Wavelet and its Applications

Ph.DDOCTOR OF PHILOSOPH

The Construction of Nonseparable Wavelet Bi-Frames and Associated Approximation Schemes

Wavelet analysis and its fast algorithms are widely used in many fields of applied mathematics such as in signal and image processing. In the present thesis, we circumvent the restrictions of orthogonal and biorthogonal wavelet bases by constructing wavelet frames. They still allow for a stable decomposition, and so-called wavelet bi-frames provide a series expansion very similar to those of pairs of biorthogonal wavelet bases. Contrary to biorthogonal bases, primal and dual wavelets are no longer supposed to satisfy any geometrical conditions, and the frame setting allows for redundancy. This provides more flexibility in their construction. Finally, we construct families of optimal wavelet bi-frames in arbitrary dimensions with arbitrarily high smoothness. Then we verify that the n-term approximation can be described by Besov spaces and we apply the theoretical findings to image denoising

Tight wavelet frame packet

Ph.DDOCTOR OF PHILOSOPH

Pseudo-splines, wavelets and framelets

Master'sMASTER OF SCIENC

On Parseval Wavelet Frames with Two or Three Generators via the Unitary Extension Principle

Abstract The unitary extension principle (UEP) by Ron and Shen yields a sufficient condition for the construction of Parseval wavelet frames with multiple generators. In this paper we characterize the UEP-type wavelet systems that can be extended to a Parseval wavelet frame by adding just one UEP-type wavelet system. We derive a condition that is necessary for the extension of a UEP-type wavelet system to any Parseval wavelet frame with any number of generators, and prove that this condition is also sufficient to ensure that an extension with just two generators is possible

Beyond B-splines: Exponential pseudo-splines and subdivision schemes reproducing exponential polynomials

The main goal of this paper is to present some generalizations of polynomial B-splines, which include exponential B-splines and the larger family of exponential pseudo-splines. We especially focus on their connections to subdivision schemes. In addition, we generalize a well-known result on the approximation order of exponential pseudo-splines, providing conditions to establish the approximation order of nonstationary subdivision schemes reproducing spaces of exponential polynomial function

Wavelet and Multiscale Methods

Various scientific models demand finer and finer resolutions of relevant features. Paradoxically, increasing computational power serves to even heighten this demand. Namely, the wealth of available data itself becomes a major obstruction. Extracting essential information from complex structures and developing rigorous models to quantify the quality of information leads to tasks that are not tractable by standard numerical techniques. The last decade has seen the emergence of several new computational methodologies to address this situation. Their common features are the nonlinearity of the solution methods as well as the ability of separating solution characteristics living on different length scales. Perhaps the most prominent examples lie in multigrid methods and adaptive grid solvers for partial differential equations. These have substantially advanced the frontiers of computability for certain problem classes in numerical analysis. Other highly visible examples are: regression techniques in nonparametric statistical estimation, the design of universal estimators in the context of mathematical learning theory and machine learning; the investigation of greedy algorithms in complexity theory, compression techniques and encoding in signal and image processing; the solution of global operator equations through the compression of fully populated matrices arising from boundary integral equations with the aid of multipole expansions and hierarchical matrices; attacking problems in high spatial dimensions by sparse grid or hyperbolic wavelet concepts. This workshop proposed to deepen the understanding of the underlying mathematical concepts that drive this new evolution of computation and to promote the exchange of ideas emerging in various disciplines