{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T23:59:56Z","timestamp":1773273596888,"version":"3.50.1"},"reference-count":20,"publisher":"Association for Computing Machinery (ACM)","issue":"1","license":[{"start":{"date-parts":[[2009,12,1]],"date-time":"2009-12-01T00:00:00Z","timestamp":1259625600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Algorithms"],"published-print":{"date-parts":[[2009,12]]},"abstract":"<jats:p>\n            Rank-convolutions have important applications in a variety of areas such as signal processing and computer vision. We define a\n            <jats:italic>mask<\/jats:italic>\n            as a function taking only values zero and infinity. Rank-convolutions with masks are of special interest to image processing.\n          <\/jats:p>\n          <jats:p>\n            We show how to compute the rank-\n            <jats:italic>k<\/jats:italic>\n            convolution of a function over an interval of length\n            <jats:italic>n<\/jats:italic>\n            with an arbitrary mask of length\n            <jats:italic>m<\/jats:italic>\n            in\n            <jats:italic>O<\/jats:italic>\n            (\n            <jats:italic>n<\/jats:italic>\n            \u221a\n            <jats:italic>m<\/jats:italic>\n            log\n            <jats:italic>m<\/jats:italic>\n            ) time. The result generalizes to the\n            <jats:italic>d<\/jats:italic>\n            -dimensional case. Previously no algorithm performing significantly better than the brute-force\n            <jats:italic>O<\/jats:italic>\n            (\n            <jats:italic>nm<\/jats:italic>\n            ) bound was known.\n          <\/jats:p>\n          <jats:p>Our algorithm seems to perform well in practice. We describe an implementation, illustrating its application to a problem in image processing. Already on relatively small images, our experiments show a signficant speedup compared to brute force.<\/jats:p>","DOI":"10.1145\/1644015.1644035","type":"journal-article","created":{"date-parts":[[2010,8,24]],"date-time":"2010-08-24T13:16:40Z","timestamp":1282655800000},"page":"1-13","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":5,"title":["Computing rank-convolutions with a mask"],"prefix":"10.1145","volume":"6","author":[{"given":"L\u00e1szl\u00f3","family":"Babai","sequence":"first","affiliation":[{"name":"University of Chicago, Chicago, IL"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pedro F.","family":"Felzenszwalb","sequence":"additional","affiliation":[{"name":"University of Chicago, Chicago, IL"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2009,12,28]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01840359"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1137\/0110042"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-0000(73)80033-9"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1007\/11841036_17"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1016\/0167-6377(94)90048-5"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1007\/11534273_28"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1109\/CVPR.2005.329"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1016\/0165-1684(90)90158-U"},{"key":"e_1_2_1_10_1","unstructured":"Felzenszwalb P. F. and Huttenlocher D. P. 2004. Distance transforms of sampled functions. Tech. rep. 2004-1963 Cornell University Faculty of Computer and Information Science.  Felzenszwalb P. F. and Huttenlocher D. P. 2004. Distance transforms of sampled functions. Tech. rep. 2004-1963 Cornell University Faculty of Computer and Information Science."},{"key":"e_1_2_1_11_1","doi-asserted-by":"publisher","DOI":"10.1023\/B:VISI.0000042934.15159.49"},{"key":"e_1_2_1_12_1","doi-asserted-by":"publisher","DOI":"10.1007\/s11263-006-7899-4"},{"key":"e_1_2_1_13_1","unstructured":"Felzenszwalb P. F. Huttenlocher D. P. and Kleinberg J. M. 2004. Fast algorithms for large-state-space HMMs with applications to Web usage analysis. In Advances in NIPS 16. MIT Press 409--416.  Felzenszwalb P. F. Huttenlocher D. P. and Kleinberg J. M. 2004. Fast algorithms for large-state-space HMMs with applications to Web usage analysis. In Advances in NIPS 16. MIT Press 409--416."},{"key":"e_1_2_1_14_1","doi-asserted-by":"publisher","DOI":"10.1109\/JPROC.2004.840301"},{"key":"e_1_2_1_15_1","doi-asserted-by":"publisher","DOI":"10.1109\/34.211471"},{"key":"e_1_2_1_16_1","doi-asserted-by":"publisher","DOI":"10.1137\/0403009"},{"key":"e_1_2_1_17_1","doi-asserted-by":"publisher","DOI":"10.1109\/78.376839"},{"key":"e_1_2_1_18_1","unstructured":"Sankoff D. and Kruskal J. 1983. Time Warps String Edits and Macromolecules: The Theory and Practice of Sequence Comparison. Addison-Wesley.  Sankoff D. and Kruskal J. 1983. Time Warps String Edits and Macromolecules: The Theory and Practice of Sequence Comparison. Addison-Wesley."},{"key":"e_1_2_1_19_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-0000(76)80029-3"},{"key":"e_1_2_1_20_1","volume-title":"Image Analysis and Mathematical Morphology","author":"Serra J.","unstructured":"Serra , J. 1982. Image Analysis and Mathematical Morphology . Academic Press . Serra, J. 1982. Image Analysis and Mathematical Morphology. 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