Spline and Spline Wavelet Methods with Applications to Signal and Image Processing

This book presents various contributions of splines to signal and image processing from a unified perspective that is based on the Zak transform (ZT). It expands the methodology from periodic splines, which were presented in the first volume, to non-periodic splines. Together, these books provide a universal toolbox accompanied by MATLAB software for manipulating polynomial and discrete splines, spline-based wavelets, wavelet packets and wavelet frames for signal/ image processing applications.

In this volume, we see that the ZT provides an integral representation of discrete and polynomial splines, which, to some extent, is similar to Fourier integral. The authors explore elements of spline theory and design, and consider different types of polynomial and discrete splines. They describe applications of spline-based wavelets to data compression. These splines are useful for real-time signal processing and, in particular, real-time wavelet and frame transforms.

Further topics addressed in this volume include: "global" splines, such as interpolating, self-dual and smoothing, whose supports are infinite; the compactly supported quasi-interpolating and smoothing splines including quasi-interpolating splines on non-uniform grids; and cubic Hermite splines as a source for the design of multiwavelets and multiwavelet frames.

Readers from various disciplines including engineering, computer science and mathematical information technology will find the descriptions of algorithms, applications and software in this book especially useful.

• Frames
• Images
• Signals
• Splines
• Wavelets

“This book is the second volume in an extensive treatment of splines, spline wavelets, and their applications to signal and image processing. … The book provides a self-contained and extensive exposition of the fundamentals of both splines and wavelets transforms … . This book belongs on the shelf of every researcher in signal and image processing. … a valuable supplement for graduate level courses.” (James Stephan Walker, Mathematical Reviews, July, 2016)

“This excellent monograph provides a universal toolbox (with numerous MATLAB codes) for methods with polynomial resp. discrete splines and spline wavelets and corresponding applications in signal and image processing. … This book is well-written and contains many examples and figures. It is very convenient for broad readership of graduate students and researchers in applied mathematics, computer science, physics, and engineering. Doubtless, this nice book will stimulate further applications of spline wavelet methods.” (Manfred Tasche, zbMATH 1338.65032, 2016)

• School of Computer Science, Tel Aviv University, Tel Aviv, Israel

  Amir Z. Averbuch, Valery A. Zheludev

• Department of Mathematical Information Technology, University of Jyväskylä, Jyväskylä, Finland

  Pekka Neittaanmäki

Amir Averbuch is a professor of Computer since 1987 in the School of Computer Science, Tel Aviv University. He is also the founder and CSO of ThetaRay LTD. He got his B.Sc and M.Sc in mathematics from the Hebrew University in Jerusalem in 1968, 1975, respectively, and a PhD in computer Science from Columbia University in 1983. He has over 30 years of experience in research and development in academia and industry. Between 1977-1986 he was research staff member in IBM T.J. Watson Research Center, Yorktown Heights. New York. His research interests include applied and computational harmonic analysis, big data processing and analysis, cyber security, signal/image processing, Wavelets, scientific computing He was visiting professor in many places. He has supervised 102 M.Sc. 19 PhD and 13 post-doc students. He has published 198 papers in refereed journals, 170 papers in conferences, 10 chapters in books and 20 patents.

Pekka Neittaanmäki is Professor in Scientific Computing and Dean of the Faculty of Information Technology and Head of COMAS (Computing and Mathematical Sciences) Graduate School at the University of Jyväskylä, Visiting Professor at the Tokyo Institute of Technology, Japan, and Adjoint Professor at the University of Houston, USA. His research interests are mathematical and numerical modeling, signal analysis, data analysis, optimization, and optimal control. He is the author or co-author of more than 300 publications in various parts of numerical analysis and applied mathematics including 14 books. He has supervised more than 80 PhD theses. He has participated in many industrial projects in Finland and Europe including among others paper machine, telecommunication, process industry as well as medical diagnostics applications.

Valery A. Zheludev, received his M.S. degree on Math. Physics from St. Petersburg State University, Russia. He received Ph.D. degree on Math. Physics from Steklov Math. Inst. of Acad. Sci. USSR and the prize "For the best Ph.D thesis" at St. Petersburg University. He received Dr. Sci. degree (habilitation) on Comput. Math. from Siberia Branch of Acad. Sci. USSR. Took positions from Lecturer at Pedagogical Univ., St. Petersburg, to full Professor at St.-Petersburg Military Institute for Construction Engineering. In 1995 he moved to Israel, where took positions from Researcher to Associate Professor at the School of Computer Science, TelAviv University, Israel. Currently, he is retired from the Associate Professor position and is working as a Researcher at the RAMOT company, which is a research branch of Tel Aviv University. Starting from 2010 he is working as a Senior Researcher at the Dept. of Mathematical Information Technology, University of Jyvaskyla, Finland.

Fields of research:|
Wavelet and Frame Analysis, Approximation Theory, Signal and Image Processing, Hyperspectral Processing, Geophysics, Pattern Recognition, Target Detection, Medical Imaging.

Participated in many industrialresearch projects in Israel and Finland in cooperation with the Israeli Ministries of Defense and Sciences and with companies such as Paradigm Geophysical Ltd, Waves Audio Ltd, Bar-Kal Systems Engineering Ltd, ELTA Systems Ltd., Applied Materials, Orbotech, Nova, ElOp, Elta, Metso, Patria Avation, TietoSaab.

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