{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,4]],"date-time":"2026-02-04T14:42:38Z","timestamp":1770216158888,"version":"3.49.0"},"reference-count":26,"publisher":"Elsevier BV","issue":"1","license":[{"start":{"date-parts":[[1998,1,1]],"date-time":"1998-01-01T00:00:00Z","timestamp":883612800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Pattern Recognition"],"published-print":{"date-parts":[[1998,1]]},"DOI":"10.1016\/s0031-3203(97)00022-8","type":"journal-article","created":{"date-parts":[[2003,4,4]],"date-time":"2003-04-04T21:45:35Z","timestamp":1049492735000},"page":"53-61","source":"Crossref","is-referenced-by-count":22,"title":["Asymptotic granulometric mixing theorem: Morphological estimation of sizing parameters and mixture proportions"],"prefix":"10.1016","volume":"31","author":[{"given":"Francis","family":"Sand","sequence":"first","affiliation":[]},{"given":"Edward R.","family":"Dougherty","sequence":"additional","affiliation":[]}],"member":"78","reference":[{"key":"10.1016\/S0031-3203(97)00022-8_BIB1","series-title":"Random Sets and Integral Geometry","author":"Matheron","year":"1975"},{"key":"10.1016\/S0031-3203(97)00022-8_BIB2","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/0146-664X(80)90006-4","article-title":"The Boolean model and random sets","volume":"12","author":"Serra","year":"1980","journal-title":"Comput. Graphics Image Process."},{"key":"10.1016\/S0031-3203(97)00022-8_BIB3","article-title":"Boolean texture analysis and synthesis","volume":"Vol. 2","author":"Preteux","year":"1988"},{"key":"10.1016\/S0031-3203(97)00022-8_BIB4","series-title":"Introduction to the Theory of Coverage Processes","author":"Hall","year":"1988"},{"issue":"4","key":"10.1016\/S0031-3203(97)00022-8_BIB5","doi-asserted-by":"crossref","first-page":"557","DOI":"10.1137\/1029111","article-title":"Random set theory and problems of modeling","volume":"29","author":"Cressie","year":"1987","journal-title":"SIAM Rev."},{"key":"10.1016\/S0031-3203(97)00022-8_BIB6","series-title":"Image Analysis and Mathematical Morphology","author":"Serra","year":"1982"},{"key":"10.1016\/S0031-3203(97)00022-8_BIB7","doi-asserted-by":"crossref","first-page":"324","DOI":"10.2307\/1427072","article-title":"Densities for stationary random sets and point processes","volume":"16","author":"Weil","year":"1984","journal-title":"Adv. Appl. Probab."},{"issue":"3","key":"10.1016\/S0031-3203(97)00022-8_BIB8","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1111\/j.1365-2818.1988.tb04684.x","article-title":"Expectation formulas and isoperimetric properties for non-isotropic Boolean models","volume":"151","author":"Weil","year":"1988","journal-title":"Microscopy"},{"issue":"1","key":"10.1016\/S0031-3203(97)00022-8_BIB9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0165-1684(94)00139-Q","article-title":"Recursive maximum-likelihood estimation in the one-dimensional discrete Boolean random set model","volume":"43","author":"Dougherty","year":"1995","journal-title":"Signal Process."},{"issue":"3","key":"10.1016\/S0031-3203(97)00022-8_BIB10","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1007\/BF01254103","article-title":"Algebraic analysis of the generating functional for discrete random sets and statistical inference for intensity in the discrete Boolean random-set model","volume":"4","author":"Sidiropoulos","year":"1994","journal-title":"Math. Imaging Vision"},{"issue":"2\/3","key":"10.1016\/S0031-3203(97)00022-8_BIB11","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/BF00118590","article-title":"On the morphological analysis of discrete random sets","volume":"2","author":"Goutsias","year":"1992","journal-title":"Math. Imaging Vision"},{"issue":"1","key":"10.1016\/S0031-3203(97)00022-8_BIB12","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1046\/j.1365-2818.1996.124405.x","article-title":"Maximum-likelihood estimation for discrete Boolean models using linear samples","volume":"182","author":"Handley","year":"1996","journal-title":"Microscopy"},{"key":"10.1016\/S0031-3203(97)00022-8_BIB13","series-title":"Mixture Models","author":"McLachlan","year":"1988"},{"key":"10.1016\/S0031-3203(97)00022-8_BIB14","series-title":"An Introduction to Morphological Image Processing","author":"Dougherty","year":"1992"},{"key":"10.1016\/S0031-3203(97)00022-8_BIB15","series-title":"Morphological Methods in Image and Signal Processing","author":"Giardina","year":"1988"},{"key":"10.1016\/S0031-3203(97)00022-8_BIB16","series-title":"Digital Image Processing Methods","article-title":"Morphological segmentation for textures and particles","author":"Vincent","year":"1994"},{"issue":"10","key":"10.1016\/S0031-3203(97)00022-8_BIB17","doi-asserted-by":"crossref","first-page":"1181","DOI":"10.1016\/0031-3203(92)90020-J","article-title":"Morphological texture-based maximum-likelihood pixel classification based on local granulometric moments","volume":"25","author":"Dougherty","year":"1992","journal-title":"Pattern Recognition"},{"issue":"8","key":"10.1016\/S0031-3203(97)00022-8_BIB18","doi-asserted-by":"crossref","first-page":"2713","DOI":"10.1117\/12.173552","article-title":"Gray-scale morphological granulometric texture classification","volume":"33","author":"Chen","year":"1994","journal-title":"Optical Eng."},{"key":"10.1016\/S0031-3203(97)00022-8_BIB19","series-title":"Mathematical Morphology and its Application to Image Processing","article-title":"Monte-carlo estimation of morphological granulometric discrete size distributions","author":"Sivakumar","year":"1994"},{"issue":"2","key":"10.1016\/S0031-3203(97)00022-8_BIB20","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1016\/1047-3203(92)90016-M","article-title":"Asymptotic normality of the morphological pattern\u2014spectrum moments and orthogonal granolumetric generators","volume":"3","author":"Sand","year":"1992","journal-title":"Visual Comm. Image Representation"},{"issue":"1","key":"10.1016\/S0031-3203(97)00022-8_BIB21","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1006\/jvci.1995.1006","article-title":"Representation of linear granulometric moments for deterministic and random binary Euclidean images","volume":"6","author":"Dougherty","year":"1995","journal-title":"Visual Comm. Image Representation"},{"key":"10.1016\/S0031-3203(97)00022-8_BIB22_1","article-title":"Report SOL 83-20R","author":"Murtagh","year":"1993"},{"key":"10.1016\/S0031-3203(97)00022-8_BIB22_2","article-title":"Report SOL 83-20R","author":"Murtagh","year":"1987"},{"key":"10.1016\/S0031-3203(97)00022-8_BIB23","series-title":"Mathematical Methods in Statistics","author":"Cramer","year":"1946"},{"key":"10.1016\/S0031-3203(97)00022-8_BIB24","article-title":"Shape characterization of silver halide crystals","volume":"Vol. 1657","author":"McCabe","year":"1992"},{"issue":"2","key":"10.1016\/S0031-3203(97)00022-8_BIB25","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1137\/0147028","article-title":"Error bounds for morphologically derived measurements","volume":"47","author":"Dougherty","year":"1987","journal-title":"SIAM J. Appl. Math."}],"container-title":["Pattern Recognition"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0031320397000228?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0031320397000228?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2019,4,27]],"date-time":"2019-04-27T13:25:21Z","timestamp":1556371521000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0031320397000228"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998,1]]},"references-count":26,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1998,1]]}},"alternative-id":["S0031320397000228"],"URL":"https:\/\/doi.org\/10.1016\/s0031-3203(97)00022-8","relation":{},"ISSN":["0031-3203"],"issn-type":[{"value":"0031-3203","type":"print"}],"subject":[],"published":{"date-parts":[[1998,1]]}}}