One of the most fundamental characteristics of natural images and multidimensional signals is the presence of geometric regularities due to smooth boundaries between smooth regions. Current representations for this class of signals such as curvelets, contourlets, shearlets, and surfacelets, are constructed in the frequency domain, which lead to basis functions with large spatial support and Gibbs oscillations.
This research develops new sparse representations for multidimensional signals with geometric regularities. These representations allow successive approximation from coarse to fine and will be digital friendly. The investigators focus on spatial domain constructions based on true multidimensional and geometric lifting schemes that would lead to a new generation of geometric wavelets. In addition, geometric tiling dictionaries with low coherence are explored. The research aims for a precise connection between the continuous domain, where geometric regularity is characterized, and the discrete domain, where input signals and transforms are defined on sampling grids. Finally, the research develops new processing algorithms that exploit the gained knowledge of geometrically regular signals and developed sparse representations.
