Hessian Schatten-Norm Regularization for Linear Inverse Problems

Abstract: We introduce a novel family of invariant, convex, and non-quadratic functionals that we employ to derive regularized solutions of ill-posed linear inverse imaging problems. The proposed regularizers involve the Schatten norms of the Hessian matrix, computed at every pixel of the image. They can be viewed as second-order extensions of the popular total-variation (TV) semi-norm since they satisfy the same invariance properties. Meanwhile, by taking advantage of second-order derivatives, they avoid the staircase effect, a common artifact of TV-based reconstructions, and perform well for a wide range of applications. To solve the corresponding optimization problems, we propose an algorithm that is based on a primal-dual formulation. A fundamental ingredient of this algorithm is the projection of matrices onto Schatten norm balls of arbitrary radius. This operation is performed efficiently based on a direct link we provide between vector projections onto $\ell_q$ norm balls and matrix projections onto Schatten norm balls. Finally, we demonstrate the effectiveness of the proposed methods through experimental results on several inverse imaging problems with real and simulated data.

• The paper introduces a novel regularization framework using Hessian Schatten norms to overcome limitations of traditional total-variation methods.
• It develops an efficient primal-dual optimization algorithm that leverages matrix projections for enhanced image reconstruction.
• Experimental results demonstrate significant SNR improvements in applications like deblurring and biomedical imaging reconstruction.

Hessian Schatten-Norm Regularization for Linear Inverse Problems: A Detailed Overview

The paper "Hessian Schatten-Norm Regularization for Linear Inverse Problems" by Stamatios Lefkimmiatis, John Paul Ward, and Michael Unser introduces a novel approach to tackle ill-posed linear inverse imaging problems through a new family of invariant, convex, and non-quadratic functionals. The core idea is centered around the use of Hessian matrix Schatten norms as regularizers for image reconstruction, which are purported to be superior due to their second-order nature and ability to circumvent issues like the staircase effect commonly associated with total-variation (TV) based methods.

Technical Contributions and Methodology

The key contributions of this research are manifold:

1. Novel Regularization Framework: The authors introduce a family of regularizers based on the Schatten norms of the Hessian matrix, analyzed per pixel of the image. These are posited as second-order extensions of the TV semi-norm, preserving the essential invariance properties like convexity and rotational invariance, while mitigating the edge oversharpening and the staircase effects typical of first-order methods.
2. Algorithm Development: The development of an efficient primal-dual optimization algorithm forms the computational backbone of this study. The introduced algorithm fundamentally relies on matrix projections onto Schatten norm balls, with the innovative linkage to vector projections onto $\ell_q$ norm balls aiding efficient computational performance. This method demonstrates flexibility in accommodating both smooth and high-dimensional problem spaces.
3. Theoretical Foundation: The paper establishes a robust theoretical framework justifying the proposed regularizers. It elaborates how these norms effectively capture curvature information, thus enabling better approximation of natural image intensity variations, enhancing the quality of reconstructions in inverse problems.
4. Experimental Validation: With rigorous experimental validations, including applications to real and simulated inverse imaging problems such as deblurring and sparse reconstruction, the paper illustrates the enhanced performance of Hessian-based Schatten norms. The comparison against traditional TV and other quadratic regularization methods highlights the significant improvements in avoiding artifacts and preserving smoothness and texture in images.

Numerical and Experimental Results

The experimental section underscores strong numerical results, showcasing significant improvements in image quality metrics such as signal-to-noise ratio (SNR) when compared against established methods. The analysis spans synthetic, standard test images, and more critically, real biomedical images, evidencing the practicality of these advanced regularizers in fields requiring precise image reconstruction.

Implications and Future Research

Implications of this research sit at the confluence of computational imaging and optimization. Practically, the enhanced regularizers could significantly impact how imaging tasks in fields such as medical imaging and remote sensing are approached—specifically in applications where preserving fine details while circumventing common reconstruction artifacts is crucial.

Theoretically, the framework extends the application of Schatten norms in new directions, encouraging further exploration into other types of multiscale geometric information capture within imaging contexts or potentially beyond in broader data science applications.

As for future developments, the trajectory of this work may well include extending applicability to non-convex settings or further optimization of computational procedures through advancements in algorithmic efficiency, perhaps leveraging emerging computational paradigms such as parallel processing or machine learning integration for adaptive regularization parameter tuning.

In summary, this paper contributes a noteworthy advance in the toolkit available to researchers tackling challenging linear inverse problems, providing new avenues for both academic inquiry and practical application in high-stakes areas of image processing.