Multivariate Wavelet Frames (Industrial and Applied Mathematics)

This book presents a systematic study of multivariate wavelet frames with matrix dilation, in particular, orthogonal and bi-orthogonal bases, which are a special case of frames. Further, it provides algorithmic methods for the construction of dual and tight wavelet frames with a desirable approximation order, namely compactly supported wavelet frames, which are commonly required by engineers. It particularly focuses on methods of constructing them. Wavelet bases and frames are actively used in numerous applications such as audio and graphic signal processing, compression and transmission of information. They are especially useful in image recovery from incomplete observed data due to the redundancy of frame systems. The construction of multivariate wavelet frames, especially bases, with desirable properties remains a challenging problem as although a general scheme of construction is well known, its practical implementation in the multidimensional setting is difficult.

Another important feature of wavelet is symmetry. Different kinds of wavelet symmetry are required in various applications, since they preserve linear phase properties and also allow symmetric boundary conditions in wavelet algorithms, which normally deliver better performance. The authors discuss how to provide H-symmetry, where H is an arbitrary symmetry group, for wavelet bases and frames. The book also studies so-called frame-like wavelet systems, which preserve many important properties of frames and can often be used in their place, as well as their approximation properties. The matrix method of computing the regularity of refinable function from the univariate case is extended to multivariate refinement equations with arbitrary dilation matrices. This makes it possible to find the exact values of the Hölder exponent of refinable functions and to make a very refine analysis of their moduli of continuity.

VLADIMIR PROTASOV is a professor at the Department of Mechanics and Mathematics, Moscow State University, where he also received his Ph.D. in 1999. He was awarded his D.Sc. at St. Petersburg Department of V.A. Steklov Institute of Mathematics, the Russian Academy of Sciences (2006). His research areas include functional analysis, Fourier analysis, wavelets, matrix theory, optimization, convex geometry, and combinatorics. He is the author of more than 70 research papers, 2 monographs, and more than 20 popular and educational publications, including 4 books. He is a member of the editorial board of Sbornik: Mathematics and Quantum Journal.

MARIA SKOPINA is a professor at the Department of Applied Mathematics and Control Processes, St. Petersburg State University. She received her Ph.D. there in 1980 and her D.Sc. from St. Petersburg Department of V.A. Steklov Institute of Mathematics, the Russian Academy of Sciences in 2000. Her main research interests are wavelets, Fourier series, approximation theory and abstract harmonic analysis. She is the author of more than 70 research papers and the monograph “Wavelet Theory,” which was published in Russian in 2005 and translated into English by the American Mathematical Society (AMS) in 2011. She is a member of the St. Petersburg Mathematical Society and the AMS. She is also an associate editor for the International Journal of Wavelets, Multiresolution and Information Processing (IJWMIP) and has organized four international conferences on “Wavelets and Applications.”

ALEKSANDR KRIVOSHEIN received his Ph.D. from St. Petersburg State University in 2013 and is currently an associate professor at the Department of Applied Mathematics and Control Processes there. His main research interests include wavelets and their applications to signal processing. He is the author of 10 research papers.