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
Lie groups of bi-M<span style="white-space:nowrap;">öbius transformations are known and their actions on non orientable n-dimensional complex manifolds have been studied. In this paper, m-M<...Lie groups of bi-M<span style="white-space:nowrap;">öbius transformations are known and their actions on non orientable n-dimensional complex manifolds have been studied. In this paper, m-M<span style="white-space:nowrap;">öbius transformations are introduced and similar problems as those related to bi-M<span style="white-space:nowrap;">öbius transformations are tackled. In particular, it is shown that the subgroup generated by every m-M<span style="white-space:nowrap;">öbius transformation is a discrete group. Cyclic subgroups are exhibited. Vector valued m-M<span style="white-space:nowrap;">öbius transformations are also studied.展开更多
Möbius transformations, which are one-to-one mappings of onto have remarkable geometric properties susceptible to be visualized by drawing pictures. Not the same thing can be said about m-Möbius tran...Möbius transformations, which are one-to-one mappings of onto have remarkable geometric properties susceptible to be visualized by drawing pictures. Not the same thing can be said about m-Möbius transformations f<sub>m</sub> mapping onto . Even for the simplest entity, the pre-image by f<sub>m</sub> of a unique point, there is no way of visualization. Pre-images by f<sub>m</sub> of figures from C are like ghost figures in C<sup>m</sup>. This paper is about handling those ghost figures. We succeeded in doing it and proving theorems about them by using their projections onto the coordinate planes. The most important achievement is the proof in that context of a theorem similar to the symmetry principle for Möbius transformations. It is like saying that the images by m-Möbius transformations of symmetric ghost points with respect to ghost circles are symmetric points with respect to the image circles. Vectors in C<sup>m </sup>are well known and vector calculus in C<sup>m</sup> is familiar, yet the pre-image by f<sub>m</sub> of a vector from C is a different entity which materializes by projections into vectors in the coordinate planes. In this paper, we study the interface between those entities and the vectors in C<sup>m</sup>. Finally, we have shown that the uniqueness theorem for Möbius transformations and the property of preserving the cross-ratio of four points by those transformations translate into similar theorems for m-Möbius transformations.展开更多
It is known that any m-Möbius transformation is an ordinary Möbius transformation in every one of its variables when the other variables do not take the values a and 1/a, where a is a parameter defin...It is known that any m-Möbius transformation is an ordinary Möbius transformation in every one of its variables when the other variables do not take the values a and 1/a, where a is a parameter defining the respective m-Möbius transformation. For ordinary Möbius transformations having distinct fixed points, the multiplier associated with one of these points completely characterizes the nature of that transformation, i.e. it tells us if it is elliptic, hyperbolic or loxodromic. The purpose of this paper is to show that fixed points exist also for m-Möbius transformations and multipliers associated with them can be computed as well. As in the classical case, the values of those multipliers describe completely the nature of the transformations. The method we used was that of a thorough study of the coefficients of the variables involved, with which occasion we discovered surprising symmetries. These were the results allowing us to prove the main theorem regarding the fixed points of a m-Möbius transformation, which is the key to further developments. Finally we were able to illustrate the geometric aspects of these transformations, making the whole theory as intuitive as possible. It was as opening a window into a space of several complex variables. This allows us to prove that if a bi-Möbius transformation is elliptic or hyperbolic in z<sub>1</sub> at a point z<sub>2</sub> it will remain the same on a circle or line passing through z<sub>2</sub>. This property remains true when we switch z<sub>1</sub> and z<sub>2</sub>. The main theorem, dealing with the fixed points of an arbitrary m-Möbius transformation made possible the extension of this result to these transformations.展开更多
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
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.
文摘Lie groups of bi-M<span style="white-space:nowrap;">öbius transformations are known and their actions on non orientable n-dimensional complex manifolds have been studied. In this paper, m-M<span style="white-space:nowrap;">öbius transformations are introduced and similar problems as those related to bi-M<span style="white-space:nowrap;">öbius transformations are tackled. In particular, it is shown that the subgroup generated by every m-M<span style="white-space:nowrap;">öbius transformation is a discrete group. Cyclic subgroups are exhibited. Vector valued m-M<span style="white-space:nowrap;">öbius transformations are also studied.
文摘Möbius transformations, which are one-to-one mappings of onto have remarkable geometric properties susceptible to be visualized by drawing pictures. Not the same thing can be said about m-Möbius transformations f<sub>m</sub> mapping onto . Even for the simplest entity, the pre-image by f<sub>m</sub> of a unique point, there is no way of visualization. Pre-images by f<sub>m</sub> of figures from C are like ghost figures in C<sup>m</sup>. This paper is about handling those ghost figures. We succeeded in doing it and proving theorems about them by using their projections onto the coordinate planes. The most important achievement is the proof in that context of a theorem similar to the symmetry principle for Möbius transformations. It is like saying that the images by m-Möbius transformations of symmetric ghost points with respect to ghost circles are symmetric points with respect to the image circles. Vectors in C<sup>m </sup>are well known and vector calculus in C<sup>m</sup> is familiar, yet the pre-image by f<sub>m</sub> of a vector from C is a different entity which materializes by projections into vectors in the coordinate planes. In this paper, we study the interface between those entities and the vectors in C<sup>m</sup>. Finally, we have shown that the uniqueness theorem for Möbius transformations and the property of preserving the cross-ratio of four points by those transformations translate into similar theorems for m-Möbius transformations.
文摘It is known that any m-Möbius transformation is an ordinary Möbius transformation in every one of its variables when the other variables do not take the values a and 1/a, where a is a parameter defining the respective m-Möbius transformation. For ordinary Möbius transformations having distinct fixed points, the multiplier associated with one of these points completely characterizes the nature of that transformation, i.e. it tells us if it is elliptic, hyperbolic or loxodromic. The purpose of this paper is to show that fixed points exist also for m-Möbius transformations and multipliers associated with them can be computed as well. As in the classical case, the values of those multipliers describe completely the nature of the transformations. The method we used was that of a thorough study of the coefficients of the variables involved, with which occasion we discovered surprising symmetries. These were the results allowing us to prove the main theorem regarding the fixed points of a m-Möbius transformation, which is the key to further developments. Finally we were able to illustrate the geometric aspects of these transformations, making the whole theory as intuitive as possible. It was as opening a window into a space of several complex variables. This allows us to prove that if a bi-Möbius transformation is elliptic or hyperbolic in z<sub>1</sub> at a point z<sub>2</sub> it will remain the same on a circle or line passing through z<sub>2</sub>. This property remains true when we switch z<sub>1</sub> and z<sub>2</sub>. The main theorem, dealing with the fixed points of an arbitrary m-Möbius transformation made possible the extension of this result to these transformations.
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.
