{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,4,2]],"date-time":"2023-04-02T04:05:44Z","timestamp":1680408344668},"reference-count":20,"publisher":"Walter de Gruyter GmbH","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,12,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Methods from integral geometry and geometric probability allow us to estimate\ngeometric size measures indirectly. In this article, a Monte Carlo algorithm for\nsimultaneous estimation of hyper-volumes and hyper-surface areas of a class of\ncompact sets in Euclidean space is developed. The algorithm is based on\nSantalo\u2019s formula and the Hadwiger formula from integral geometry, and employs a\ncomparison principle to assign geometric probabilities. An essential component\nof the method is to be able to generate uniform sets of random lines on the\nsphere. We utilize an empirically established method to generate these random\nchords, and we describe a geometric randomness model associated with it. We\nverify our results by computing measures for hyper-ellipsoids and certain\nnon-convex sets.<\/jats:p>","DOI":"10.1515\/mcma-2020-2071","type":"journal-article","created":{"date-parts":[[2020,8,18]],"date-time":"2020-08-18T07:03:42Z","timestamp":1597734222000},"page":"315-323","source":"Crossref","is-referenced-by-count":1,"title":["On the density of lines and Santalo\u2019s formula for computing geometric size measures"],"prefix":"10.1515","volume":"26","author":[{"given":"Khaldoun","family":"El Khaldi","sequence":"first","affiliation":[{"name":"Department of Computer Science , Notre Dame University-Louaize , Zouk Mosbeh , Lebanon"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Elias G.","family":"Saleeby","sequence":"additional","affiliation":[{"name":"Mount Lebanon , Lebanon"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2020,8,5]]},"reference":[{"key":"2023040102281281984_j_mcma-2020-2071_ref_001_w2aab3b7d321b1b6b1ab2b1b1Aa","doi-asserted-by":"crossref","unstructured":"S.  Ahmed and E. G.  Saleeby,\nOn volumes of hyper-ellipsoids,\nMath. Mag. 91 (2018), no. 1, 43\u201350.","DOI":"10.1080\/0025570X.2018.1404834"},{"key":"2023040102281281984_j_mcma-2020-2071_ref_002_w2aab3b7d321b1b6b1ab2b1b2Aa","doi-asserted-by":"crossref","unstructured":"A. L. D.  Beckers and A. W. M.  Smeulders,\nThe probability of a random straight line in two and three dimensions,\nPattern Recognit. Let. 11 (1990), no. 4, 233\u2013240.","DOI":"10.1016\/0167-8655(90)90061-6"},{"key":"2023040102281281984_j_mcma-2020-2071_ref_003_w2aab3b7d321b1b6b1ab2b1b3Aa","unstructured":"D.  Berengut,\nRandom chords of a sphere,\nTechnical report, Stanford University, 1972."},{"key":"2023040102281281984_j_mcma-2020-2071_ref_004_w2aab3b7d321b1b6b1ab2b1b4Aa","unstructured":"J.  Edwards,\nA Treatise on the Integral Calculus: With Applications, Examples and Problems. Volume 2,\nMacmillan, London, 1922."},{"key":"2023040102281281984_j_mcma-2020-2071_ref_005_w2aab3b7d321b1b6b1ab2b1b5Aa","doi-asserted-by":"crossref","unstructured":"K.  El Khaldi and E. G.  Saleeby,\nOn the tangent model for the density of lines and a Monte Carlo method for computing hypersurface area,\nMonte Carlo Methods Appl. 23 (2017), no. 1, 13\u201320.","DOI":"10.1515\/mcma-2017-0100"},{"key":"2023040102281281984_j_mcma-2020-2071_ref_006_w2aab3b7d321b1b6b1ab2b1b6Aa","doi-asserted-by":"crossref","unstructured":"D. S. K.  Fok and D.  Crevier,\nVolume estimation by Monte Carlo methods,\nJ. Statist. Comput. Simul. 31 (1989), no. 4, 223\u2013235.","DOI":"10.1080\/00949658908811145"},{"key":"2023040102281281984_j_mcma-2020-2071_ref_007_w2aab3b7d321b1b6b1ab2b1b7Aa","unstructured":"J.  Hopcroft and R.  Kannan,\nFoundations of data science, 2014, https:\/\/www.cs.cornell.edu\/jeh\/book.pdf."},{"key":"2023040102281281984_j_mcma-2020-2071_ref_008_w2aab3b7d321b1b6b1ab2b1b8Aa","doi-asserted-by":"crossref","unstructured":"M.  Hyk\u0161ov\u00e1, A.  Kalousov\u00e1 and I.  Saxl,\nEarly history of geometric probability and stereology,\nImage Anal. Stereol. 31 (2012), no. 1, 1\u201316.","DOI":"10.5566\/ias.v31.p1-16"},{"key":"2023040102281281984_j_mcma-2020-2071_ref_009_w2aab3b7d321b1b6b1ab2b1b9Aa","doi-asserted-by":"crossref","unstructured":"M.  Kiderlen and D.  Meschenmoser,\nError bounds for surface area estimators based on Crofton\u2019s formula,\nImage Anal. Stereol. 28 (2009), no. 3, 165\u2013177.","DOI":"10.5566\/ias.v28.p165-177"},{"key":"2023040102281281984_j_mcma-2020-2071_ref_010_w2aab3b7d321b1b6b1ab2b1c10Aa","doi-asserted-by":"crossref","unstructured":"J. F. C.  Kingman,\nRandom secants of a convex body,\nJ. Appl. Probability 6 (1969), 660\u2013672.","DOI":"10.1017\/S0021900200026693"},{"key":"2023040102281281984_j_mcma-2020-2071_ref_011_w2aab3b7d321b1b6b1ab2b1c11Aa","doi-asserted-by":"crossref","unstructured":"X.  Li, W.  Wang, R. R.  Martin and A.  Bowyer,\nUsing low-discrepancy sequences and the crofton formula to compute surface areas of geometric models,\nComput.-Aided Design 35 (2003), no. 9, 771\u2013782.","DOI":"10.1016\/S0010-4485(02)00100-8"},{"key":"2023040102281281984_j_mcma-2020-2071_ref_012_w2aab3b7d321b1b6b1ab2b1c12Aa","doi-asserted-by":"crossref","unstructured":"P.  Mattila,\nGeometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability,\nCambridge Stud. Adv. Math. 44,\nCambridge University, Cambridge, 1995.","DOI":"10.1017\/CBO9780511623813"},{"key":"2023040102281281984_j_mcma-2020-2071_ref_013_w2aab3b7d321b1b6b1ab2b1c13Aa","doi-asserted-by":"crossref","unstructured":"M. E.  Muller,\nA note on a method for generating points uniformly on n-dimensional spheres,\nCommun. ACM 2 (1959), no. 4, 19\u201320.","DOI":"10.1145\/377939.377946"},{"key":"2023040102281281984_j_mcma-2020-2071_ref_014_w2aab3b7d321b1b6b1ab2b1c14Aa","doi-asserted-by":"crossref","unstructured":"N.  Rabiei and E. G.  Saleeby,\nOn the sample-mean method for computing hyper-volumes,\nMonte Carlo Methods Appl. 25 (2019), no. 2, 163\u2013176.","DOI":"10.1515\/mcma-2019-2034"},{"key":"2023040102281281984_j_mcma-2020-2071_ref_015_w2aab3b7d321b1b6b1ab2b1c15Aa","unstructured":"D. L.  Ren,\nTopics in Integral Geometry,\nSer. Pure Math. 19,\nWorld Scientific, River Edge, 1994."},{"key":"2023040102281281984_j_mcma-2020-2071_ref_016_w2aab3b7d321b1b6b1ab2b1c16Aa","doi-asserted-by":"crossref","unstructured":"L. A.  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