Weighted Schatten $p$-Norm Minimization for Image Denoising and Background Subtraction (1512.01003v1)

Abstract: Low rank matrix approximation (LRMA), which aims to recover the underlying low rank matrix from its degraded observation, has a wide range of applications in computer vision. The latest LRMA methods resort to using the nuclear norm minimization (NNM) as a convex relaxation of the nonconvex rank minimization. However, NNM tends to over-shrink the rank components and treats the different rank components equally, limiting its flexibility in practical applications. We propose a more flexible model, namely the Weighted Schatten $p$-Norm Minimization (WSNM), to generalize the NNM to the Schatten $p$-norm minimization with weights assigned to different singular values. The proposed WSNM not only gives better approximation to the original low-rank assumption, but also considers the importance of different rank components. We analyze the solution of WSNM and prove that, under certain weights permutation, WSNM can be equivalently transformed into independent non-convex $l_p$-norm subproblems, whose global optimum can be efficiently solved by generalized iterated shrinkage algorithm. We apply WSNM to typical low-level vision problems, e.g., image denoising and background subtraction. Extensive experimental results show, both qualitatively and quantitatively, that the proposed WSNM can more effectively remove noise, and model complex and dynamic scenes compared with state-of-the-art methods.

• The paper proposes a novel Weighted Schatten p-Norm Minimization method that refines low-rank matrix approximation by adaptively weighting singular values.
• It demonstrates superior image denoising performance with higher PSNR and clearer perceptual quality compared to BM3D, EPLL, and WNNM.
• For background subtraction, WSNM effectively distinguishes foreground from dynamic scenes, achieving better similarity metrics than traditional NNM-based approaches.

Analysis of Weighted Schatten p-Norm Minimization for Image Denoising and Background Subtraction

The paper at hand introduces a novel approach to Low Rank Matrix Approximation (LRMA) through a method termed Weighted Schatten p-Norm Minimization (WSNM). The primary goal of WSNM is to enhance the flexibility and accuracy of LRMA, widely used in computer vision applications such as image denoising and background subtraction. In contrast to traditional Nuclear Norm Minimization (NNM), the WSNM method addresses the limitations of NNM by overcoming the over-shrinkage of singular values, facilitating better adaptability under measurement noise scenarios.

Core Concepts of WSNM

The WSNM model generalizes the NNM by incorporating the Schatten p-norm, thereby allowing differential weighting of the singular values of a matrix. This approach facilitates a finer control over the approximation process, enabling more distinct treatment of various rank components, which is particularly beneficial when different components hold varying degrees of importance. The WSNM model elegantly transforms into independent non-convex lp-norm subproblems that can be solved to global optimality using the Generalized Iterated Shrinkage Algorithm (GST).

Practical Implementation and Results

The proposed WSNM technique is applied to image processing tasks, primarily focusing on image denoising and background subtraction. The experimental results underscore WSNM's superior performance over existing state-of-the-art methods. Several key results include:

1. Image Denoising:
   • WSNM outperformed benchmarks such as BM3D, EPLL, and WNNM in terms of Peak Signal-to-Noise Ratio (PSNR) and perceptual quality across varying noise levels.
   • The strategy of dynamically adjusting the weights in accordance with singular values allows for a more precise restoration, exhibiting especially strong performance at higher levels of noise.
2. Background Subtraction:
   • On benchmark video sequences, the WSNM-based model demonstrated more accurate and distinct foreground background separation compared to methods like NNM-RPCA and RegL1ALM.
   • These results validate WSNM’s capacity to handle dynamic scenes effectively, often achieving the best similarity metrics between the computed and true foreground regions.

Theoretical Implications

Theoretically, WSNM advances the understanding of LRMA by offering an approach that balances convexity and flexibility. The generalized framework not only encompasses existing models such as WNNM as specific cases but also extends them to a broader scope of applications. The introduction of weights sorted in non-descending order and employing the Schatten p-norm contributes to a robust approximation that aligns more closely with the true low-rank structure of the data.

Future Directions

The insights gained from WSNM suggest several promising avenues for future research. Expanding the model’s application to broader classes of problems in signal processing and computer vision could be worthwhile. Additionally, the investigation of other forms of weighted norms or mixed-norm approaches could further enhance the ability to capture complexities inherent in large-scale visual data. Future studies might also delve into computational optimizations to extend the applicability of WSNM to real-time or very large-scale problems, potentially incorporating parallel computational strategies for efficiency gains.

In conclusion, the Weighted Schatten p-Norm Minimization model represents a significant step forward in the field of robust matrix approximations. Its flexibility in managing singular values and its competitive performance in challenging practical scenarios underscore its utility and potential for future advancements in image processing and beyond.