongside image reconstruction using a unified optimization scheme.
• To validate the method on realistic CT reconstruction tasks with noisy and mismatched projection angles, demonstrating robustness and performance close to the oracle case.

Proposed method

• Cal-RED extends the RED framework by introducing an iterative alternating update scheme: first updating the measurement operator parameters θ, then updating the image x using standard RED.
• The method uses a data-fidelity term based on the ℓ2-norm of the residual between measurements and the forward model, and incorporates a denoising prior via the DnCNN network trained on AWGN at multiple noise levels.
• The image update step leverages the gradient of the data-fidelity term and the denoising operator Dσ(x), with the key equation ∇gθ(x*) + τ(x* − Dσ(x*)) = 0.
• The parameter update step is derived from minimizing a joint objective that balances data consistency and prior-based regularization, enabling simultaneous calibration and reconstruction.
• The algorithm is implemented using an ADMM-like alternating scheme, with the denoiser acting as a proximal operator for the prior, and is compatible with both RED and plug-and-play priors.
• The method is validated on simulated CT data with perturbed projection angles, using a Radon transform as the forward model and evaluating performance via SNR and angular RMSE.

Experimental results

Research questions

• RQ1 Can a joint reconstruction and calibration framework be designed that leverages deep denoisers as priors without requiring explicit regularization functions?
• RQ2 How effective is Cal-RED in reducing angular errors in CT reconstruction when the projection angles are severely mismatched (e.g., 5° error)?
• RQ3 To what extent does Cal-RED improve image quality (measured by SNR) compared to uncalibrated RED and other baselines under noisy and mismatched conditions?
• RQ4 Can Cal-RED achieve performance close to the oracle RED method that knows the true angles, even when the initial parameters are highly inaccurate?

Key findings

• Cal-RED reduces the angular error from an initial 5° to 0.65°, representing a reduction by a factor of more than 7, demonstrating strong calibration capability.
• The method improves the imaging SNR from 10.16 dB (uncalibrated RED) to 21.13 dB under 30 dB input sinogram SNR and 5° angular error, approaching the oracle RED performance.
• Cal-RED achieves an SNR of 21.40 dB when input SNR is 30 dB and angular error is 5°, while uncalibrated RED only reaches 10.01 dB, highlighting the critical role of calibration.
• Even when no calibration error is present, Cal-RED degrades SNR by only 0.03 dB compared to standard RED, indicating minimal performance loss from the calibration extension.
• Cal-RED significantly outperforms Cal-FISTA and Cal-LSM in both SNR and angular calibration accuracy, showing the superiority of deep denoising priors in calibration tasks.
• Visual results show that Cal-RED reconstructs high-resolution images with minimal artifacts and preserves fine details, outperforming U-Net (which oversmoothes) and Cal-FISTA (which introduces blocking artifacts).