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Adaptive multiresolution analysis structures and shearlet systems. (English) Zbl 1283.42053

The authors introduce the concept of an adaptive multiresolution analysis (AMRA). This concept is a variant of the classical multiresolution analysis (MRA) and is motivated by the desire to design a fast and flexible decomposition algorithm that adapts to the data at each level of decomposition. This new concept is studied within the framework of affine-like systems and for filter design purposes a unitary extension principle is derived. Based on their theoretical results, the authors then present a comprehensive theory of shearlet systems that includes tight shearlet frames with spatially compactly supported generators. The underlying fast adaptive decomposition and reconstruction algorithms are explicitly described and illustrated by hand of an example.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42C15 General harmonic expansions, frames
65T60 Numerical methods for wavelets
65T99 Numerical methods in Fourier analysis
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory

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