1053:
836:
1074:
1042:
1111:
1084:
1064:
107:
proposed digital connectivity such as 4-connectivity and 8-connectivity in two dimensions as well as 6-connectivity and 26-connectivity in three dimensions. The labeling method for inferring a connected component was studied in the 1970s. Theodosios
Pavlidis (1982) suggested the use of
84:(1931–2004), whose publications on the subject played a major role in establishing and developing the field. The term "digital topology" was itself invented by Rosenfeld, who used it in a 1973 publication for the first time.
168:. Grid cell topology also applies to multilevel (e.g., color) 2D or 3D images, for example based on a total order of possible image values and applying a 'maximum-label rule' (see the book by Klette and Rosenfeld, 2004).
139:
were studied. David
Morgenthaler and Rosenfeld (1981) gave a mathematical definition of surfaces in three-dimensional digital space. This definition contains a total of nine types of digital surfaces. The
144:
was studied in the 1990s. A recursive definition of the digital k-manifold was proposed intuitively by Chen and Zhang in 1993. Many applications were found in image processing and computer vision.
296:
175:. The main differences between them are: (1) digital topology mainly studies digital objects that are formed by grid cells (the cells of integer lattices), rather than more general
423:
567:
327:
361:
1114:
132:. The book from 2008 contains new definitions of topological balls and spheres independent of a metric and numerous applications to digital image analysis.
116:(1989) extended the Alexandrov–Hopf 2D grid cell topology to three and higher dimensions. He also proposed (2008) a more general axiomatic theory of
717:
499:
160:) to ensure the basic topological duality of separation and connectedness. This alternative use corresponds to open or closed sets in the 2D
748:
623:
1102:
1097:
687:
633:
612:
585:
526:
1092:
704:
113:
164:, and the result generalizes to 3D: the alternative use of 6- or 26-adjacency corresponds to open or closed sets in the 3D
152:
A basic (early) result in digital topology says that 2D binary images require the alternative use of 4- or 8-adjacency or "
994:
645:
1002:
1135:
458:
187:
117:
219:
202:
1073:
801:
1087:
121:
77:
38:
1022:
1017:
943:
820:
808:
781:
741:
453:
448:
443:
366:
172:
92:
864:
791:
65:
1052:
543:
1012:
964:
938:
786:
468:
426:
191:
198:
is a special kind of combinatorial manifold which is defined in digital space i.e. grid cell space.
1063:
859:
473:
129:
54:
643:
Morgenthaler, David G.; Rosenfeld, Azriel (1981). "Surfaces in three-dimensional digital images".
1057:
1007:
928:
918:
796:
776:
165:
161:
153:
109:
88:
1027:
60:
Concepts and results of digital topology are used to specify and justify important (low-level)
1045:
911:
869:
734:
713:
683:
629:
608:
581:
522:
495:
771:
654:
573:
438:
195:
141:
96:
81:
697:
666:
595:
509:
305:
884:
879:
693:
662:
591:
505:
136:
34:
340:
974:
906:
676:
538:
488:
125:
61:
42:
27:
Properties of 2D or 3D digital images that correspond to classic topological properties
682:. Lecture Notes in Mathematics. Vol. 877. Rockville, MD: Computer Science Press.
658:
17:
1129:
984:
894:
874:
50:
1077:
969:
889:
835:
605:
Discrete
Surfaces and Manifolds: A Theory of Digital-Discrete Geometry and Topology
176:
68:, border or surface tracing, counting of components or tunnels, or region-filling.
186:
is a kind of manifold which is a discretization of a manifold. It usually means a
1067:
979:
494:. Applied and Numerical Harmonic Analysis. Boston, MA: Birkhäuser Boston, Inc.
923:
854:
813:
577:
213:
in direct adjacency (i.e., a (6,26)-surface in 3D). The formula for genus is
100:
948:
933:
901:
850:
757:
463:
210:
46:
572:. Algorithms and Combinatorics. Vol. 11. Berlin: Springer-Verlag.
157:
179:, and (2) digital topology also deals with non-Jordan manifolds.
730:
333:
adjacent points on the surface (Chen and Rong, ICPR 2008). If
76:
Digital topology was first studied in the late 1960s by the
726:
712:. Berlin: Publishing House Dr. Baerbel Kovalevski. 2008.
329:
indicates the set of surface-points each of which has
546:
369:
343:
308:
222:
993:
957:
843:
764:
519:
Topological
Algorithms for Digital Image Processing
675:
561:
537:
487:
417:
355:
321:
290:
91:, which could be considered as a link to classic
517:Kong, Tat Yung; Rosenfeld, Azriel, eds. (1996).
742:
8:
678:Algorithms for graphics and image processing
540:Discrete Images, Objects, and Functions in
1110:
1083:
749:
735:
727:
291:{\displaystyle g=1+(M_{5}+2M_{6}-M_{3})/8}
553:
549:
548:
545:
409:
393:
374:
368:
342:
313:
307:
280:
271:
258:
242:
221:
112:method for finding connected components.
108:graph-theoretic algorithms such as the
622:Klette, R.; Rosenfeld, Azriel (2004).
171:Digital topology is highly related to
64:algorithms, including algorithms for
33:deals with properties and features of
7:
418:{\displaystyle M_{3}=8+M_{5}+2M_{6}}
25:
710:Geometry of Locally Finite Spaces
118:locally finite topological spaces
103:, Topologie I (1935). Rosenfeld
53:) or topological features (e.g.,
1109:
1082:
1072:
1062:
1051:
1041:
1040:
834:
562:{\displaystyle \mathbb {Z} ^{n}}
156:" (for "object" or "non-object"
277:
235:
1:
659:10.1016/S0019-9958(81)90290-4
337:is simply connected, i.e.,
1152:
1003:Banach fixed-point theorem
490:Geometry of Digital Spaces
95:, appeared in the book of
87:A related work called the
1036:
832:
578:10.1007/978-3-642-46779-0
486:Herman, Gabor T. (1998).
459:Topological data analysis
188:piecewise linear manifold
705:Vladimir A. Kovalevsky.
674:Pavlidis, Theo (1982).
646:Information and Control
209:be a closed digital 2D
122:abstract cell complexes
78:computer image analysis
1058:Mathematics portal
958:Metrics and properties
944:Second-countable space
563:
454:Computational topology
449:Computational geometry
444:Combinatorial topology
419:
357:
323:
292:
201:A digital form of the
184:combinatorial manifold
173:combinatorial topology
124:formerly suggested by
114:Vladimir A. Kovalevsky
93:combinatorial topology
18:Combinatorial manifold
564:
420:
358:
324:
322:{\displaystyle M_{i}}
293:
1013:Invariance of domain
965:Euler characteristic
939:Bundle (mathematics)
544:
536:Voss, Klaus (1993).
469:Discrete mathematics
427:Euler characteristic
367:
341:
306:
220:
203:Gauss–Bonnet theorem
192:simplicial complexes
135:In the early 1980s,
1023:Tychonoff's theorem
1018:Poincaré conjecture
772:General (point-set)
628:. Morgan Kaufmann.
474:Geospatial topology
356:{\displaystyle g=0}
130:Alexandrov topology
45:that correspond to
1008:De Rham cohomology
929:Polyhedral complex
919:Simplicial complex
559:
415:
353:
319:
288:
166:grid cell topology
162:grid cell topology
154:pixel connectivity
128:(1908). It is the
110:depth-first search
89:grid cell topology
49:properties (e.g.,
1123:
1122:
912:fundamental group
719:978-3-9812252-0-4
603:Chen, L. (2004).
501:978-0-8176-3897-9
39:three-dimensional
16:(Redirected from
1143:
1136:Digital topology
1113:
1112:
1086:
1085:
1076:
1066:
1056:
1055:
1044:
1043:
838:
751:
744:
737:
728:
723:
701:
681:
670:
639:
625:Digital Geometry
618:
607:. SP Computing.
599:
571:
568:
566:
565:
560:
558:
557:
552:
532:
513:
493:
439:Digital geometry
424:
422:
421:
416:
414:
413:
398:
397:
379:
378:
362:
360:
359:
354:
328:
326:
325:
320:
318:
317:
297:
295:
294:
289:
284:
276:
275:
263:
262:
247:
246:
196:digital manifold
142:digital manifold
137:digital surfaces
97:Pavel Alexandrov
82:Azriel Rosenfeld
31:Digital topology
21:
1151:
1150:
1146:
1145:
1144:
1142:
1141:
1140:
1126:
1125:
1124:
1119:
1050:
1032:
1028:Urysohn's lemma
989:
953:
839:
830:
802:low-dimensional
760:
755:
720:
708:
690:
673:
642:
636:
621:
615:
602:
588:
547:
542:
541:
535:
529:
516:
502:
485:
482:
435:
405:
389:
370:
365:
364:
339:
338:
309:
304:
303:
267:
254:
238:
218:
217:
150:
74:
35:two-dimensional
28:
23:
22:
15:
12:
11:
5:
1149:
1147:
1139:
1138:
1128:
1127:
1121:
1120:
1118:
1117:
1107:
1106:
1105:
1100:
1095:
1080:
1070:
1060:
1048:
1037:
1034:
1033:
1031:
1030:
1025:
1020:
1015:
1010:
1005:
999:
997:
991:
990:
988:
987:
982:
977:
975:Winding number
972:
967:
961:
959:
955:
954:
952:
951:
946:
941:
936:
931:
926:
921:
916:
915:
914:
909:
907:homotopy group
899:
898:
897:
892:
887:
882:
877:
867:
862:
857:
847:
845:
841:
840:
833:
831:
829:
828:
823:
818:
817:
816:
806:
805:
804:
794:
789:
784:
779:
774:
768:
766:
762:
761:
756:
754:
753:
746:
739:
731:
725:
724:
718:
702:
688:
671:
653:(3): 227–247.
640:
634:
619:
613:
600:
586:
556:
551:
533:
527:
514:
500:
481:
478:
477:
476:
471:
466:
461:
456:
451:
446:
441:
434:
431:
412:
408:
404:
401:
396:
392:
388:
385:
382:
377:
373:
352:
349:
346:
316:
312:
300:
299:
287:
283:
279:
274:
270:
266:
261:
257:
253:
250:
245:
241:
237:
234:
231:
228:
225:
177:cell complexes
149:
146:
126:Ernst Steinitz
73:
70:
62:image analysis
57:) of objects.
43:digital images
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1148:
1137:
1134:
1133:
1131:
1116:
1108:
1104:
1101:
1099:
1096:
1094:
1091:
1090:
1089:
1081:
1079:
1075:
1071:
1069:
1065:
1061:
1059:
1054:
1049:
1047:
1039:
1038:
1035:
1029:
1026:
1024:
1021:
1019:
1016:
1014:
1011:
1009:
1006:
1004:
1001:
1000:
998:
996:
992:
986:
985:Orientability
983:
981:
978:
976:
973:
971:
968:
966:
963:
962:
960:
956:
950:
947:
945:
942:
940:
937:
935:
932:
930:
927:
925:
922:
920:
917:
913:
910:
908:
905:
904:
903:
900:
896:
893:
891:
888:
886:
883:
881:
878:
876:
873:
872:
871:
868:
866:
863:
861:
858:
856:
852:
849:
848:
846:
842:
837:
827:
824:
822:
821:Set-theoretic
819:
815:
812:
811:
810:
807:
803:
800:
799:
798:
795:
793:
790:
788:
785:
783:
782:Combinatorial
780:
778:
775:
773:
770:
769:
767:
763:
759:
752:
747:
745:
740:
738:
733:
732:
729:
721:
715:
711:
706:
703:
699:
695:
691:
689:0-914894-65-X
685:
680:
679:
672:
668:
664:
660:
656:
652:
648:
647:
641:
637:
635:1-55860-861-3
631:
627:
626:
620:
616:
614:0-9755122-1-8
610:
606:
601:
597:
593:
589:
587:0-387-55943-4
583:
579:
575:
570:
569:
554:
534:
530:
528:0-444-89754-2
524:
520:
515:
511:
507:
503:
497:
492:
491:
484:
483:
479:
475:
472:
470:
467:
465:
462:
460:
457:
455:
452:
450:
447:
445:
442:
440:
437:
436:
432:
430:
428:
410:
406:
402:
399:
394:
390:
386:
383:
380:
375:
371:
350:
347:
344:
336:
332:
314:
310:
285:
281:
272:
268:
264:
259:
255:
251:
248:
243:
239:
232:
229:
226:
223:
216:
215:
214:
212:
208:
204:
199:
197:
193:
189:
185:
180:
178:
174:
169:
167:
163:
159:
155:
148:Basic results
147:
145:
143:
138:
133:
131:
127:
123:
119:
115:
111:
106:
102:
98:
94:
90:
85:
83:
79:
71:
69:
67:
63:
58:
56:
52:
51:connectedness
48:
44:
40:
36:
32:
19:
1115:Publications
980:Chern number
970:Betti number
853: /
844:Key concepts
825:
792:Differential
709:
677:
650:
644:
624:
604:
539:
521:. Elsevier.
518:
489:
425:. (See also
334:
330:
301:
206:
200:
183:
181:
170:
151:
134:
104:
86:
75:
59:
30:
29:
1078:Wikiversity
995:Key results
80:researcher
47:topological
924:CW complex
865:Continuity
855:Closed set
814:cohomology
480:References
101:Heinz Hopf
55:boundaries
1103:geometric
1098:algebraic
949:Cobordism
885:Hausdorff
880:connected
797:Geometric
787:Continuum
777:Algebraic
265:−
205:is: Let
1130:Category
1068:Wikibook
1046:Category
934:Manifold
902:Homotopy
860:Interior
851:Open set
809:Homology
758:Topology
707:(2008).
464:Topology
433:See also
211:manifold
190:made by
66:thinning
37:(2D) or
1093:general
895:uniform
875:compact
826:Digital
698:0643798
667:0686842
596:1224678
510:1711168
363:, then
72:History
1088:Topics
890:metric
765:Fields
716:
696:
686:
665:
632:
611:
594:
584:
525:
508:
498:
302:where
158:pixels
105:et al.
870:Space
41:(3D)
714:ISBN
684:ISBN
630:ISBN
609:ISBN
582:ISBN
523:ISBN
496:ISBN
194:. A
120:and
99:and
655:doi
574:doi
429:.)
1132::
694:MR
692:.
663:MR
661:.
651:51
649:.
592:MR
590:.
580:.
506:MR
504:.
182:A
750:e
743:t
736:v
722:.
700:.
669:.
657::
638:.
617:.
598:.
576::
555:n
550:Z
531:.
512:.
411:6
407:M
403:2
400:+
395:5
391:M
387:+
384:8
381:=
376:3
372:M
351:0
348:=
345:g
335:M
331:i
315:i
311:M
298:,
286:8
282:/
278:)
273:3
269:M
260:6
256:M
252:2
249:+
244:5
240:M
236:(
233:+
230:1
227:=
224:g
207:M
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.