{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T15:57:23Z","timestamp":1773244643489,"version":"3.50.1"},"reference-count":23,"publisher":"World Scientific Pub Co Pte Lt","issue":"02","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Wavelets Multiresolut Inf. Process."],"published-print":{"date-parts":[[2008,3]]},"abstract":"<jats:p> Finding optimal representations of signals in higher dimensions, in particular directional representations, is currently the subject of intensive research. Since the classical wavelet transform does not provide precise directional information in the sense of resolving the wavefront set, several new representation systems were proposed in the past, including ridgelets, curvelets and, more recently, Shearlets. In this paper we study and visualize the continuous Shearlet transform. Moreover, we aim at deriving mother Shearlet functions which ensure optimal accuracy of the parameters of the associated transform. For this, we first show that this transform is associated with a unitary group representation coming from the so-called Shearlet group and compute the associated admissibility condition. This enables us to employ the general uncertainty principle in order to derive mother Shearlet functions that minimize the uncertainty relations derived for the infinitesimal generators of the Shearlet group: scaling, shear and translations. We further discuss methods to ensure square-integrability of the derived minimizers by considering weighted L<jats:sup>2<\/jats:sup>-spaces. Moreover, we study whether the minimizers satisfy the admissibility condition, thereby proposing a method to balance between the minimizing and the admissibility property. <\/jats:p>","DOI":"10.1142\/s021969130800229x","type":"journal-article","created":{"date-parts":[[2008,3,25]],"date-time":"2008-03-25T08:05:59Z","timestamp":1206432359000},"page":"157-181","source":"Crossref","is-referenced-by-count":102,"title":["THE UNCERTAINTY PRINCIPLE ASSOCIATED WITH THE CONTINUOUS SHEARLET TRANSFORM"],"prefix":"10.1142","volume":"06","author":[{"given":"STEPHAN","family":"DAHLKE","sequence":"first","affiliation":[{"name":"Philipps-Universit\u00e4t Marburg, FB12 Mathematik und Informatik, Hans-Meerwein Stra\u00dfe, Lahnberge, 35032 Marburg, Germany"}]},{"given":"GITTA","family":"KUTYNIOK","sequence":"additional","affiliation":[{"name":"Program in Applied and Computational Mathematics, Princeton University, Washington Road, Fine Hall, Princeton, NJ 08544-1000, USA"}]},{"given":"PETER","family":"MAASS","sequence":"additional","affiliation":[{"name":"Fachbereich 3, Universit\u00e4t Bremen, Postfach 33 04 40, 28334 Bremen, Germany"}]},{"given":"CHEN","family":"SAGIV","sequence":"additional","affiliation":[{"name":"Fachbereich 3, Universit\u00e4t Bremen, Postfach 33 04 40, 28334 Bremen, Germany"}]},{"given":"HANS-GEORG","family":"STARK","sequence":"additional","affiliation":[{"name":"Fachhochschule Aschaffenburg, Mathematik\/Informatik\/Technomathematik, W\u00fcrzburger Str. 45, 63743 Aschaffenburg, Germany"}]},{"given":"GERD","family":"TESCHKE","sequence":"additional","affiliation":[{"name":"Research Group Inverse Problems in Science and Technology, Konrad-Zuse-Zentrum f\u00fcr Informationstechnik Berlin (ZIB), Takustr. 7, 14195 Berlin-Dahlem, Germany"}]}],"member":"219","published-online":{"date-parts":[[2011,11,21]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1258-4"},{"key":"rf2","first-page":"3987","volume":"39","author":"Ali S. T.","journal-title":"Rev. Math. Phys."},{"key":"rf3","first-page":"314","volume":"5","author":"Antoine J.-P.","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511543395"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1137\/S0036141093256265"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1098\/rsta.1999.0444"},{"key":"rf7","first-page":"219","volume":"56","author":"Cand\u00e8s E. J.","journal-title":"Comm. Pure Appl. Math."},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1016\/0898-1221(95)00108-5"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1109\/18.57199"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2005.859376"},{"key":"rf13","doi-asserted-by":"publisher","DOI":"10.1515\/9781400882427"},{"key":"rf14","first-page":"429","volume":"93","author":"Gabor D.","journal-title":"J. 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A.\u00a0Unser (SPIE, San Diego, CA, 2005)\u00a0pp. 254\u2013262."},{"key":"rf22","doi-asserted-by":"publisher","DOI":"10.1007\/11408031_30"},{"key":"rf23","first-page":"1633","volume":"15","author":"Sagiv C.","journal-title":"J. Math. Imaging Vision"},{"key":"rf24","doi-asserted-by":"publisher","DOI":"10.1142\/S0219691305000841"},{"key":"rf25","first-page":"215","volume":"56","author":"Torresani B.","journal-title":"Ann. Inst. H. Poincar\u00e9 Phys. 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