Digital Gabor Filters Do Generate MRA-based Wavelet Tight Frames

Gabor frames, especially digital Gabor filters, have long been known as indispensable tools for local time-frequency analysis of discrete signals. With strong orientation selectivity, tensor products discrete (tight) Gabor frames also see their applications in image analysis and restoration. However, the lack of multi-scale structures existing in MRA-based wavelet (tight) frames makes discrete Gabor frames less effective on modeling local structures of signals with varying sizes. Indeed, historically speaking, it was the motivation of studying wavelet systems. By applying the unitary extension principle on some most often seen digital Gabor filters (e.g. local discrete Fourier transform and discrete Cosine transform), we are surprised to find out that these digital filter banks generate MRA-based wavelet tight frames in square integrable function space, and the corresponding refinable functions and wavelets can be explicitly given. In other words, the discrete tight frames associated with these digital Gabor filters can be used as the filter banks of MRA wavelet tight frames, which introduce both multi-scale structures and fast cascade implementation of discrete signal decomposition/reconstruction. Discrete tight frames generated by such filters with both wavelet and Gabor structures can see their potential applications in image processing and recovery.