Frequency-Constrained QR: Signal and Image Reconstruction

Because a finite set of measurements is limited in the amount of spectral content it can represent, the reconstruction process from discrete samples is inherently band-limited. In the case of 1D sampling using ideal measurements, the maximum bandwidth of regular and irregular sampling is well known using Nyquist and Gröchenig sampling theorems and lemmas, respectively. However, determining the appropriate reconstruction bandwidth becomes difficult when considering 2D sampling geometries, samples with variable apertures, or signal to noise ratio limitations. Instead of determining the maximum bandwidth a priori, we derive an inverse method to simultaneously reconstruct a signal and determine its effective bandwidth. This inverse method is equivalent to incrementally computing a band-limited inverse using a frequency-constrained QR decomposition (FQR). Comparisons between reconstruction results using FQR and QR decompositions illustrate how FQR is less sensitive to noisy measurement errors, but it is more sensitive to high-frequency components. These methods are particularly useful in the reconstruction of remote sensing images from such as microwave radiometers and scatterometers.