[Paper Review] Color Image Recovery Using Generalized Matrix Completion over Higher-Order Finite Dimensional Algebra

Liang Liao, Zhuang Guo+5 more|arXiv (Cornell University)|Aug 4, 2023

Image and Signal Denoising MethodsComputer Science3 citations

TL;DR

This paper proposes a generalized higher-order matrix completion framework using t-matrices—representations built via finite-dimensional algebra and direct sum decomposition—to improve color image recovery from missing pixels. By extending low-rank matrix completion to higher-order structures through a pixel neighborhood strategy and convex optimization, the method achieves superior performance over conventional matrix and tensor completion techniques on both synthetic and real-world images.

ABSTRACT

To improve the accuracy of color image completion with missing entries, we present a recovery method based on generalized higher-order scalars. We extend the traditional second-order matrix model to a more comprehensive higher-order matrix equivalent, called the "t-matrix" model, which incorporates a pixel neighborhood expansion strategy to characterize the local pixel constraints. This "t-matrix" model is then used to extend some commonly used matrix and tensor completion algorithms to their higher-order versions. We perform extensive experiments on various algorithms using simulated data and algorithms on simulated data and publicly available images and compare their performance. The results show that our generalized matrix completion model and the corresponding algorithm compare favorably with their lower-order tensor and conventional matrix counterparts.

Motivation & Objective

• To address the limitations of traditional low-rank matrix and tensor completion in handling complex, high-dimensional color image data with missing entries.
• To develop a generalized higher-order matrix model that captures local pixel constraints more effectively than second-order models.
• To extend standard matrix and tensor completion algorithms into higher-order versions using t-matrices and finite-dimensional algebra.
• To improve image recovery accuracy by leveraging the structural properties of t-matrices and their spectral decomposition.
• To provide a mathematically rigorous framework for t-matrix operations, including rank, nuclear norm, and inner product definitions.

Proposed method

• Introduces the t-matrix model as a higher-order extension of matrices, constructed via direct sum decomposition of K complex matrices.
• Defines the t-matrix rank and nuclear norm based on the sum of ranks and traces of component matrices, enabling low-rank approximation.
• Applies a pixel neighborhood expansion strategy to encode local spatial constraints within the t-matrix structure.
• Uses convex optimization with a Lagrangian multiplier formulation involving t-matrix variables and real-valued inner products.
• Employs a real-valued isomorphism of complex matrices via 2×2 real blocks to ensure real-valued inner products and numerical stability.
• Derives the t-matrix singular value decomposition (TSVD) using spectral-slice-wise operations and inverse mapping from the direct sum representation.

Experimental results

Research questions

• RQ1 Can a higher-order matrix model based on finite-dimensional algebra improve the accuracy of color image completion with missing pixels compared to traditional matrix and tensor methods?
• RQ2 How does incorporating local pixel neighborhood constraints through the t-matrix structure enhance low-rank recovery performance?
• RQ3 What is the mathematical foundation for defining t-matrix rank, nuclear norm, and inner product in a way that supports convex optimization?
• RQ4 To what extent do t-matrix-based algorithms outperform conventional matrix and tensor completion techniques on real-world and simulated image data?
• RQ5 Can the direct sum representation of t-matrices be leveraged to design efficient, scalable optimization algorithms for image recovery?

Key findings

• The proposed t-matrix model achieves better image recovery performance than conventional low-rank matrix and tensor completion methods on both simulated and real-world color image datasets.
• The pixel neighborhood strategy embedded in the t-matrix framework effectively captures local spatial correlations, improving reconstruction fidelity.
• The generalized matrix completion algorithm based on t-matrices outperforms existing methods in terms of PSNR and SSIM metrics, with quantitative improvements reported on benchmark datasets.
• The use of finite-dimensional algebra and direct sum decomposition provides a solid mathematical foundation for defining t-matrix operations, including rank and nuclear norm.
• The isomorphism of complex matrices into real matrices enables stable, real-valued optimization, avoiding issues with complex inner products in convex solvers.
• The proposed method demonstrates robustness and scalability in handling high-dimensional, missing-pixel scenarios common in color image restoration.