Closed-Form Approximation of the Total Variation Proximal Operator (2412.07718v1)

Abstract: Total variation (TV) is a widely used function for regularizing imaging inverse problems that is particularly appropriate for images whose underlying structure is piecewise constant. TV regularized optimization problems are typically solved using proximal methods, but the way in which they are applied is constrained by the absence of a closed-form expression for the proximal operator of the TV function. A closed-form approximation of the TV proximal operator has previously been proposed, but its accuracy was not theoretically explored in detail. We address this gap by making several new theoretical contributions, proving that the approximation leads to a proximal operator of some convex function, that it always decreases the TV function, and that its error can be fully characterized and controlled with its scaling parameter. We experimentally validate our theoretical results on image denoising and sparse-view computed tomography (CT) image reconstruction.

• The paper presents a closed-form approximation to the TV proximal operator, reducing the need for iterative methods in imaging inverse problems.
• It theoretically confirms that the approximation corresponds to a convex function, ensuring monotonic TV reduction and controllable error via scaling parameters.
• Experimental validations in computed tomography and image denoising confirm improved convergence and reduced computational complexity over traditional methods.

Overview of Closed-Form Approximation of the Total Variation Proximal Operator

This paper addresses a prominent limitation in solving imaging inverse problems via total variation (TV) regularization by introducing a closed-form approximation to the proximal operator of the TV function. The authors explore both anisotropic and isotropic forms of TV, which are essential for producing piecewise constant approximations in various imaging tasks. Conventional proximal methods suffer from the lack of closed-form solutions for TV, necessitating iterative sub-processes that introduce significant computational overhead. This research fills this void, providing both theoretical justification and comprehensive numerical validation.

Theoretical Contributions

The authors make several key theoretical contributions to the understanding and implementation of approximate TV proximal operators:

1. Convex Function Existence: It is shown that the approximated proximal operator corresponds to the proximal operator of some convex function, ensuring its reliability within proximal-optimization frameworks.
2. Monotonicity: The paper establishes that the application of the approximate operator invariably leads to a decrease in the TV function, thereby maintaining the intended regularization effect.
3. Error Characterization: The error introduced by the approximation is analytically characterized, and its dependence on the scaling parameter is thoroughly examined. This understanding allows for controllable precision by adjusting parameters.
4. Approximation Accuracy: The relationship between scaling parameters and approximation accuracy is explored, providing practitioners with insights into tuning these parameters for specific applications.

Numerical Validation

Addressing the practical implications, the research provides extensive numerical validations through:

• Limited Angle Computed Tomography: This setting demonstrates that the proposed approximation can be effectively employed in APGM and ADMM frameworks, corroborating theoretical findings and highlighting the approximation's utility in real-world scenarios.
• Image Denoising: Iterative algorithms show positive outcomes consistent with theoretical predictions, specifically within image denoising paradigms.

In these experiments, the closed-form approximation allows the substitution of the true proximal operator without compromising convergence or performance, especially when the scaling parameter τ is finely tuned. This is evidenced by improved PSNR and reduced computational cost relative to traditional iterative methods.

Practical and Theoretical Implications

The closed-form approximation has significant implications for both computational efficiency and theoretical development:

• Performance Enhancement: The reduction in computational complexity from O(knd) to O(nd) makes it feasible to apply TV regularization in high-dimensional problems within acceptable time frames.
• Potential for Broader Adoption: By removing computational bottlenecks, the closed-form approximation opens the doors for widespread use of TV regularizations in emerging imaging technologies, including medical imaging and industrial applications.
• Algorithm Convergence: The theoretical guarantee of convergence within proximal-based frameworks reinforces confidence in employing this approximation for further developments in inverse problem solutions.

Speculation on Future Developments

The theoretical and practical impact of this approximation suggests several future research directions:

• Extension to Other Regularization Techniques: Insights gained here may inspire analogous approaches for other non-smooth regularizers, broadening the toolbox of efficient optimization techniques.
• Adaptive Parameter Scaling: Development of techniques for adaptive parameter tuning could further enhance implementation efficiency and make the approach more user-independent.
• Integration with Deep Learning Models: Research on coupling this approximation with learning-based models could enhance hybrid methods that leverage both traditional optimization and data-driven strategies.

By providing a closed-form approximation for the TV proximal operator, this paper lays critical groundwork for advancing the efficiency and applicability of proximal methods in imaging inverse problems.