This paper addresses Pinching problems in Möbius geometry for hypersurfaces with Möbius isotropy in the unit sphere.By implementing the minimum norm tensor principle,we rigorously estimate the squared norm o...This paper addresses Pinching problems in Möbius geometry for hypersurfaces with Möbius isotropy in the unit sphere.By implementing the minimum norm tensor principle,we rigorously estimate the squared norm of the quadratic gradient term associated with the Möbius second fundamental form.This analysis yields a critical inequality governing the geometric config-uration.Leveraging this inequality,we subsequently prove a Pinching theorem characterizing the eigenvalues of the Blaschke tensor.展开更多
Ensuring digital media security through robust image watermarking is essential to prevent unauthorized distribution,tampering,and copyright infringement.This study introduces a novel hybrid watermarking framework that...Ensuring digital media security through robust image watermarking is essential to prevent unauthorized distribution,tampering,and copyright infringement.This study introduces a novel hybrid watermarking framework that integrates Discrete Wavelet Transform(DWT),Redundant Discrete Wavelet Transform(RDWT),and Möbius Transformations(MT),with optimization of transformation parameters achieved via a Genetic Algorithm(GA).By combining frequency and spatial domain techniques,the proposed method significantly enhances both the imper-ceptibility and robustness of watermark embedding.The approach leverages DWT and RDWT for multi-resolution decomposition,enabling watermark insertion in frequency subbands that balance visibility and resistance to attacks.RDWT,in particular,offers shift-invariance,which improves performance under geometric transformations.Möbius transformations are employed for spatial manipulation,providing conformal mapping and spatial dispersion that fortify watermark resilience against rotation,scaling,and translation.The GA dynamically optimizes the Möbius parameters,selecting configurations that maximize robustness metrics such as Peak Signal-to-Noise Ratio(PSNR),Structural Similarity Index Measure(SSIM),Bit Error Rate(BER),and Normalized Cross-Correlation(NCC).Extensive experiments conducted on medical and standard benchmark images demonstrate the efficacy of the proposed RDWT-MT scheme.Results show that PSNR exceeds 68 dB,SSIM approaches 1.0,and BER remains at 0.0000,indicating excellent imperceptibility and perfect watermark recovery.Moreover,the method exhibits exceptional resilience to a wide range of image processing attacks,including Gaussian noise,JPEG compression,histogram equalization,and cropping,achieving NCC values close to or equal to 1.0.Comparative evaluations with state-of-the-art watermarking techniques highlight the superiority of the proposed method in terms of robustness,fidelity,and computational efficiency.The hybrid framework ensures secure,adaptive watermark embedding,making it highly suitable for applications in digital rights management,content authentication,and medical image protection.The integration of spatial and frequency domain features with evolutionary optimization presents a promising direction for future watermarking technologies.展开更多
文摘This paper addresses Pinching problems in Möbius geometry for hypersurfaces with Möbius isotropy in the unit sphere.By implementing the minimum norm tensor principle,we rigorously estimate the squared norm of the quadratic gradient term associated with the Möbius second fundamental form.This analysis yields a critical inequality governing the geometric config-uration.Leveraging this inequality,we subsequently prove a Pinching theorem characterizing the eigenvalues of the Blaschke tensor.
文摘Ensuring digital media security through robust image watermarking is essential to prevent unauthorized distribution,tampering,and copyright infringement.This study introduces a novel hybrid watermarking framework that integrates Discrete Wavelet Transform(DWT),Redundant Discrete Wavelet Transform(RDWT),and Möbius Transformations(MT),with optimization of transformation parameters achieved via a Genetic Algorithm(GA).By combining frequency and spatial domain techniques,the proposed method significantly enhances both the imper-ceptibility and robustness of watermark embedding.The approach leverages DWT and RDWT for multi-resolution decomposition,enabling watermark insertion in frequency subbands that balance visibility and resistance to attacks.RDWT,in particular,offers shift-invariance,which improves performance under geometric transformations.Möbius transformations are employed for spatial manipulation,providing conformal mapping and spatial dispersion that fortify watermark resilience against rotation,scaling,and translation.The GA dynamically optimizes the Möbius parameters,selecting configurations that maximize robustness metrics such as Peak Signal-to-Noise Ratio(PSNR),Structural Similarity Index Measure(SSIM),Bit Error Rate(BER),and Normalized Cross-Correlation(NCC).Extensive experiments conducted on medical and standard benchmark images demonstrate the efficacy of the proposed RDWT-MT scheme.Results show that PSNR exceeds 68 dB,SSIM approaches 1.0,and BER remains at 0.0000,indicating excellent imperceptibility and perfect watermark recovery.Moreover,the method exhibits exceptional resilience to a wide range of image processing attacks,including Gaussian noise,JPEG compression,histogram equalization,and cropping,achieving NCC values close to or equal to 1.0.Comparative evaluations with state-of-the-art watermarking techniques highlight the superiority of the proposed method in terms of robustness,fidelity,and computational efficiency.The hybrid framework ensures secure,adaptive watermark embedding,making it highly suitable for applications in digital rights management,content authentication,and medical image protection.The integration of spatial and frequency domain features with evolutionary optimization presents a promising direction for future watermarking technologies.
