In this paper we study the nonsubsampled contourlet transform. We address the corresponding filter design problem using the McClellan transformation. We show how zeroes can be imposed in the filters so that the iterated structure produces regular basis functions. The proposed design framework yields filters that can be implemented efficiently through a lifting factorization. We apply the constructed transform in image noise removal where the results obtained are comparable to the state-of-the art, being superior in some cases.

The rational design, synthesis, and characterization of the building blocks obtained by the esterification of the first generation of tapered monodendrons 3,4,5-tris(p-dodecan-1-yloxy)benzoic acid (12-AG) and 3,4,5-tris[p-(n-dodecan-1-yloxy)benzyloxy]benzoic acid (12-ABG) containing semifluorinated dodecyl groups [i.e., 12Fn-AG-15C5 (n = 0, 4, 6, 8), 12Fn-AG-B15C5, 12Fn-ABG- 15C5, and 12Fn-ABG-B15C5 (n = 0 and 8) where n following the letter F represents the number of outer perfluorinated methylenic units of the dodecyl group] with 4‘-hydroxymethyl(benzo-15-crown-5) (B15C5) and 1-hydroxymethyl(15-crown-5) (15C5) are described. All building blocks self-assemble into supramolecular cylindrical or rod-like dendrimers via ion-mediated complexation processes. These rod-like supermolecules form a thermotropic hexagonal columnar (Φh) liquid crystalline (LC) phase. The fluorination of the dodecyl groups of these tapered building blocks enhances dramatically their self-assembly ability. The building blocks based on n = 6 and 8 self-assemble into supramolecular columns solely via the fluorophobic effect. Direct structural characterization of the supramolecular columns obtained via these two molecular recognition processes by a combination of techniques consisting of differential scanning calorimetry, X-ray diffraction, and thermal optical polarized microscopy, and of the columns obtained solely via the fluorophobic effect allowed the construction of molecular models for the supramolecular columns obtained via these two organizing forces. An increase in the column diameter with increasing n and with the complexation of metal salts (i.e., alkali metal trifluoromethanesulfonates) accounts for a structural model in which the uncomplexed and complexed crown ethers are placed side-by-side in the center of the column with the melted tapered side groups radiating toward its periphery. The perfluorinated segments of the building blocks are microsegregated from the perhydrogenated and aromatic segments of the column. The supramolecular columns obtained from building blocks with n = 8 align homeotropically in the Φh LC phase on untreated glass slides, i.e., form single crystal liquid crystals in which the long axes of their columns are perpendicular to the glass surface. Both the self-assembly of supramolecular columns induced solely by the fluorophobic effect and the homeotropic alignment of these columns in their Φh LC phase open extremely interesting new synthetic and technologic opportunities in the area of self-assembly of well-defined supramolecular architectures obtained from monodendrons and other building blocks.

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