Beamlets are densely embedded in H −1

Abstract

The insufficiency of using ordinary measurable functions to model complex natural images was first emphasized by David Mumford (Q Appl Math 59:85–111, 2001). The idea was later rediscovered by Yves Meyer (Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, vol. 22, University Lecture Series, AMS, Providence, 2001) who introduced proper texture models based on generalized functions or distributions. The simpler but effective Sobolev texture model of H − 1 was subsequently explored by Osher et al. (Multiscale Model Simul 1:349–370, 2003) to facilitate practical computation. H − 1 textures have also been further employed in the recent works of Daubechies and Teschke (Appl Comput Harmon Anal 19(1):1–16, 2005), Lieu and Vese (UCLA CAM Tech Report, 05–33, 2005), Shen (Appl Math Res Express 4:143–167, 2005), and many others, leading to a new generation of models for image processing and analysis. On the other hand, beamlets are the unconventional class of geometric wavelets invented by Donoho and Huo (Multiscale and Multiresolution Methods, Lect Notes Comput Sci Eng, vol. 20, pp. 149–196. Springer, Berlin, 2002) to efficiently represent and detect lower dimensional singular image features. In the current work, we make an intriguing connection between the above two realms by demonstrating that H − 1 is the natural space (of generalized functions) that hosts beamlets, and in return can be completely described by them. Computational evidences existing in the literature also help confirm this newly discovered bond.