{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T18:31:46Z","timestamp":1772649106397,"version":"3.50.1"},"reference-count":24,"publisher":"World Scientific Pub Co Pte Lt","issue":"03","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Wavelets Multiresolut Inf. Process."],"published-print":{"date-parts":[[2014,5]]},"abstract":"<jats:p> The only quadrature operator of order two on L<jats:sub>2<\/jats:sub>(\u211d<jats:sup>2<\/jats:sup>) which covaries with orthogonal transforms, in particular rotations is (up to the sign) the Riesz transform. This property was used for the construction of monogenic wavelets and curvelets. Recently, shearlets were applied for various signal processing tasks. Unfortunately, the Riesz transform does not correspond with the shear operation. In this paper, we propose a novel quadrature operator called linearized Riesz transform which is related to the shear operator. We prove properties of this transform and analyze its performance vs. the usual Riesz transform numerically. Furthermore, we demonstrate the relation between the corresponding optical filters. Based on the linearized Riesz transform we introduce finite discrete quasi-monogenic shearlets and prove that they form a tight frame. Numerical experiments show the good fit of the directional information given by the shearlets and the orientation obtained from the quasi-monogenic shearlet coefficients. Finally, we provide experiments on the directional analysis of textures using our quasi-monogenic shearlets. <\/jats:p>","DOI":"10.1142\/s0219691314500271","type":"journal-article","created":{"date-parts":[[2014,1,15]],"date-time":"2014-01-15T06:38:05Z","timestamp":1389767885000},"page":"1450027","source":"Crossref","is-referenced-by-count":10,"title":["Linearized Riesz transform and quasi-monogenic shearlets"],"prefix":"10.1142","volume":"12","author":[{"given":"S.","family":"H\u00e4user","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Kaiserslautern, Paul-Ehrlich-Str. 31, 67663 Kaiserslautern, Germany"}]},{"given":"B.","family":"Heise","sequence":"additional","affiliation":[{"name":"CDL MS-MACH, Center of Surface and Nanoanalytics, Department of Knowledge-Based Mathematical Systems, FLLL, ZONA, Johannes Kepler University, Altenberger Str. 69, Linz, A-4040, Austria"}]},{"given":"G.","family":"Steidl","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Kaiserslautern, Paul-Ehrlich-Str. 31, 67663 Kaiserslautern, Germany"}]}],"member":"219","published-online":{"date-parts":[[2014,5,26]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1364\/OE.14.003792"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-0603-9_11"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1137\/05064182X"},{"key":"rf4","unstructured":"E. 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