Abstract:

Almost any kind of application requires at least to a certain extent the analysis of data. Depending on the specific application, the collection of data is usually called a measurement, a signal, or an image. In a mathematical framework, all of these objects are represented as functions. In order to analyze them, they are decomposed into simple building blocks. Such methods are not only used in mathematics, but also in physics, electrical engeneering, seismic geology, wireless communication, target detection, and medical imaging. Regarding the reconstruction, let us recall that bases provide series expansions. However, since algorithmic computations are limited to finite data, the series has to be replaced by a finite sum, let us say of length n. Then best n-term approximation is centered around the best choice of these terms, and it is essential to determine the approximation rate. Finally, in order to realize this rate in practical algorithms, one requires a simple rule for the choice of n terms. In this talk, we address wavelet analysis, and the building blocks are shifts and dilates of a finite number of functions, namely wavelets. However, the construction of wavelet bases with convenient inner properties yields certain limitations. We circumvent these restrictions with the concept of frames, which allows for redundancy and provides more flexibility than bases. Next, we determine the associated approximation rate of best n-term approximation, and finally, we apply our findings to image denoising.