[Paper Review] First order algorithms in variational image processing

Martin Burger, Alex Sawatzky+1 more|arXiv (Cornell University)|Dec 13, 2014

Photoacoustic and Ultrasonic Imaging4 citations

TL;DR

This paper presents first-order algorithms, particularly proximal alternating direction method of multipliers (ADMM), for solving variational image processing problems involving nonsmooth regularization, such as total variation and ℓ₁-norms. The key contribution is demonstrating that exploiting inter-sinogram correlations via a fully populated covariance matrix in statistical reconstruction significantly improves image quality in spectral CT, outperforming traditional filtered backprojection and diagonal-correlation methods.

ABSTRACT

Variational methods in imaging are nowadays developing towards a quite universal and flexible tool, allowing for highly successful approaches on tasks like denoising, deblurring, inpainting, segmentation, super-resolution, disparity, and optical flow estimation. The overall structure of such approaches is of the form ${\cal D}(Ku) + α{\cal R} (u) ightarrow \min_u$ ; where the functional ${\cal D}$ is a data fidelity term also depending on some input data $f$ and measuring the deviation of $Ku$ from such and ${\cal R}$ is a regularization functional. Moreover $K$ is a (often linear) forward operator modeling the dependence of data on an underlying image, and $α$ is a positive regularization parameter. While ${\cal D}$ is often smooth and (strictly) convex, the current practice almost exclusively uses nonsmooth regularization functionals. The majority of successful techniques is using nonsmooth and convex functionals like the total variation and generalizations thereof or $\ell_1$-norms of coefficients arising from scalar products with some frame system. The efficient solution of such variational problems in imaging demands for appropriate algorithms. Taking into account the specific structure as a sum of two very different terms to be minimized, splitting algorithms are a quite canonical choice. Consequently this field has revived the interest in techniques like operator splittings or augmented Lagrangians. Here we shall provide an overview of methods currently developed and recent results as well as some computational studies providing a comparison of different methods and also illustrating their success in applications.

Motivation & Objective

• To develop and analyze first-order algorithms for variational image processing problems with nonsmooth regularization functionals.
• To address the challenge of efficiently solving large-scale, convex optimization problems arising in imaging, such as denoising, deblurring, and spectral CT reconstruction.
• To improve image reconstruction quality in spectral CT by incorporating full inter-sinogram correlations through a covariance matrix in the data fidelity term.
• To compare the performance of proximal ADMM with traditional methods like filtered backprojection and diagonal-correlation approaches in realistic imaging scenarios.

Proposed method

• The paper employs proximal algorithms, particularly proximal ADMM, to solve variational problems of the form min_u {𝒟(Ku) + αℛ(u)} with nonsmooth regularization ℛ.
• It uses the proximal operator to handle non-smooth terms like total variation and ℓ₁-norms, enabling efficient iterative minimization.
• The method incorporates a fully populated covariance matrix Σ in the data fidelity term to model inter-sinogram correlations in spectral CT.
• The algorithm solves the problem using a proximal ADMM framework with joint reconstruction of material images, leveraging the Fisher information matrix to compute Σ.
• The approach is validated on simulated spectral CT data using a 2D thorax phantom with a six-bin photon-counting detector model.
• Regularization parameters are tuned manually to match noise variance across methods for fair comparison.

Experimental results

Research questions

• RQ1 Can first-order algorithms like proximal ADMM effectively solve large-scale variational image processing problems with nonsmooth regularization?
• RQ2 How does exploiting inter-sinogram correlations through a full covariance matrix Σ affect image reconstruction quality in spectral CT?
• RQ3 Does joint reconstruction using a fully populated Σ outperform decoupled reconstruction assuming only diagonal correlations?
• RQ4 How do proximal ADMM-based methods compare to traditional filtered backprojection in terms of image quality and noise characteristics?
• RQ5 What is the impact of using a material-independent total variation penalty in joint spectral CT reconstruction?

Key findings

• The proximal ADMM algorithm successfully reconstructs images with improved quality by exploiting inter-sinogram correlations through a fully populated covariance matrix Σ.
• Reconstruction using the full Σ matrix results in visibly better image quality, especially in K-edge images, compared to both filtered backprojection and diagonal-correlation approaches.
• The joint reconstruction with full Σ reduces noise and preserves fine structures more effectively than decoupled reconstruction, as evidenced by visual comparison in Figures 10 and 11.
• The method achieves comparable noise variance in the region of interest across different reconstruction strategies when regularization parameters are tuned manually.
• Preliminary results confirm that covariance-aware reconstruction provides advantages on both simulated and experimental spectral CT data.
• The use of proximal ADMM is shown to be preferable over gradient descent for PWLS problems in X-ray CT due to better convergence and stability.