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In order to protect the digital image copyright, it is necessary to design a robust watermarking algorithm. To achieve this purpose, a novel color image watermarking scheme based on an improved QR decomposition is proposed in this paper. The proposed method gives a new algorithm to find elements of Q and R matrices instead of using the Gram–Schmidt algorithm for QR factorization. First, the R matrix is performed by solving a set of linear equations where diagonal elements of R are checked and modified if they are zero or negative. After that, the Q matrix is computed based on the R matrix. In addition, a novel formula is proposed to improve the extracting time where the first element R(1, 1) of the R matrix is found instead of computing QR decomposition as the previous proposals. Experimental results show that the proposed method outperforms other considered methods in this paper in terms of the quality of the watermarked images. Furthermore, the execution time is significantly improved, and the extracted watermark is more robust against almost tested attacks. [ABSTRACT FROM AUTHOR]

The classical Gram-Schmidt algorithm for computing the QR factorization of a matrix X requires at least one pass over the current orthogonalized matrix Q as each column of X is added to the factorization. When Q becomes so large that it must be maintained on a backing store, each pass involves the costly transfer of data from the backing store to main memory. However, if one orthogonalizes the columns of X in blocks of m columns, the number of passes is reduced by a factor of 1/m. Moreover, matrix-vector products are converted into matrix-matrix products, allowing level-3 BLAS cache performance. In this paper we derive such a block algorithm and give some experimental results that suggest it can be quite effective for large scale problems, even when the matrix X is rank degenerate. [ABSTRACT FROM AUTHOR]