@article {IOPORT.05586439,
    author       = {Shen, Jianhong},
    title        = {Beamlets are densely embedded in $H^{-1}$.},
    year         = {2009},
    journal      = {Advances in Computational Mathematics},
    volume       = {31},
    number       = {1-3},
    issn         = {1019-7168},
    pages        = {329-348},
    publisher    = {Springer, Dordrecht},
    doi          = {10.1007/s10444-008-9094-3},
    abstract     = {Summary: The insufficiency of using ordinary measurable functions to model complex natural images was first emphasized by {\it D. Mumford} and {\it B. Gidas} [Q. Appl. Math. 59, No.~1, 85--111 (2001; Zbl 1159.68598)]. The idea was later rediscovered by [{\it Y. Meyer}, Oscillating patterns in image processing and nonlinear evolution equations. University Lecture Series. 22. Providence, RI: American Mathematical Society (AMS) (2001; Zbl 0987.35003)] 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 {\it St. Osher, A. Sol\'e} and {\it L. Vese} model [Multiscale Model. Simul. 1, No.~3, 349--370 (2003; Zbl 1051.49026)] to facilitate practical computation. $H^{-1}$ textures have also been further employed in the recent works of {\it I. Daubechies} and {\it G. Teschke} [Appl. Comput. Harmon. Anal. 19, No. 1, 1--16 (2005; Zbl 1079.68104)], {\it L. H. Lieu} and {\it L. A. Vese} [Appl. Math. Optim. 58, No. 2, 167--193 (2008; Zbl 1191.68789)], the author [AMRX, Appl. Math. Res. Express 2005, No. 4, 143--167 (2005; Zbl 1092.49028)], 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, ) 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.},
    identifier   = {05586439},
}