Department of Mathematics

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The period doubling renormalization operator was introduced by Feigenbaum and by Coullet and Tresser in the 1970s to study the asymptotic small-scale geometry of the attractor of one-dimensional systems that are at the transition from simple to chaotic dynamics. This geometry turns out not to depend on the choice of the map under rather mild smoothness conditions. The existence of a unique renormalization fixed point that is also hyperbolic among generic smooth-enough maps plays a crucial role in the corresponding renormalization theory. The uniqueness and hyperbolicity of the renormalization fixed point were first shown in the holomorphic context, by means that generalize to other renormalization operators. It was then proved that, in the space of C 2+α unimodal maps, for α &gt; 0, the period doubling renormalization fixed point is hyperbolic as well. In this paper we study what happens when one approaches from below the minimal smoothness thresholds for the uniqueness and for the hyperbolicity of the period doubling renormalization generic fixed point. Indeed, our main result states that in the space of C 2 unimodal maps the analytic fixed point is not hyperbolic and that the same remains true when adding enough smoothness to get a priori bounds. In this smoother class, called C2+, the failure of hyperbolicity is tamer than in C2. Things get much worse with just a bit less smoothness than C2, as then even the uniqueness is lost and other asymptotic behavior becomes possible. We show that the period doubling renormalization operator acting on the space of C1+Lip unimodal maps has infinite topological entropy. © 2008 Cambridge University Press.

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Ergodic Theory and Dynamical Systems

Chaotic period doubling

In this paper, a multipurpose watermarking scheme is proposed. The meaning of the word multipurpose is to make the proposed scheme as single watermarking scheme (SWS) or multiple watermarking scheme (MWS) according to our requirement and convenience. We first segment the host image into blocks by means of Hilbert space filling curve and based on amount of DCT energy in the blocks, the threshold values are selected which make proposed scheme multipurpose. For embedding of n watermarks (n -1) thresholds are selected. If the amount of DCT energy of the block is less than the threshold value then ENOPV decomposition is performed and watermark is embedded in either low or high or all frequency sub-bands by modifying the singular values. If the amount of DCT energy of the block is greater than the threshold value then embedding is done by modifying the singular values. This process of embedding through ENOPV-SVD and SVD is applied alternatively to all (n - 1) threshold values. Finally, modified blocks are mapped back to their original positions using inverse Hilbert space filling curve to get the watermarked image. A reliable extraction process is developed for extracting all watermarks from attacked image. Experiments are done on different standard gray scale images and robustness is carried out by a variety of attacks. © 2009 SPIE-IS&T.

Proceedings of SPIE - The International Society for Optical Engineering

Multipurpose watermarking scheme using essentially non-oscillatory point-value decomposition

In this paper a novel watermarking scheme based on Wavelet Packet Transform (WPT) with best tree is explored. The basic idea behind the proposed scheme is to decompose the preprocessed host image via wavelet packet transform and then find the best tree by entropy based algorithm. Watermark is embedded in all frequency bands of best tree. A reliable watermark extraction scheme is developed for the extraction of watermark from the distorted images. Experimental evaluationsdemonstrate that the proposed scheme is robust against variety of attacks. © 2009 IEEE.

2009 IEEE International Advance Computing Conference, IACC 2009

A robust watermarking scheme using best tree wavelet packet transform

Al-Cu thin films were deposited by DC magnetron sputtering. The films are characterized by atomic force microscopy and its surface morphologies are analyzed by wavelet technique. Multiresolution signal decomposition wavelet technique was employed to extract the surface roughness from the AFM images of Al-Cu thin films. It is observed that the Al-Cu thin films exhibit higher surface roughness value with increasing deposition time. The calculated surface roughness of the thin films, using wavelet technique, is comparable with that of its experimental values. © 2009 World Scientific Publishing Company.

International Journal of Wavelets, Multiresolution and Information Processing

Wavelet analysis of surface morphologies of magnetron sputtered Al-Cu thin films

We present a robust stereo-image coding algorithm using digital watermarking in fractional Fourier transform (FrFT) and singular value decomposition (SVD). For the purpose of the security, the original (left stereo) image has been degraded and watermark (right disparity map) is embedded in the degraded image. This watermarked degraded stereo image is processed in an insecure channel. At the receiver's end, both the watermarked image (left stereo image) and watermark images are found by the decoding process. The use of the FrFT, SVD, and degradation process of the stereo image add much more complexity to decode the information about the stereo images and disparity map extraction. Moreover, processing of the watermarked image only provides the stereo as well as 3-D information of the scene/object. Experimental results show that the proposed algorithm is efficient to achieve stereo image security. © 2009 SPIE and IS & T.

Journal of Electronic Imaging

Stereo image coding via digital watermarking

Digital Watermarking is used for copyright protection and authentication of digital media-image/audio/video data. In this paper, a novel dual watermarking scheme is described based on Discrete Wavelet Transforms (DWT) and Singular Value Decompositions (SVD). Unlike the existing dual watermarking schemes, both the watermark embedded using DWT and SVD are invisible in the proposed scheme. For embedding, both the images are transformed into Wavelet domain and SVD is performed further on both the images. Subsequently the singular values are summed up to find the new set of singular values. Both the watermarks get embedded in the same manner. A reliable watermark extraction scheme has been developed for the extraction of the watermarks from the distorted image. Experimental evaluation demonstrates that the proposed scheme is capable of withstanding variety of attacks.

Journal of Digital Information Management

Dual watermarking scheme for copyright protection and authentication

This paper presents a new semi-blind reference watermarking scheme based on discrete wavelet transform(DWT) and singular value decomposition(SVD) for copyright protection and authenticity. We are using a gray scale logo image as watermark instead of randomly generated Gaussian noise type watermark. For watermark embedding, the original image is transformed into wavelet domain and a reference sub-image is formed using directive contrast and wavelet coefficients. We embed watermark into reference image by modifying the singular values of reference image using the singular values of the watermark. A reliable watermark extraction scheme is developed for the extraction of watermark from distorted image. Experimental evaluation demonstrates that the proposed scheme is able to withstand a variety of attacks. We show that the proposed scheme also stands with the ambiguity attack also. © 2008 Elsevier B.V. All rights reserved.

Computer Standards and Interfaces

A new robust reference watermarking scheme based on DWT-SVD

In this paper, a newer version of Walsh-Hadamard Transform namely multiresolution Walsh-Hadamard Transform (MR-WHT) is proposed for images. Further, a robust watermarking scheme is proposed for copyright protection using MR- WHT and singular value decomposition. The core idea of the proposed scheme is to decompose an image using MR-WHT and then middle singular values of high frequency sub-band at the coarsest and the finest level are modified with the singular values of the watermark. Finally, a reliable watermark extraction scheme is developed for the extraction of the watermark from the distorted image. The experimental results show better visual imperceptibility and resiliency of the proposed scheme against intentional or un-intentional variety of attacks. © 2008 IEEE.

Robust watermarking in multiresolution walsh-hadamard transform

In this paper, a robust watermarking scheme is proposed for copyright protection using Distributed multiresolution discrete cosine transform(D-MR-DCT) and singular value decomposition. The core idea of the proposed scheme is to decompose an image into four frequency sub-bands using D-MR-DCT and then singular values of every sub-band are modified with the singular values of the watermark. The experimental results show better visual imperceptibility and resiliency of the proposed scheme against intentional or unintentional variety of attacks. © 2009 IEEE.

Proceedings of the 7th International Conference on Advances in Pattern Recognition, ICAPR 2009

Robust watermarking using distributed MR-DCT and SVD

Analysis of large dimensional contingency tables is rather difficult. Fienberg and Kim (1999, Journal of American Statistical Association, 94, 229-239) studied the problem of combining conditional (on single variable) log-linear structures for graphical models to obtain partial information about the full graphical log-linear model. In this paper, we consider the general log-linear models and obtain explicit representation for the log-linear parameters of the full model based on that of conditional structures. As a consequence, we give conditions under which a particular log-linear parameter is present or not in the full model. Some of the main results of Fienberg and Kim follow from our results. The explicit relationships between full model and the conditional structures are also presented. The connections between conditional structures and the layer structures are pointed out. We investigate also the hierarchical nature of the full model, based on conditional structures. Kim (2006, Computational Statistics and Data Analysis, 50, 2044-2064) analyzed graphical log-linear models based on conditional log-linear structures, when a set of variables is conditioned. For this case, we employ the Möbius inversion technique to obtain the interaction parameters of the full log-linear model, and discuss their properties. The hierarchical nature of the full model is also studied based on conditional structures. This result could be effectively used for the model selection also. As applications of our results, we have discussed several typical examples, including a real-life example. © 2007 The Institute of Statistical Mathematics, Tokyo.