Compactly Supported Tensor Product Complex Tight Framelets with Directionality

Although tensor product real-valued wavelets have been successfully applied to many high-dimensional problems, they can only capture well edge singularities along the coordinate axis directions. As an alternative and improvement of tensor product real-valued wavelets and dual tree complex wavelet transform, recently tensor product complex tight framelets with increasing directionality have been introduced in [8] and applied to image denoising in [13]. Despite several desirable properties, the directional tensor product complex tight framelets constructed in [8,13] are bandlimited and do not have compact support in the space/time domain. Since compactly supported wavelets and framelets are of great interest and importance in both theory and application, it remains as an unsolved problem whether there exist compactly supported tensor product complex tight framelets with directionality. In this paper, we shall satisfactorily answer this question by proving a theoretical result on directionality of tight framelets and by introducing an algorithm to construct compactly supported complex tight framelets with directionality. Our examples show that compactly supported complex tight framelets with directionality can be easily derived from any given eligible low-pass filters and refinable functions. Several examples of compactly supported tensor product complex tight framelets with directionality have been presented

Some Smooth Compactly Supported Tight Wavelet Frames with Vanishing Moments

Let A∈Rd×d, d≥1 be a dilation matrix with integer entries and |detA|=2. We construct several families of compactly supported Parseval framelets associated to A having any desired number of vanishing moments. The first family has a single generator and its construction is based on refinable functions associated to Daubechies low pass filters and a theorem of Bownik. For the construction of the second family we adapt methods employed by Chui and He and Petukhov for dyadic dilations to any dilation matrix A. The third family of Parseval framelets has the additional property that we can find members of that family having any desired degree of regularity. The number of generators is 2d+d and its construction involves some compactly supported refinable functions, the Oblique Extension Principle and a slight generalization of a theorem of Lai and Stöckler. For the particular case d=2 and based on the previous construction, we present two families of compactly supported Parseval framelets with any desired number of vanishing moments and degree of regularity. None of these framelet families have been obtained by means of tensor products of lower-dimensional functions. One of the families has only two generators, whereas the other family has only three generators. Some of the generators associated with these constructions are even and therefore symmetric. All have even absolute values.The first author was partially supported by MEC/MICINN Grant #MTM2011-27998 (Spain)

Wavelet frame bijectivity on Lebesgue and Hardy spaces

We prove a sufficient condition for frame-type wavelet series in $L^p$, the Hardy space $H^1$, and BMO. For example, functions in these spaces are shown to have expansions in terms of the Mexican hat wavelet, thus giving a strong answer to an old question of Meyer. Bijectivity of the wavelet frame operator acting on Hardy space is established with the help of new frequency-domain estimates on the Calder\'on-Zygmund constants of the frame kernel.Comment: 23 pages, 7 figure

A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity

The richness of natural images makes the quest for optimal representations in image processing and computer vision challenging. The latter observation has not prevented the design of image representations, which trade off between efficiency and complexity, while achieving accurate rendering of smooth regions as well as reproducing faithful contours and textures. The most recent ones, proposed in the past decade, share an hybrid heritage highlighting the multiscale and oriented nature of edges and patterns in images. This paper presents a panorama of the aforementioned literature on decompositions in multiscale, multi-orientation bases or dictionaries. They typically exhibit redundancy to improve sparsity in the transformed domain and sometimes its invariance with respect to simple geometric deformations (translation, rotation). Oriented multiscale dictionaries extend traditional wavelet processing and may offer rotation invariance. Highly redundant dictionaries require specific algorithms to simplify the search for an efficient (sparse) representation. We also discuss the extension of multiscale geometric decompositions to non-Euclidean domains such as the sphere or arbitrary meshed surfaces. The etymology of panorama suggests an overview, based on a choice of partially overlapping "pictures". We hope that this paper will contribute to the appreciation and apprehension of a stream of current research directions in image understanding.Comment: 65 pages, 33 figures, 303 reference

Non-stationary Sibling Wavelet Frames on Bounded Intervals

Frame Theory is a modern branch of Harmonic Analysis. It has its roots in Communication Theory and Quantum Mechanics. Frames are overcomplete and stable families of functions which provide non-unique and non-orthogonal series representations for each element of the space. The first milestone was set 1946 by Gabor with the paper ''Theory of communications''. He formulated a fundamental approach to signal decomposition in terms of elementary signals generated by translations and modulations of a Gaussian. The frames for Hilbert spaces were formally defined for the first time 1952 by Duffin and Schaeffer in their fundamental paper ''A class of nonharmonic Fourier series''. The breakthrough of frames came 1986 with Daubechies, Grossmann and Meyer's paper ''Painless nonorthogonal expansions''. Since then a lot of scientists have been investigating frames from different points of view. In this thesis we study non-stationary sibling frames, in general, and the possibility to construct such function families in spline spaces, in particular. Our work follows a theoretical, constructive track. Nonetheless, as demonstrated by several papers by Daubechies and other authors, frames are very useful in various areas of Applied Mathematics, including Signal and Image Processing, Data Compression and Signal Detection. The overcompleteness of the system incorporates redundant information in the frame coefficients. In certain applications one can take advantage of these correlations. The content of this thesis can be split naturally into three parts: Chapters 1-3 introduce basic definitions, necessary notations and classical results from the General Frame Theory, from B-Spline Theory and on non-stationary tight wavelet spline frames. Chapters 4-5 describe the theory we developed for sibling frames on an abstract level. The last chapter presents our explicit construction of a certain class of non-stationary sibling spline frames with vanishing moments in L_2[a,b] which exemplifies and thus proves the applicability of our theoretical results from Chapters 4-5. As a principle of writing we did the best possible to make this thesis self-contained. Classical handbooks, recent monographs, fundamental research papers and survey articles from Wavelet, Frame and Spline Theory are cited for further - more detailed - reading. In Chapter 3 we summarize the considerations on normalized tight spline frames of Chui, He and Stoeckler and some of the results from their article ''Nonstationary tight wavelet frames. I: Bounded intervals'' (see Appl. Comp. Harm. Anal. 17 (2004), 141-197). Our work detailed in Chapters 4-6 is meant to extend and supplement their theory for bounded intervals. Chapter 4 deals with our extension of the general construction principle of non-stationary wavelet frames from the tight case to the non-tight (= sibling) case on which our present work focuses. We apply this principle in Chapter 6 in order to give a general construction scheme for certain non-stationary sibling spline frames of order m with L vanishing moments (m \in N, m >= 2, 1 <= L <= m), as well as some concrete illustrative examples. Chapter 4 presents in Subsection 4.3.3 the motivation for our detailed investigations in Chapter 5. We need some sufficient conditions on the function families defined by coefficient matrices which are as simple as possible, in order to be able to verify easily if some concrete spline families are Bessel families (and thus sibling frames) or not. In Chapter 5 we develop general strategies for proving the boundedness of certain linear operators. These will enable us to check in Chapter 6 the Bessel condition for concrete spline systems which are our candidates for sibling spline frames. Some results concerning multivariate Bessel families are also included

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