mycoreobjectrosdok_document_0000028040rosdok_document_0000028040rosdokdocumentrosdok_derivate_00002299412026-10-21T14:05:22.973Z2026-10-21T13:46:22.303ZMCRJANITOReditorKEpublished1rosdok_derivate_0000229941truetruederivate_types:fulltextVolltextfulltextwf_edit_epub wf_register_epubstate:publishedstate:publishedveröffentlichtpublishedrosdok_id000059031987274040Oau2026-10-212026-10-21T14:05:02ZrdaConverted from PICA to MODS using Pica2MODS XSLT Transformer 2.14-SNAPSHOT [SCM: "" "" ""] with mode 'DEFAULT'.WorkingpaperThealgorithmic complexity of the minimization of the number of segments in multileaf collimator field segmentationIntensity maps are nonnegative matrices describing the intensity modulation of beams in radiotherapy. In order to use a multileaf collimator in the static mode for the realization of the intensity modulation one has to determine a segmentation, i.e. a representation of an intensity map as a positive combination of special matrices corresponding to fixed positions of the multileaf collimator, called segments. We consider the problem to construct segmentations with the minimal total number of monitor units and the minimal number of segments. Neglecting machine-dependent constraints like the interleaf collision constraint and assuming that the entries of the intensity map are bounded by a constant, we prove that a segmentation with minimal number of segments under the condition that the number of monitor units is minimal, can be determined in time polynomial in the matrix dimensions. The results of our algorithm are compared with Engel's [9] heuristic for the reduction of the number of segments.ThomasKalinowski1980 -VerfasserInaut1284202510000-0002-8444-6848https://purl.uni-rostock.de/rosdok/id00005903urn:nbn:de:gbv:28-rosdok_id00005903-810.18453/rosdok_id00005903510 MathematikMathematisch-Naturwissenschaftliche Fakultätalle Rechte vorbehaltenRechteinhaber Universität Rostockfrei zugänglich (Open Access)Lizenz Metadaten: CC0en1 Online-Ressource (31 Seiten)2004University of Rostock, Institute of MathematicsRostockmonographic20042026Universitätsbibliothek RostockRostock2026Universitätsbibliothek Rostockhttps://purl.uni-rostock.de/rosdok/id00005903Thomas Kalinowskirosdok_id000058161986139832Odvhttps://purl.uni-rostock.de/rosdok/id00005816Preprints aus dem Institut für Mathematik3255797-804/10004-0001doctype:epubdoctype:epubDokumenttypDocument typedoctype:epub.workingpaperdoctype:epub.workingpaperWorkingpaperworking paperdiniPublType:workingPaper diniPublType2022:WorkingPaperinfo:eu-repo/semantics/workingPaperdocumentdiniPublType2022:WorkingPaperdiniPublType2022:WorkingPaperArbeitspapierWorking paperKDSF (Pu37)SDNB:510SDNB:510510 Mathematik510 Mathematicsinstitution:unirostockinstitution:unirostockUniversität RostockUniversity of RostockUniversität RostockUniversität RostockUni.Rostockhttp://d-nb.info/gnd/38329-6institution:unirostock.mnfinstitution:unirostock.mnfMathematisch-Naturwissenschaftliche FakultätFaculty of Mathematics and Natural SciencesUniversität Rostock. Mathematisch-Naturwissenschaftliche FakultätMathematisch-Natur-<br />wissenschaftliche FakultätUni.Rostock.Fakultaet.MNFhttp://d-nb.info/gnd/2147083-2licenseinfo:worklicenseinfo:workWerkworklicenseinfo:work.rightsreservedlicenseinfo:work.rightsreservedalle Rechte vorbehaltenall rights reserved/creativecommons/r/reserved/0.9/88x31.png[DE-28]Urheberrechtsschutz 1.0$gRights Statements$uhttp://rightsstatements.org/vocab/InC/1.0/http://rightsstatements.org/vocab/InC/1.0/http://rightsstatements.org/vocab/InC/1.0/licenseinfo:depositlicenseinfo:depositVeröffentlichungsgenehmigungpermission to storelicenseinfo:deposit.rightsunirostocklicenseinfo:deposit.rightsunirostockRechteinhaber Universität Rostockrightsholder Rostock Universityaccesscondition:openaccessaccesscondition:openaccessfrei zugänglich (Open Access)open accesshttp://purl.org/coar/access_right/c_abf2OAfreeinfo:eu-repo/semantics/openAccess[DE-28]Open Access$gControlled Vocabulary for Access Rights$uhttp://purl.org/coar/access_right/c_abf2licenseinfo:metadatalicenseinfo:metadataLizenzen für Metadatenlicenseinfo:metadata.cc0licenseinfo:metadata.cc0Lizenz Metadaten: CC0license metadata: CC0/creativecommons/p/zero/1.0/88x31.pnghttps://creativecommons.org/publicdomain/zero/1.0/rfc5646:enrfc5646:enEnglischEnglishengengThealgorithmic complexity of the minimization of the number of segments in multileaf collimator field segmentationThe algorithmic complexity of the minimization of the number of segments in multileaf collimator field segmentationgnd:128420251orcid:0000-0002-8444-6848gnd:128420251orcid:0000-0002-8444-6848rosdok_document_0000028040-d3976069e50gnd:128420251orcid:0000-0002-8444-6848Thomas KalinowskiThomas KalinowskiThomas KalinowskiThomas KalinowskiThomas KalinowskiRostockUniversity of Rostock, Institute of Mathematicsepub.workingpaperhttps://purl.uni-rostock.de/rosdok/id00005903urn:nbn:de:gbv:28-rosdok_id00005903-810.18453/rosdok_id00005903Intensity maps are nonnegative matrices describing the intensity modulation of beams in radiotherapy. In order to use a multileaf collimator in the static mode for the realization of the intensity modulation one has to determine a segmentation, i.e. a representation of an intensity map as a positive combination of special matrices corresponding to fixed positions of the multileaf collimator, called segments. We consider the problem to construct segmentations with the minimal total number of monitor units and the minimal number of segments. Neglecting machine-dependent constraints like the interleaf collision constraint and assuming that the entries of the intensity map are bounded by a constant, we prove that a segmentation with minimal number of segments under the condition that the number of monitor units is minimal, can be determined in time polynomial in the matrix dimensions. The results of our algorithm are compared with Engel's [9] heuristic for the reduction of the number of segments.20042004Preprints aus dem Institut für Mathematikrosdok_id000058161986139832Odvhttps://purl.uni-rostock.de/rosdok/id00005816Preprints aus dem Institut für Mathematik3255797-804/10004-0001https://purl.uni-rostock.de/rosdok/id000058163255797-8Thomas Kalinowskiepub.workingpaper1 Kalinowski_algorithmic_2004.pdf 226296 4593ab4f3382d694fcb0bfbb0928fe4f rosdok_id000059031987274040Oau2026-10-212026-10-21T14:05:02ZrdaConverted from PICA to MODS using Pica2MODS XSLT Transformer 2.14-SNAPSHOT [SCM: "" "" ""] with mode 'DEFAULT'.WorkingpaperThealgorithmic complexity of the minimization of the number of segments in multileaf collimator field segmentationIntensity maps are nonnegative matrices describing the intensity modulation of beams in radiotherapy. In order to use a multileaf collimator in the static mode for the realization of the intensity modulation one has to determine a segmentation, i.e. a representation of an intensity map as a positive combination of special matrices corresponding to fixed positions of the multileaf collimator, called segments. We consider the problem to construct segmentations with the minimal total number of monitor units and the minimal number of segments. Neglecting machine-dependent constraints like the interleaf collision constraint and assuming that the entries of the intensity map are bounded by a constant, we prove that a segmentation with minimal number of segments under the condition that the number of monitor units is minimal, can be determined in time polynomial in the matrix dimensions. The results of our algorithm are compared with Engel's [9] heuristic for the reduction of the number of segments.ThomasKalinowski1980 -VerfasserInaut1284202510000-0002-8444-6848https://purl.uni-rostock.de/rosdok/id00005903urn:nbn:de:gbv:28-rosdok_id00005903-810.18453/rosdok_id00005903510 MathematikMathematisch-Naturwissenschaftliche Fakultätalle Rechte vorbehaltenRechteinhaber Universität Rostockfrei zugänglich (Open Access)Lizenz Metadaten: CC0en1 Online-Ressource (31 Seiten)2004University of Rostock, Institute of MathematicsRostockmonographic20042026Universitätsbibliothek RostockRostock2026Universitätsbibliothek Rostockhttps://purl.uni-rostock.de/rosdok/id00005903Thomas Kalinowskirosdok_id000058161986139832Odvhttps://purl.uni-rostock.de/rosdok/id00005816Preprints aus dem Institut für Mathematik3255797-804/10004-0001 2026-10-21T13:46:22.303Z 2026-10-21T14:05:22.973Z 2026-10-31T14:05:22.987Z editorKE {"identifier":"rosdok/id00005903","type":"local_id","additional":"","service":"MCRLocalID","created":"2026-10-21T13:46:22.312Z"} {"identifier":"urn:nbn:de:gbv:28-rosdok_id00005903-8","type":"dnbUrn","additional":"","service":"RosDokURN","created":"2026-10-21T13:46:22.318Z"} {"identifier":"https://purl.uni-rostock.de/rosdok/id00005903","type":"purl","additional":"","service":"RosDokPURL","created":"2026-10-21T13:46:22.323Z","registrationStarted":"2026-10-21T14:05:21.615Z"} {"identifier":"10.18453/rosdok_id00005903","type":"doi","additional":"","service":"RosDokDOI","created":"2026-10-21T13:46:22.325Z","registrationStarted":"2026-10-21T14:05:21.383Z","registered":"2026-10-21T14:05:22.970Z"} MCRJANITORfulltextfile/rosdok_document_0000028040/rosdok_derivate_0000229941/Kalinowski_algorithmic_2004.pdfThealgorithmic complexity of the minimization of the number of segments in multileaf collimator field segmentationThe algorithmic complexity of the minimization of the number of segments in multileaf collimator field segmentationgnd:128420251orcid:0000-0002-8444-6848gnd:128420251orcid:0000-0002-8444-6848rosdok_document_0000028040-d3976069e50gnd:128420251orcid:0000-0002-8444-6848Thomas KalinowskiThomas KalinowskiThomas KalinowskiThomas KalinowskiThomas KalinowskiRostockUniversity of Rostock, Institute of Mathematicsepub.workingpaperhttps://purl.uni-rostock.de/rosdok/id00005903urn:nbn:de:gbv:28-rosdok_id00005903-810.18453/rosdok_id00005903Intensity maps are nonnegative matrices describing the intensity modulation of beams in radiotherapy. In order to use a multileaf collimator in the static mode for the realization of the intensity modulation one has to determine a segmentation, i.e. a representation of an intensity map as a positive combination of special matrices corresponding to fixed positions of the multileaf collimator, called segments. We consider the problem to construct segmentations with the minimal total number of monitor units and the minimal number of segments. Neglecting machine-dependent constraints like the interleaf collision constraint and assuming that the entries of the intensity map are bounded by a constant, we prove that a segmentation with minimal number of segments under the condition that the number of monitor units is minimal, can be determined in time polynomial in the matrix dimensions. The results of our algorithm are compared with Engel's [9] heuristic for the reduction of the number of segments.20042004Preprints aus dem Institut für Mathematikrosdok_id000058161986139832Odvhttps://purl.uni-rostock.de/rosdok/id00005816Preprints aus dem Institut für Mathematik3255797-804/10004-0001https://purl.uni-rostock.de/rosdok/id000058163255797-8Thomas Kalinowski[xslt]Saxonrosdok_id00005903https://purl.uni-rostock.de/rosdok/id00005903198727404010.18453/rosdok_id00005903urn:nbn:de:gbv:28-rosdok_id00005903-8Thomas KalinowskiKalinowski ThomasThe algorithmic complexity of the minimization of the number of segments in multileaf collimator field segmentationWorkingpaperworking paperUniversity of Rostock, Institute of Mathematics, 2004Intensity maps are nonnegative matrices describing the intensity modulation of beams in radiotherapy. In order to use a multileaf collimator in the static mode for the realization of the intensity modulation one has to determine a segmentation, i.e. a representation of an intensity map as a positive…rosdok_id000058160004-000104/1Thomas KalinowskiThealgorithmic complexity of the minimization of the number of segments in multileaf collimator field segmentationPreprints aus dem Institut für MathematikUniversitätsbibliothek Rostockhttps://purl.uni-rostock.de/rosdok/id00005903ThomasKalinowski1980 -VerfasserInaut1284202510000-0002-8444-6848[purl]https://purl.uni-rostock.de/rosdok/id00005903[urn]urn:nbn:de:gbv:28-rosdok_id00005903-8[doi]10.18453/rosdok_id00005903open_access20042004epoch:21th_centuryrfc5646:endoctype:epub.workingpaperaccesscondition:openaccessSDNB:510institution:unirostock.mnfstate:published