1006:
55:
473:
0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 1 1 1 1 1 0 0 0
490:
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 0 0
1115:
1009:
Example of dilation on a grayscale image using a 5x5 flat structuring element. The top figure demonstrates the application of the structuring element window to the individual pixels of the original image. The bottom figure shows the resulting dilated
305:. The dilation of a square of size 10, centered at the origin, by a disk of radius 2, also centered at the origin, is a square of side 14, with rounded corners, centered at the origin. The radius of the rounded corners is 2.
1727:
1357:
388:
276:
699:
780:
1766:
184:
986:
576:
460:
1030:
1639:
630:
1567:
1146:
802:
538:
1600:
1503:
849:
1523:
826:
1543:
1647:
1159:
311:
214:
642:
742:
1733:
131:
1014:
It is common to use flat structuring elements in morphological applications. Flat structuring functions are functions
1850:
1836:
1822:
889:
1866:
1110:{\displaystyle b(x)=\left\{{\begin{array}{ll}0,&x\in B,\\-\infty ,&{\text{otherwise}},\end{array}}\right.}
543:
405:
1435:
filters, returning the maximum value within a moving window (the symmetric of the structuring function support
1612:
603:
1641:
that distributes over the supremum, and preserves the least element. That is, the following are true:
854:
Grayscale structuring elements are also functions of the same format, called "structuring functions".
58:
The dilation of a dark-blue square by a disk, resulting in the light-blue square with rounded corners.
1871:
504:
63:
1552:
1125:
785:
517:
1792:
27:
1572:
1476:
725:
831:
1508:
1451:
811:
705:
1797:
1787:
1782:
1528:
8:
395:
43:
1802:
67:
1846:
1832:
1818:
39:
1467:
1447:
1432:
1005:
597:
1722:{\displaystyle \bigvee _{i}\delta (X_{i})=\delta \left(\bigvee _{i}X_{i}\right),}
729:
78:
470:
Suppose A is the following 11 x 11 matrix and B is the following 3 x 3 matrix:
1054:
1860:
1463:
733:
31:
1352:{\displaystyle (f\oplus b)(x)=\sup _{y\in E}=\sup _{z\in E}=\sup _{z\in B}.}
1777:
805:
383:{\displaystyle A\oplus B=\{z\in E\mid (B^{s})_{z}\cap A\neq \varnothing \}}
1829:
Image
Analysis and Mathematical Morphology, Volume 2: Theoretical Advances
484:
Each pixel of every superimposed B is included in the dilation of A by B.
636:
582:
511:
721:
709:
35:
1459:
1424:
994:
481:
B, with the center of B aligned with the corresponding pixel in A.
46:
for probing and expanding the shapes contained in the input image.
1545:, respectively. Its universe and least element are symbolized by
1455:
1428:
1505:
be a complete lattice, with infimum and supremum symbolized by
1151:
In this case, the dilation is greatly simplified, and given by
74:
1104:
54:
271:{\displaystyle A\oplus B=B\oplus A=\bigcup _{a\in A}B_{a}}
499:
Here are some properties of the binary dilation operator
73:
A binary image is viewed in mathematical morphology as a
487:
The dilation of A by B is given by this 11 x 11 matrix.
293:
can be understood as the locus of the points covered by
694:{\displaystyle (A\oplus B)\oplus C=A\oplus (B\oplus C)}
474:0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1736:
1650:
1615:
1575:
1555:
1531:
1511:
1479:
1162:
1128:
1033:
892:
834:
814:
788:
775:{\displaystyle \mathbb {R} \cup \{\infty ,-\infty \}}
745:
645:
606:
546:
520:
408:
314:
217:
134:
62:
In binary morphology, dilation is a shift-invariant (
1761:{\displaystyle \delta (\varnothing )=\varnothing .}
1760:
1721:
1633:
1594:
1561:
1537:
1517:
1497:
1351:
1140:
1109:
980:
843:
820:
796:
774:
693:
624:
570:
532:
454:
382:
270:
178:
1843:An Introduction to Morphological Image Processing
285:has a center on the origin, then the dilation of
179:{\displaystyle A\oplus B=\bigcup _{b\in B}A_{b},}
1858:
1442:
1307:
1249:
1191:
921:
828:is an element greater than any real number, and
494:
981:{\displaystyle (f\oplus b)(x)=\sup _{y\in E},}
108:a structuring element regarded as a subset of
1000:
1589:
1576:
769:
754:
571:{\displaystyle A\oplus B\subseteq C\oplus B}
455:{\displaystyle B^{s}=\{x\in E\mid -x\in B\}}
449:
422:
377:
327:
477:For each pixel in A that has a value of 1,
1815:Image Analysis and Mathematical Morphology
1431:. Thus, dilation is a particular case of
851:is an element less than any real number.
790:
747:
96:be a Euclidean space or an integer grid,
1004:
53:
42:. The dilation operation usually uses a
211:Dilation is commutative, also given by
1859:
1634:{\displaystyle \delta :L\rightarrow L}
715:
308:The dilation can also be obtained by
625:{\displaystyle A\subseteq A\oplus B}
26:) is one of the basic operations in
16:Operation in mathematical morphology
592:belongs to the structuring element
13:
1084:
865:) and the structuring function by
838:
815:
766:
757:
49:
14:
1883:
1752:
1743:
1602:be a collection of elements from
374:
1427:operator can be replaced by the
34:, it has been expanded first to
1808:
1462:. In particular, it contains a
1415:In the bounded, discrete case (
1746:
1740:
1677:
1664:
1625:
1569:, respectively. Moreover, let
1492:
1480:
1343:
1340:
1328:
1322:
1300:
1297:
1291:
1282:
1270:
1264:
1242:
1239:
1227:
1218:
1212:
1206:
1184:
1178:
1175:
1163:
1043:
1037:
972:
969:
957:
948:
942:
936:
914:
908:
905:
893:
688:
676:
658:
646:
356:
342:
1:
1443:Dilation on complete lattices
873:), the grayscale dilation of
495:Properties of binary dilation
1562:{\displaystyle \varnothing }
1454:, where every subset has an
1141:{\displaystyle B\subseteq E}
797:{\displaystyle \mathbb {R} }
533:{\displaystyle A\subseteq C}
7:
1771:
1609:A dilation is any operator
1470:(also denoted "universe").
30:. Originally developed for
10:
1888:
1001:Flat structuring functions
465:
66:) operator, equivalent to
1595:{\displaystyle \{X_{i}\}}
1498:{\displaystyle (L,\leq )}
1845:by Edward R. Dougherty,
993:where "sup" denotes the
844:{\displaystyle -\infty }
22:(usually represented by
1867:Mathematical morphology
1793:Mathematical morphology
1518:{\displaystyle \wedge }
821:{\displaystyle \infty }
724:morphology, images are
28:mathematical morphology
1762:
1723:
1635:
1596:
1563:
1539:
1519:
1499:
1452:partially ordered sets
1353:
1142:
1111:
1011:
982:
845:
822:
798:
776:
695:
626:
572:
534:
456:
384:
272:
200:is the translation of
180:
59:
1763:
1724:
1636:
1597:
1564:
1540:
1538:{\displaystyle \vee }
1520:
1500:
1354:
1143:
1112:
1008:
983:
857:Denoting an image by
846:
823:
799:
777:
696:
627:
573:
535:
505:translation invariant
457:
385:
273:
181:
88:, for some dimension
64:translation invariant
57:
1798:Opening (morphology)
1788:Erosion (morphology)
1783:Closing (morphology)
1734:
1648:
1613:
1573:
1553:
1529:
1509:
1477:
1160:
1126:
1031:
890:
832:
812:
786:
743:
643:
604:
544:
518:
406:
312:
215:
132:
84:or the integer grid
38:images, and then to
297:when the center of
44:structuring element
1803:Minkowski addition
1758:
1719:
1700:
1660:
1631:
1592:
1559:
1535:
1515:
1495:
1349:
1321:
1263:
1205:
1138:
1107:
1102:
1012:
978:
935:
841:
818:
794:
772:
716:Grayscale dilation
691:
622:
568:
530:
452:
380:
268:
257:
176:
162:
100:a binary image in
68:Minkowski addition
60:
1691:
1651:
1448:Complete lattices
1423:is bounded), the
1306:
1248:
1190:
1095:
920:
588:If the origin of
242:
147:
40:complete lattices
1879:
1872:Digital geometry
1767:
1765:
1764:
1759:
1728:
1726:
1725:
1720:
1715:
1711:
1710:
1709:
1699:
1676:
1675:
1659:
1640:
1638:
1637:
1632:
1601:
1599:
1598:
1593:
1588:
1587:
1568:
1566:
1565:
1560:
1544:
1542:
1541:
1536:
1524:
1522:
1521:
1516:
1504:
1502:
1501:
1496:
1468:greatest element
1433:order statistics
1358:
1356:
1355:
1350:
1320:
1262:
1204:
1147:
1145:
1144:
1139:
1116:
1114:
1113:
1108:
1106:
1103:
1096:
1093:
987:
985:
984:
979:
934:
850:
848:
847:
842:
827:
825:
824:
819:
803:
801:
800:
795:
793:
781:
779:
778:
773:
750:
700:
698:
697:
692:
631:
629:
628:
623:
577:
575:
574:
569:
539:
537:
536:
531:
461:
459:
458:
453:
418:
417:
389:
387:
386:
381:
364:
363:
354:
353:
277:
275:
274:
269:
267:
266:
256:
185:
183:
182:
177:
172:
171:
161:
115:The dilation of
1887:
1886:
1882:
1881:
1880:
1878:
1877:
1876:
1857:
1856:
1831:by Jean Serra,
1817:by Jean Serra,
1811:
1774:
1735:
1732:
1731:
1705:
1701:
1695:
1690:
1686:
1671:
1667:
1655:
1649:
1646:
1645:
1614:
1611:
1610:
1583:
1579:
1574:
1571:
1570:
1554:
1551:
1550:
1530:
1527:
1526:
1510:
1507:
1506:
1478:
1475:
1474:
1445:
1310:
1252:
1194:
1161:
1158:
1157:
1127:
1124:
1123:
1101:
1100:
1092:
1090:
1078:
1077:
1063:
1053:
1049:
1032:
1029:
1028:
1003:
924:
891:
888:
887:
833:
830:
829:
813:
810:
809:
789:
787:
784:
783:
746:
744:
741:
740:
730:Euclidean space
718:
644:
641:
640:
605:
602:
601:
545:
542:
541:
519:
516:
515:
497:
492:
475:
468:
413:
409:
407:
404:
403:
359:
355:
349:
345:
313:
310:
309:
262:
258:
246:
216:
213:
212:
199:
167:
163:
151:
133:
130:
129:
79:Euclidean space
52:
50:Binary dilation
17:
12:
11:
5:
1885:
1875:
1874:
1869:
1855:
1854:
1840:
1826:
1810:
1807:
1806:
1805:
1800:
1795:
1790:
1785:
1780:
1773:
1770:
1769:
1768:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1729:
1718:
1714:
1708:
1704:
1698:
1694:
1689:
1685:
1682:
1679:
1674:
1670:
1666:
1663:
1658:
1654:
1630:
1627:
1624:
1621:
1618:
1591:
1586:
1582:
1578:
1558:
1534:
1514:
1494:
1491:
1488:
1485:
1482:
1444:
1441:
1419:is a grid and
1396: = (
1380: = (
1368: = (
1362:
1361:
1360:
1359:
1348:
1345:
1342:
1339:
1336:
1333:
1330:
1327:
1324:
1319:
1316:
1313:
1309:
1305:
1302:
1299:
1296:
1293:
1290:
1287:
1284:
1281:
1278:
1275:
1272:
1269:
1266:
1261:
1258:
1255:
1251:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1203:
1200:
1197:
1193:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1165:
1137:
1134:
1131:
1120:
1119:
1118:
1117:
1105:
1099:
1091:
1089:
1086:
1083:
1080:
1079:
1076:
1073:
1070:
1067:
1064:
1062:
1059:
1056:
1055:
1052:
1048:
1045:
1042:
1039:
1036:
1022:) in the form
1002:
999:
991:
990:
989:
988:
977:
974:
971:
968:
965:
962:
959:
956:
953:
950:
947:
944:
941:
938:
933:
930:
927:
923:
919:
916:
913:
910:
907:
904:
901:
898:
895:
840:
837:
817:
804:is the set of
792:
771:
768:
765:
762:
759:
756:
753:
749:
717:
714:
713:
712:
702:
690:
687:
684:
681:
678:
675:
672:
669:
666:
663:
660:
657:
654:
651:
648:
633:
621:
618:
615:
612:
609:
586:
579:
567:
564:
561:
558:
555:
552:
549:
529:
526:
523:
514:, that is, if
508:
496:
493:
489:
472:
467:
464:
451:
448:
445:
442:
439:
436:
433:
430:
427:
424:
421:
416:
412:
379:
376:
373:
370:
367:
362:
358:
352:
348:
344:
341:
338:
335:
332:
329:
326:
323:
320:
317:
265:
261:
255:
252:
249:
245:
241:
238:
235:
232:
229:
226:
223:
220:
195:
189:
188:
187:
186:
175:
170:
166:
160:
157:
154:
150:
146:
143:
140:
137:
123:is defined by
51:
48:
15:
9:
6:
4:
3:
2:
1884:
1873:
1870:
1868:
1865:
1864:
1862:
1852:
1851:0-8194-0845-X
1848:
1844:
1841:
1838:
1837:0-12-637241-1
1834:
1830:
1827:
1824:
1823:0-12-637240-3
1820:
1816:
1813:
1812:
1804:
1801:
1799:
1796:
1794:
1791:
1789:
1786:
1784:
1781:
1779:
1776:
1775:
1755:
1749:
1737:
1730:
1716:
1712:
1706:
1702:
1696:
1692:
1687:
1683:
1680:
1672:
1668:
1661:
1656:
1652:
1644:
1643:
1642:
1628:
1622:
1619:
1616:
1607:
1605:
1584:
1580:
1556:
1548:
1532:
1512:
1489:
1486:
1483:
1471:
1469:
1465:
1464:least element
1461:
1457:
1453:
1449:
1440:
1438:
1434:
1430:
1426:
1422:
1418:
1413:
1411:
1408: −
1407:
1403:
1400: −
1399:
1395:
1392: −
1391:
1387:
1383:
1379:
1375:
1371:
1367:
1346:
1337:
1334:
1331:
1325:
1317:
1314:
1311:
1303:
1294:
1288:
1285:
1279:
1276:
1273:
1267:
1259:
1256:
1253:
1245:
1236:
1233:
1230:
1224:
1221:
1215:
1209:
1201:
1198:
1195:
1187:
1181:
1172:
1169:
1166:
1156:
1155:
1154:
1153:
1152:
1149:
1135:
1132:
1129:
1097:
1087:
1081:
1074:
1071:
1068:
1065:
1060:
1057:
1050:
1046:
1040:
1034:
1027:
1026:
1025:
1024:
1023:
1021:
1017:
1007:
998:
996:
975:
966:
963:
960:
954:
951:
945:
939:
931:
928:
925:
917:
911:
902:
899:
896:
886:
885:
884:
883:
882:
880:
876:
872:
868:
864:
860:
855:
852:
835:
807:
763:
760:
751:
738:
735:
731:
727:
723:
711:
707:
703:
685:
682:
679:
673:
670:
667:
664:
661:
655:
652:
649:
638:
634:
619:
616:
613:
610:
607:
599:
596:, then it is
595:
591:
587:
584:
580:
565:
562:
559:
556:
553:
550:
547:
527:
524:
521:
513:
509:
506:
502:
501:
500:
488:
485:
482:
480:
471:
463:
446:
443:
440:
437:
434:
431:
428:
425:
419:
414:
410:
401:
397:
393:
371:
368:
365:
360:
350:
346:
339:
336:
333:
330:
324:
321:
318:
315:
306:
304:
301:moves inside
300:
296:
292:
288:
284:
279:
263:
259:
253:
250:
247:
243:
239:
236:
233:
230:
227:
224:
221:
218:
209:
207:
203:
198:
194:
173:
168:
164:
158:
155:
152:
148:
144:
141:
138:
135:
128:
127:
126:
125:
124:
122:
118:
113:
111:
107:
103:
99:
95:
91:
87:
83:
80:
76:
71:
69:
65:
56:
47:
45:
41:
37:
33:
32:binary images
29:
25:
21:
1842:
1828:
1814:
1809:Bibliography
1778:Buffer (GIS)
1608:
1603:
1546:
1472:
1446:
1436:
1420:
1416:
1414:
1409:
1405:
1401:
1397:
1393:
1389:
1385:
1381:
1377:
1373:
1369:
1365:
1363:
1150:
1121:
1019:
1015:
1013:
992:
881:is given by
878:
874:
870:
866:
862:
858:
856:
853:
736:
719:
706:distributive
593:
589:
498:
486:
483:
478:
476:
469:
399:
394:denotes the
391:
307:
302:
298:
294:
290:
286:
282:
280:
210:
205:
201:
196:
192:
190:
120:
116:
114:
109:
105:
101:
97:
93:
89:
85:
81:
72:
61:
23:
19:
18:
637:associative
583:commutative
479:superimpose
402:, that is,
1861:Categories
728:mapping a
512:increasing
1753:∅
1744:∅
1738:δ
1693:⋁
1684:δ
1662:δ
1653:⋁
1626:→
1617:δ
1557:∅
1533:∨
1513:∧
1490:≤
1364:(Suppose
1335:−
1315:∈
1277:−
1257:∈
1234:−
1199:∈
1170:⊕
1133:⊆
1094:otherwise
1085:∞
1082:−
1069:∈
964:−
929:∈
900:⊕
839:∞
836:−
816:∞
767:∞
764:−
758:∞
752:∪
726:functions
722:grayscale
710:set union
683:⊕
674:⊕
662:⊕
653:⊕
617:⊕
611:⊆
598:extensive
563:⊕
557:⊆
551:⊕
525:⊆
444:∈
438:−
435:∣
429:∈
396:symmetric
375:∅
372:≠
366:∩
340:∣
334:∈
319:⊕
251:∈
244:⋃
234:⊕
222:⊕
156:∈
149:⋃
139:⊕
36:grayscale
1772:See also
1460:supremum
1425:supremum
1388:), then
995:supremum
782:, where
639:, i.e.,
600:, i.e.,
390:, where
20:Dilation
1456:infimum
1429:maximum
1404:,
1384:,
1372:,
540:, then
466:Example
1853:(1992)
1849:
1839:(1988)
1835:
1825:(1982)
1821:
1466:and a
1458:and a
1122:where
1010:image.
704:It is
635:It is
581:It is
510:It is
503:It is
191:where
104:, and
92:. Let
75:subset
806:reals
739:into
708:over
77:of a
1847:ISBN
1833:ISBN
1819:ISBN
1549:and
1525:and
1473:Let
1450:are
1412:).)
734:grid
1439:).
1376:),
1308:sup
1250:sup
1192:sup
997:.
922:sup
877:by
732:or
720:In
398:of
289:by
281:If
204:by
119:by
70:.
1863::
1606:.
1410:qz
1406:qx
1402:pz
1398:px
1386:qz
1382:pz
1374:qx
1370:px
1148:.
808:,
462:.
278:.
208:.
112:.
1756:.
1750:=
1747:)
1741:(
1717:,
1713:)
1707:i
1703:X
1697:i
1688:(
1681:=
1678:)
1673:i
1669:X
1665:(
1657:i
1629:L
1623:L
1620::
1604:L
1590:}
1585:i
1581:X
1577:{
1547:U
1493:)
1487:,
1484:L
1481:(
1437:B
1421:B
1417:E
1394:z
1390:x
1378:z
1366:x
1347:.
1344:]
1341:)
1338:z
1332:x
1329:(
1326:f
1323:[
1318:B
1312:z
1304:=
1301:]
1298:)
1295:z
1292:(
1289:b
1286:+
1283:)
1280:z
1274:x
1271:(
1268:f
1265:[
1260:E
1254:z
1246:=
1243:]
1240:)
1237:y
1231:x
1228:(
1225:b
1222:+
1219:)
1216:y
1213:(
1210:f
1207:[
1202:E
1196:y
1188:=
1185:)
1182:x
1179:(
1176:)
1173:b
1167:f
1164:(
1136:E
1130:B
1098:,
1088:,
1075:,
1072:B
1066:x
1061:,
1058:0
1051:{
1047:=
1044:)
1041:x
1038:(
1035:b
1020:x
1018:(
1016:b
976:,
973:]
970:)
967:y
961:x
958:(
955:b
952:+
949:)
946:y
943:(
940:f
937:[
932:E
926:y
918:=
915:)
912:x
909:(
906:)
903:b
897:f
894:(
879:b
875:f
871:x
869:(
867:b
863:x
861:(
859:f
791:R
770:}
761:,
755:{
748:R
737:E
701:.
689:)
686:C
680:B
677:(
671:A
668:=
665:C
659:)
656:B
650:A
647:(
632:.
620:B
614:A
608:A
594:B
590:E
585:.
578:.
566:B
560:C
554:B
548:A
528:C
522:A
507:.
450:}
447:B
441:x
432:E
426:x
423:{
420:=
415:s
411:B
400:B
392:B
378:}
369:A
361:z
357:)
351:s
347:B
343:(
337:E
331:z
328:{
325:=
322:B
316:A
303:A
299:B
295:B
291:B
287:A
283:B
264:a
260:B
254:A
248:a
240:=
237:A
231:B
228:=
225:B
219:A
206:b
202:A
197:b
193:A
174:,
169:b
165:A
159:B
153:b
145:=
142:B
136:A
121:B
117:A
110:R
106:B
102:E
98:A
94:E
90:d
86:Z
82:R
24:⊕
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
