Abstract
In this contribution, we consider the sequence {Hn(x;q)}n≥0 of monic polynomials orthogonal with respect to a Sobolev-type inner product involving forward difference operators For the first time in the literature, we apply the non-standard properties of {Hn(x;q)}n≥0 in a watermarking problem. Several differences are found in this watermarking application for the non-standard cases (when j>0) with respect to the standard classical Krawtchouk case λ=μ=0.
Highlights
This work is devoted to an application of the Krawtchouk–Sobolev type polynomials, previously introduced in [1] and orthogonal with respect to the inner productSoria-Lorente A
Dither Modulation (DM) is a special form of quantization index molulation that is applied in an image watermarking system in order to assign one bit to each transformation coefficient
Kernel size: 2 × 2, 4 × 4, 6 × 6, 8 × 7, 10 × 10, 12 × 12, 14 × 14, 16 × 16, 18 × 18, 20 × 20. It is shown in the results displayed in Figures 6–8, that robustness of the watermarking scheme based on KS1Ms, KS2Ms, and KS3Ms is much higher than that of the schemes based on KMs, KMs, and Fractional Moments of Charlier (FrCMs)
Summary
This work is devoted to an application of the Krawtchouk–Sobolev type polynomials, previously introduced in [1] and orthogonal with respect to the inner product. To the best of our knowledge, it is the first time that any Sobolev-type orthogonal polynomial family has been considered in such an application, achieving reasonably different, and in some cases, positive results In this concern, new open problems come up from this novel direction of research which are subject of a future study, such as the optimality of the parameters in the definition of the Krawtchouk–Sobolev polynomials, which compromise a secure watermarking scheme and similarity of the cover and the watermarked image. New open problems come up from this novel direction of research which are subject of a future study, such as the optimality of the parameters in the definition of the Krawtchouk–Sobolev polynomials, which compromise a secure watermarking scheme and similarity of the cover and the watermarked image Such parameters involve the boundary points of the support of Krawtchouk measure and the level in which such points interfere in the Sobolev inner product, and the order of the difference operators involved. We include a last section of directions of future work at the end of the paper
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
