Watermarking for multimedia security using complex wavelets

This paper investigates the application of complex wavelet transforms to the field of digital data hiding. Complex wavelets offer improved directional selectivity and shift invariance over their discretely sampled counterparts allowing for better adaptation of watermark distortions to the host media. Two methods of deriving visual models for the watermarking system are adapted to the complex wavelet transforms and their performances are compared. To produce improved capacity a spread transform embedding algorithm is devised, this combines the robustness of spread spectrum methods with the high capacity of quantization based methods. Using established information theoretic methods, limits of watermark capacity are derived that demonstrate the superiority of complex wavelets over discretely sampled wavelets. Finally results for the algorithm against commonly used attacks demonstrate its robustness and the improved performance offered by complex wavelet transforms

A class of M-Channel linear-phase biorthogonal filter banks and their applications to subband coding

This correspondence presents a new factorization for linearphase biorthogonal perfect reconstruction (PR) FIR filter banks. Using this factorization, we propose a new family of lapped transform called the generalized lapped transform (GLT). Since the analysis and synthesis filters of the GLT are not restricted to be the time reverses of each other, they can offer more freedom to avoid blocking artifacts and improve coding gain in subband coding applications. The GLT is found to have higher coding gain and smoother synthesis basis functions than the lapped orthogonal transform (LOT). Simulation results also demonstrated that the GLT has significantly less blocking artifacts, higher peak signal-tonoise ratio (PSNR), and better visual quality than the LOT in image coding. Simplified GLT with different complexity/performance tradeoff is also studied. © 1999 IEEE.published_or_final_versio

Wavelets and their use

This review paper is intended to give a useful guide for those who want to apply discrete wavelets in their practice. The notion of wavelets and their use in practical computing and various applications are briefly described, but rigorous proofs of mathematical statements are omitted, and the reader is just referred to corresponding literature. The multiresolution analysis and fast wavelet transform became a standard procedure for dealing with discrete wavelets. The proper choice of a wavelet and use of nonstandard matrix multiplication are often crucial for achievement of a goal. Analysis of various functions with the help of wavelets allows to reveal fractal structures, singularities etc. Wavelet transform of operator expressions helps solve some equations. In practical applications one deals often with the discretized functions, and the problem of stability of wavelet transform and corresponding numerical algorithms becomes important. After discussing all these topics we turn to practical applications of the wavelet machinery. They are so numerous that we have to limit ourselves by some examples only. The authors would be grateful for any comments which improve this review paper and move us closer to the goal proclaimed in the first phrase of the abstract.Comment: 63 pages with 22 ps-figures, to be published in Physics-Uspekh

Lossless and low-cost integer-based lifting wavelet transform

Discrete wavelet transform (DWT) is a powerful tool for analyzing real-time signals, including aperiodic, irregular, noisy, and transient data, because of its capability to explore signals in both the frequency- and time-domain in different resolutions. For this reason, they are used extensively in a wide number of applications in image and signal processing. Despite the wide usage, the implementation of the wavelet transform is usually lossy or computationally complex, and it requires expensive hardware. However, in many applications, such as medical diagnosis, reversible data-hiding, and critical satellite data, lossless implementation of the wavelet transform is desirable. It is also important to have more hardware-friendly implementations due to its recent inclusion in signal processing modules in system-on-chips (SoCs). To address the need, this research work provides a generalized implementation of a wavelet transform using an integer-based lifting method to produce lossless and low-cost architecture while maintaining the performance close to the original wavelets. In order to achieve a general implementation method for all orthogonal and biorthogonal wavelets, the Daubechies wavelet family has been utilized at first since it is one of the most widely used wavelets and based on a systematic method of construction of compact support orthogonal wavelets. Though the first two phases of this work are for Daubechies wavelets, they can be generalized in order to apply to other wavelets as well. Subsequently, some techniques used in the primary works have been adopted and the critical issues for achieving general lossless implementation have solved to propose a general lossless method. The research work presented here can be divided into several phases. In the first phase, low-cost architectures of the Daubechies-4 (D4) and Daubechies-6 (D6) wavelets have been derived by applying the integer-polynomial mapping. A lifting architecture has been used which reduces the cost by a half compared to the conventional convolution-based approach. The application of integer-polynomial mapping (IPM) of the polynomial filter coefficient with a floating-point value further decreases the complexity and reduces the loss in signal reconstruction. Also, the “resource sharing” between lifting steps results in a further reduction in implementation costs and near-lossless data reconstruction. In the second phase, a completely lossless or error-free architecture has been proposed for the Daubechies-8 (D8) wavelet. Several lifting variants have been derived for the same wavelet, the integer mapping has been applied, and the best variant is determined in terms of performance, using entropy and transform coding gain. Then a theory has been derived regarding the impact of scaling steps on the transform coding gain (GT). The approach results in the lowest cost lossless architecture of the D8 in the literature, to the best of our knowledge. The proposed approach may be applied to other orthogonal wavelets, including biorthogonal ones to achieve higher performance. In the final phase, a general algorithm has been proposed to implement the original filter coefficients expressed by a polyphase matrix into a more efficient lifting structure. This is done by using modified factorization, so that the factorized polyphase matrix does not include the lossy scaling step like the conventional lifting method. This general technique has been applied on some widely used orthogonal and biorthogonal wavelets and its advantages have been discussed. Since the discrete wavelet transform is used in a vast number of applications, the proposed algorithms can be utilized in those cases to achieve lossless, low-cost, and hardware-friendly architectures

Image Compression using Discrete Cosine Transform & Discrete Wavelet Transform

Image Compression addresses the problem of reducing the amount of data required to represent the digital image. Compression is achieved by the removal of one or more of three basic data redundancies: (1) Coding redundancy, which is present when less than optimal (i.e. the smallest length) code words are used; (2) Interpixel redundancy, which results from correlations between the pixels of an image & (3) psycho visual redundancy which is due to data that is ignored by the human visual system (i.e. visually nonessential information). Huffman codes contain the smallest possible number of code symbols (e.g., bits) per source symbol (e.g., grey level value) subject to the constraint that the source symbols are coded one at a time. So, Huffman coding when combined with technique of reducing the image redundancies using Discrete Cosine Transform (DCT) helps in compressing the image data to a very good extent. The Discrete Cosine Transform (DCT) is an example of transform coding. The current JPEG standard uses the DCT as its basis. The DC relocates the highest energies to the upper left corner of the image. The lesser energy or information is relocated into other areas. The DCT is fast. It can be quickly calculated and is best for images with smooth edges like photos with human subjects. The DCT coefficients are all real numbers unlike the Fourier Transform. The Inverse Discrete Cosine Transform (IDCT) can be used to retrieve the image from its transform representation. The Discrete wavelet transform (DWT) has gained widespread acceptance in signal processing and image compression. Because of their inherent multi-resolution nature, wavelet-coding schemes are especially suitable for applications where scalability and tolerable degradation are important. Recently the JPEG committee has released its new image coding standard, JPEG-2000, which has been based upon DWT

The development of the quaternion wavelet transform

The purpose of this article is to review what has been written on what other authors have called quaternion wavelet transforms (QWTs): there is no consensus about what these should look like and what their properties should be. We briefly explain what real continuous and discrete wavelet transforms and multiresolution analysis are and why complex wavelet transforms were introduced; we then go on to detail published approaches to QWTs and to analyse them. We conclude with our own analysis of what it is that should define a QWT as being truly quaternionic and why all but a few of the “QWTs” we have described do not fit our definition

Unified Theory for Biorthogonal Modulated Filter Banks

Modulated filter banks (MFBs) are practical signal decomposition tools for M -channel multirate systems. They combine high subfilter selectivity with efficient realization based on polyphase filters and block transforms. Consequently, the O(M 2 ) burden of computations in a general filter bank (FB) is reduced to O(M log2 M ) - the latter being a complexity order comparable with the FFT-like transforms.Often hiding from the plain sight, these versatile digital signal processing tools have important role in various professional and everyday life applications of information and communications technology, including audiovisual communications and media storage (e.g., audio codecs for low-energy music playback in portable devices, as well as communication waveform processing and channelization). The algorithmic efficiency implies low cost, small size, and extended battery life, bringing the devices close to our skins.The main objective of this thesis is to formulate a generalized and unified approach to the MFBs, which includes, in addition to the deep theoretical background behind these banks, both their design by using appropriate optimization techniques and efficient algorithmic realizations. The FBs discussed in this thesis are discrete-time time-frequency decomposition/reconstruction, or equivalently, analysis-synthesis systems, where the subfilters are generated through modulation from either a single or two prototype filters. The perfect reconstruction (PR) property is a particularly important characteristics of the MFBs and this is the core theme of this thesis. In the presented biorthogonal arbitrary-delay exponentially modulated filter bank (EMFB), the PR property can be maintained also for complex-valued signals.The EMFB concept is quite flexible, since it may respond to the various requirements given to a subband processing system: low-delay PR prototype design, subfilters having symmetric impulse responses, efficient algorithms, and the definition covers odd and even-stacked cosine-modulated FBs as special cases. Oversampling schemes for the subsignals prove out to be advantageous in subband processing problems requiring phase information about the localized frequency components. In addition, the MFBs have strong connections with the lapped transform (LT) theory, especially with the class of LTs grounded in parametric window functions.<br/

Learning Sparse Orthogonal Wavelet Filters

The wavelet transform is a well studied and understood analysis technique used in signal processing. In wavelet analysis, signals are represented by a sum of self-similar wavelet and scaling functions. Typically, the wavelet transform makes use of a fixed set of wavelet functions that are analytically derived. We propose a method for learning wavelet functions directly from data. We impose an orthogonality constraint on the functions so that the learned wavelets can be used to perform both analysis and synthesis. We accomplish this by using gradient descent and leveraging existing automatic differentiation frameworks. Our learned wavelets are able to capture the structure of the data by exploiting sparsity. We show that the learned wavelets have similar structure to traditional wavelets. Machine learning has proven to be a powerful tool in signal processing and computer vision. Recently, neural networks have become a popular and successful method used to solve a variety of tasks. However, much of the success is not well understood, and the neural network models are often treated as black boxes. This thesis provides insight into the structure of neural networks. In particular, we consider the connection between convolutional neural networks and multiresolution analysis. We show that the wavelet transform shares similarities to current convolutional neural network architectures. We hope that viewing neural networks through the lens of multiresolution analysis may provide some useful insights. We begin the thesis by motivating our method for one-dimensional signals. We then show that we can easily extend the framework to multidimensional signals. Our learning method is evaluated on a variety of supervised and unsupervised tasks, such as image compression and audio classification. The tasks are chosen to compare the usefulness of the learned wavelets to traditional wavelets, as well as provide a comparison to existing neural network architectures. The wavelet transform used in this thesis has some drawbacks and limitations, caused in part by the fact that we make use of separable real filters. We address these shortcomings by exploring an extension of the wavelet transform known as the dual-tree complex wavelet transform. Our wavelet learning model is extended into the dual-tree domain with few modifications, overcoming the limitations of our standard model. With this new model we are able to show that localized, oriented filters arise from natural images