The construction of nonseparable wavelet bi-frames and associated approximation schemes

Almost any kind of application requires the analysis of functions or data. In order to analyze them, they are decomposed into simple building blocks. Such methods are not only used in mathematics, but also in physics, eletrical 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. Best N-term approximation is centered around the best choice of these terms, and it is essential to determine the approximation rate. In the present work, 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 wavelet frames, which allows for redundancy and provides more flexibility for the construction. In particular, we derive a family of arbitrarily smooth wavelet bi-frames in arbitrary dimensions with only two wavelets satisfying a variety of optimality conditions. Next, we determine the associated approximation rate of best N-term approximation, and finally, we apply our findings to image denoising.