Robust and Blind Audio Watermarking Scheme Based on Genetic Algorithm in Dual Transform Domain

In order to protect the copyright of audio media in cyberspace, a robust and blind audio watermarking scheme based on the genetic algorithm (GA) is proposed in a dual transform domain. A formula for calculating the embedding depth is developed, and two embedding depths with different values are used to represent the “1” and “0” states of the binary watermark, respectively. In the extracting process, the embedding depth in each audio fragment will be calculated and compared with the average embedding depth to determine the watermark bit by bit, so this scheme can blindly extract the watermark without the original audio. GA will be applied to optimize the algorithm parameters for meeting the performance requirements in different applications. Besides, the embedding rule is further optimized to enhance the transparency based on the principle of minimal modification to the audio. Experimental results prove that the payload capacity reaches 172.27 bps, the bit error rate (BER) is 0.1% under the premise that its transparency is higher than 25 dB, and its robustness is strong against many attacks. Significantly, this scheme can adaptively select the algorithm parameters to satisfy the specific performance requirements.

Modified Sumudu Transform and Its Properties

Saif et al. (J. Math. Comput. Sci. 21 (2020) 127-135) considered modified Laplace transform and developed some of their certain properties and relations. Motivated by this work, in this paper, we define modified Sumudu transform and investigate many properties and relations including modified Sumudu transforms of the power function, sine, cosine, hyperbolic sine, hyperbolic cosine, exponential function, and function derivatives. Moreover, we attain two shifting properties and a scale preserving theorem for the modified Sumudu transform. We give modified inverse Sumudu transform and investigate some relations and examples. Furthermore, we show that the modified Sumudu transform is the theoretical dual transform to the modified Laplace transform.

Degenerate Sumudu Transform and Its Properties

Kim-Kim (Russ. J. Math. Phys. 2017, 24, 241-248) defined the degenerate Laplace transform and investigated some of their certain properties. Motivated by this study, in this paper, we introduce the degenerate Sumudu transform and establish some properties and relations. We derive degenerate Sumudu transforms of power functions, degenerate sine, degenerate cosine, degenerate hyperbolic sine, degenerate hyperbolic cosine, degenerate exponential function, and function derivatives. We also acquire a relationship between degenerate Sumudu transform and degenerate gamma function. Moreover, we investigate a scale preserving theorem for the degenerate Sumudu transform. Furthermore, we show that the degenerate Sumudu transform is the theoretical dual transform to the degenerate Laplace transform.

On Radon transforms between lines and hyperplanes

We obtain new inversion formulas for the Radon transform and its dual between lines and hyperplanes in [Formula: see text]. The Radon transform in this setting is non-injective and the consideration is restricted to the so-called quasi-radial functions that are constant on symmetric clusters of lines. For the corresponding dual transform, which is injective, explicit inversion formulas are obtained both in the symmetric case and in full generality. The main tools are the Funk transform on the sphere, the Radon-John [Formula: see text]-plane transform in [Formula: see text], the Grassmannian modification of the Kelvin transform, and the Erdélyi–Kober fractional integrals.