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In this paper, the deficiency of Schur decomposition (SD) for image watermarking has been comprehensively investigated. A remedial scheme is developed not only to fix the existing problems but also to reinforce the performance in robustness and imperceptibility. This scheme starts with partitioning the host image into non-overlapping blocks of $4\times 4$ pixels and then applying SD to each block individually. Level shifting serves as a controlling gauge for embedding strength. The use of intentional perturbation prevents nil orthonormal vectors derived from zero eigenvalues. The dominant vector is identified by analyzing the entire Schur matrix instead of just diagonal elements. To achieve effective watermark embedding and extraction, the orthonormality of the acquired unitary matrix ought to be preserved while the resulting distortion can be compensated via the systematic modification of the Schur matrix. Finally, a recursive regulation guarantees the retrieval of watermark bits. Experiment results indicate that the proposed scheme is free of errors in the absence of attacks and can withstand a variety of image processing attacks as well. Moreover, the inclusion of distortion compensation contributes an improvement of 2.47 dB in terms of peak signal-to-noise ratio. In comparison with previous SD-based schemes, the proposed one exhibits superior robustness and imperceptibility while operating at the same payload capacity.

A Reinforced Blind Color Image Watermarking Scheme Based on Schur Decomposition