Highly Symmetric Multiple Bi-Frames for Curve and Surface Multiresolution Processing

Wavelets and wavelet frames are important and useful mathematical tools in numerous applications, such as signal and image processing, and numerical analysis. Recently, the theory of wavelet frames plays an essential role in signal processing, image processing, sampling theory, and harmonic analysis. However, multiwavelets and multiple frames are more flexible and have more freedom in their construction which can provide more desired properties than the scalar case, such as short compact support, orthogonality, high approximation order, and symmetry. These properties are useful in several applications, such as curve and surface noise-removing as studied in this dissertation. Thus, the study of multiwavelets and multiple frames construction has more advantages for many applications. Recently, the construction of highly symmetric bi-frames for curve and surface multiresolution processing has been investigated. The 6-fold symmetric bi-frames, which lead to highly symmetric analysis and synthesis bi-frame algorithms, have been introduced. Moreover, these multiple bi-frame algorithms play an important role on curve and surface multiresolution processing. This dissertation is an extension of the study of construction of univariate biorthogonal wavelet frames (bi-frames for short) or dual wavelet frames with each framelet being symmetric in the scalar case. We will expand the study of biorthogonal wavelets and bi-frames construction from the scalar case to the vector case to construct biorthogonal multiwavelets and multiple bi-frames in one-dimension. In addition, we will extend the study of highly symmetric bi-frames for triangle surface multiresolution processing from the scalar case to the vector case. More precisely, the objective of this research is to construct highly symmetric biorthogonal multiwavelets and multiple bi-frames in one and two dimensions for curve and surface multiresolution processing. It runs in parallel with the scalar case. We mainly present the methods of constructing biorthogonal multiwavelets and multiple bi-frames in both dimensions by using the idea of lifting scheme. On the whole, we discuss several topics include a brief introduction and discussion of multiwavelets theory, multiresolution analysis, scalar wavelet frames, multiple frames, and the lifting scheme. Then, we present and discuss some results of one-dimensional biorthogonal multiwavelets and multiple bi-frames for curve multiresolution processing with uniform symmetry: type I and type II along with biorthogonality, sum rule orders, vanishing moments, and uniform symmetry for both types. In addition, we present and discuss some results of two-dimensional biorthogonal multiwavelets and multiple bi-frames and the multiresolution algorithms for surface multiresolution processing. Finally, we show experimental results on curve and surface noise-removing by applying our multiple bi-frame algorithms

08492 Abstracts Collection -- Structured Decompositions and Efficient Algorithms

From 30.11. to 05.12.2008, the Dagstuhl Seminar 08492 ``Structured Decompositions and Efficient Algorithms \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

An Investigation of Orthogonal Wavelet Division Multiplexing Techniques as an Alternative to Orthogonal Frequency Division Multiplex Transmissions and Comparison of Wavelet Families and Their Children

Recently, issues surrounding wireless communications have risen to prominence because of the increase in the popularity of wireless applications. Bandwidth problems, and the difficulty of modulating signals across carriers, represent significant challenges. Every modulation scheme used to date has had limitations, and the use of the Discrete Fourier Transform in OFDM (Orthogonal Frequency Division Multiplex) is no exception. The restriction on further development of OFDM lies primarily within the type of transform it uses in the heart of its system, Fourier transform. OFDM suffers from sensitivity to Peak to Average Power Ratio, carrier frequency offset and wasting some bandwidth to guard successive OFDM symbols. The discovery of the wavelet transform has opened up a number of potential applications from image compression to watermarking and encryption. Very recently, work has been done to investigate the potential of using wavelet transforms within the communication space. This research will further investigate a recently proposed, innovative, modulation technique, Orthogonal Wavelet Division Multiplex, which utilises the wavelet transform opening a new avenue for an alternative modulation scheme with some interesting potential characteristics. Wavelet transform has many families and each of those families has children which each differ in filter length. This research consider comprehensively investigates the new modulation scheme, and proposes multi-level dynamic sub-banding as a tool to adapt variable signal bandwidths. Furthermore, all compactly supported wavelet families and their associated children of those families are investigated and evaluated against each other and compared with OFDM. The linear computational complexity of wavelet transform is less than the logarithmic complexity of Fourier in OFDM. The more important complexity is the operational complexity which is cost effectiveness, such as the time response of the system, the memory consumption and the number of iterative operations required for data processing. Those complexities are investigated for all available compactly supported wavelet families and their children and compared with OFDM. The evaluation reveals which wavelet families perform more effectively than OFDM, and for each wavelet family identifies which family children perform the best. Based on these results, it is concluded that the wavelet modulation scheme has some interesting advantages over OFDM, such as lower complexity and bandwidth conservation of up to 25%, due to the elimination of guard intervals and dynamic bandwidth allocation, which result in better cost effectiveness

Representation and design of wavelets using unitary circuits

The representation of discrete, compact wavelet transformations (WTs) as circuits of local unitary gates is discussed. We employ a similar formalism as used in the multiscale representation of quantum many-body wave functions using unitary circuits, further cementing the relation established in the literature between classical and quantum multiscale methods. An algorithm for constructing the circuit representation of known orthogonal, dyadic, discrete WTs is presented, and the explicit representation for Daubechies wavelets, coiflets, and symlets is provided. Furthermore, we demonstrate the usefulness of the circuit formalism in designing WTs, including various classes of symmetric wavelets and multiwavelets, boundary wavelets, and biorthogonal wavelets

Correspondence between Multiwavelet Shrinkage/Multiple Wavelet Frame Shrinkage and Nonlinear Diffusion

There are numerous methodologies for signal and image denoising. Wavelet, wavelet frame shrinkage, and nonlinear diffusion are effective ways for signal and image denoising. Also, multiwavelet transforms and multiple wavelet frame transforms have been used for signal and image denoising. Multiwavelets have important property that they can possess the orthogonality, short support, good performance at the boundaries, and symmetry simultaneously. The advantage of multiwavelet transform for signal and image denoising was illustrated by Bui et al. in 1998. They showed that the evaluation of thresholding on a multiwavelet basis has produced good results. Further, Strela et al. have showed that the decimated multiwavelet denoising provides superior results than decimated conventional (scalar) wavelet denoising. Mrazek, Weickert, and Steidl in 2003 examined the association between one-dimensional nonlinear diffusion and undecimated Haar wavelet shrinkage. They proved that nonlinear diffusion could be presented by using wavelet shrinkage. High-order nonlinear diffusion in terms of one-dimensional frame shrinkage and two-dimensional frame shrinkage were presented in 2012 by Jiang, and in 2013 by Dong, Jiang, and Shen, respectively. They obtained that the correspondence between both approaches leads to a different form of diffusion equation that mixes benefits from both approaches. The objective of this dissertation is to study the correspondence between one-dimensional multiwavelet shrinkage and high-order nonlinear diffusion, and to study high-order nonlinear diffusion in terms of one-dimensional multiple frame shrinkage also well. Further, this dissertation formulates nonlinear diffusion in terms of 2D multiwavelet shrinkage and 2D multiple wavelet frame shrinkage. From the experiment results, it can be inferred that nonlinear diffusion in terms of multiwavelet shrinkage/multiple frame shrinkage gives better results than a scalar case. On the whole, this dissertation expands nonlinear diffusion in terms of wavelet shrinkage and nonlinear diffusion in terms of frame shrinkage from the scalar wavelets and frames to the multiwavelets and multiple frames