Directional Haar wavelet frames on triangles
The composite wavelets such as those considered by \textit{K. Guo} and \textit{D. Labate} [SIAM J. Math. Anal. 39, No. 1, 298--318 (2007; Zbl 1197.42017)] and by \textit{I. A. Krishtal} et al. [J. Geom. Anal. 17, No. 1, 87--96 (2007; Zbl 1124.42026)] are modified in one sense and extended in another to produce Haar wavelets on triangles. They are wavelets with respect to composite dilations with expanding matrix \(A=2I\) and the dihedral group \(D_8\) of isometries of the square \([-1,1]^2\), considered as a group of shear matrices. The geometry works as follows: \([0,1]^2\) is subdivided into four triangles, each with a vertex at \((0,0)\) such that each triangle is obtained from the other under one of the shear matrices. The triangle with vertices \(T_0\sim\{(0,0), (1,0), (1,1/2)\}\) and the triangle \(T_1\) with vertex set \(\{(0,0), (1,1/2) , (1,1)\}\) generate a subdivision of \([-1,1]^2\) consisting of 16 triangles under the action of \(D_8\). The characteristic functions of these triangles are thought of as the basic scaling functions, and it is shown how to subdivide \(T_0\) and \(T_1\) each into 4 further triangles such that the composite dilations generated by \(A=2I\) and \(D_8\) generate a ``self-affine'' tiling of the plane under integer translations. The subdivision gives rise to a multiresolution structure and thus to a tight frame of ``directional Haar wavelets'' supported on the subdivision triangles with values \(\pm 1\). They can be regarded as having an orientation defined by their center of mass. It is proved that the wavelet system forms a Parseval frame for \(L^2(\mathbb{R})\). Filter bank and wavelet transform algorithms are described. Compression and denoising algorithms that take into account directionality are described and numerical comparisons are made with \textit{E. J. Candès} and \textit{D. L. Donoho}'s curvelets [Appl. Comput. Harmon. Anal. 19, No. 2, 198--222 (2005; Zbl 1086.42023)] and Do and Vetterli's contourlets [\textit{M. N. Do} and \textit{M. Vetterli}, Contourlets. Welland, Grant V. (ed.), Beyond wavelets. San Diego, CA: Academic Press/Elsevier Science. Studies in Computational Mathematics 10, 83--105 (2003; Zbl 1086.65125)].
scientific article; zbMATH DE number 5597505
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