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    Area of Science:

    • Computer Vision
    • Image Processing
    • Machine Learning

    Background:

    • Sparse coding is crucial for image processing but lacks a sparsity measurement benchmark.
    • Existing methods often rely on NP-hard sparse coding formulations.
    • Rank minimization offers a potential alternative for sparsity analysis.

    Purpose of the Study:

    • To develop a benchmark for measuring image patch/group sparsity.
    • To establish an equivalence between group-based sparse coding (GSC) and rank minimization.
    • To introduce a new norm minimization method for improved sparse coding performance.

    Main Methods:

    • Designed an adaptive dictionary to link GSC and rank minimization.
    • Utilized Singular Value Decomposition (SVD) to estimate singular values for sparsity measurement.
    • Investigated four rank minimization methods, identifying weighted Schatten p-norm minimization (WSNM) as most effective.
    • Translated WSNM into a non-convex weighted ℓp-norm minimization problem in GSC.

    Main Results:

    • Established an equivalence between GSC and rank minimization under the designed dictionary.
    • Developed a benchmark for sparsity measurement based on singular values.
    • WSNM demonstrated the closest approximation to real singular values.
    • The proposed weighted ℓp-norm minimization outperformed ℓ1-norm, ℓp-norm, and weighted ℓ1-norm in experiments.

    Conclusions:

    • The developed benchmark is feasible and effective for evaluating sparse coding methods.
    • The proposed weighted ℓp-norm minimization method shows significant improvements in image restoration.
    • This work bridges the gap between sparse coding and rank minimization, offering new avenues for research.