{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:35:45Z","timestamp":1787322945540,"version":"build-2736575974"},"reference-count":32,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Math. Anal."],"published-print":{"date-parts":[[2009,1]]},"abstract":"<jats:p>In this paper, we propose a solution for a fundamental problem in computational harmonic analysis, namely, the construction of a multiresolution analysis with directional components. We will do so by constructing subdivision schemes which provide a means to incorporate directionality into the data and thus the limit function. We develop a new type of nonstationary bivariate subdivision scheme, which allows us to adapt the subdivision process depending on directionality constraints during its performance, and we derive a complete characterization of those masks for which these adaptive directional subdivision schemes converge. In addition, we present several numerical examples to illustrate how this scheme works. Secondly, we describe a fast decomposition associated with a sparse directional representation system for two-dimensional data, where we focus on the recently introduced sparse directional representation system of shearlets. In fact, we show that the introduced adaptive directional subdivision schemes can be used as a framework for deriving a shearlet multiresolution analysis with finitely supported filters, thereby leading to a fast shearlet decomposition.<\/jats:p>","DOI":"10.1137\/08072276x","type":"journal-article","created":{"date-parts":[[2009,9,16]],"date-time":"2009-09-16T18:06:52Z","timestamp":1253124412000},"page":"1436-1471","source":"Crossref","is-referenced-by-count":39,"title":["Adaptive Directional Subdivision Schemes and Shearlet Multiresolution Analysis"],"prefix":"10.1137","volume":"41","author":[{"given":"Gitta","family":"Kutyniok","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tomas","family":"Sauer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2009,9,16]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1006\/acha.1998.0255"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.20078"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1098\/rsta.1999.0444"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.10116"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2005.02.003"},{"key":"R6","doi-asserted-by":"crossref","unstructured":"A. S. Cavaretta, W. Dahmen, and C. A. Micchelli,\n                      Stationary subdivision,\n                      Mem. Amer. Math. Soc. 93 (1991).","DOI":"10.1090\/memo\/0453"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1016\/j.apnum.2003.12.012"},{"key":"R8","unstructured":"M. Charina, C. Conti, and T. Sauer,\n                      $L_p$\u2013convergence of subdivision schemes: Joint spectral radius versus restricted spectral radius\n                      , in Approximation Theory XI (Gatlinburg, TN, 2004), M. Neamtu and L. L. Schumaker, eds., Nashboro Press, Nashville, TN, 2005, pp. 129\u2013150."},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1137\/S003614109427429X"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"D. Cox, J. Little, and D. O'Shea,\n                      Ideals, Varieties and Algorithms\n                      , Undergrad. Texts Math., Springer-Verlag, New York, 1992.","DOI":"10.1007\/978-1-4757-2181-2"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1142\/S021969130800229X"},{"key":"R12","unstructured":"S. Dahlke, G. Kutyniok, G. Steidl, and G. Teschke,\n                      Shearlet coorbit spaces and associated banach frames\n                      , Appl. Comput. Harmon. Anal., to appear."},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1007\/s003659900045"},{"key":"R14","doi-asserted-by":"crossref","unstructured":"I. Daubechies,\n                      Ten Lectures on Wavelets\n                      , SIAM, Philadelphia, 1992.","DOI":"10.1137\/1.9781611970104"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1007\/BF01889598"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2005.859376"},{"key":"R17","unstructured":"D. L. Donoho,\n                      Interpolating Wavelet Transforms\n                      , Technical report, Department of Statistics, Stanford University, Stanford, CA, 1992."},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1007\/BF01450912"},{"key":"R19","unstructured":"K. Guo, G. Kutyniok, and D. Labate,\n                      Sparse multidimensional representations using anisotropic dilation and shear operators,\n                      in Proceedings of the International Conference Between Wavelets and Splines (Athens, GA, 2005), G. Chen and M. J. Lai, eds., Nashboro Press, Nashville, TN, 2006, pp. 189\u2013201."},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1137\/060649781"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1006\/acha.2000.0343"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-08-04700-4"},{"key":"R23","unstructured":"G. Kutyniok and T. Sauer,\n                      From wavelets to shearlets and back again\n                      , in Approximation Theory XII (San Antonio, TX, 2007), C. K. Chui, M. Neamtu, and L. Schumaker, eds., Nashboro Press, Nashville, TN, 2008, pp. 201\u2013209."},{"key":"R24","doi-asserted-by":"crossref","unstructured":"D. Labate, W.Q. Lim, G. Kutyniok, and G. Weiss,\n                      Sparse Multidimensional Representation Using Shearlets,\n                      in Wavelets XI (San Diego, CA, 2005), SPIE Proc. 5914, M. Papadakis, A. F. Laine, and M. A. Unser, eds., SPIE, Bellingham, WA, 2005, pp. 254\u2013262.","DOI":"10.1117\/12.613494"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1023\/A:1018937723499"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1023\/A:1025889132677"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-00-02646-5"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-02-06678-9"},{"key":"R29","first-page":"257","volume":"96","author":"Sauer T.","year":"2002","journal-title":"Rev. R. Acad. Cien. Serie A. Mat."},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1007\/s10444-004-7635-y"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1006\/acha.1996.0015"},{"key":"R32","unstructured":"M. Vetterli and J. Kova\u010devi\u0107,\n                      Wavelets and Subband Coding\n                      , Prentice-Hall, Englewood, NJ, 1995."}],"container-title":["SIAM Journal on Mathematical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/08072276X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:35:02Z","timestamp":1787319302000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/08072276X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,1]]},"references-count":32,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2009,1]]}},"alternative-id":["10.1137\/08072276X"],"URL":"https:\/\/doi.org\/10.1137\/08072276x","relation":{},"ISSN":["0036-1410","1095-7154"],"issn-type":[{"value":"0036-1410","type":"print"},{"value":"1095-7154","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,1]]}}}