892:
695:
560:
397:
384:"Marks and Cheung focused on images with a given spectral support region and an initial base sampling lattice such that the induced spectral replicas of this support region do not overlap. They then showed that cosets of some sublattice could be removed from the base lattice until the sampling density was minimal … or approached minimal ... allows the sampling rate to be reduced until it equals or approaches … minimum." In this context, the limit is the area of the support of the spectrum."
899:
The corresponding reduction in sampling density is shown in Figure 4 where the red dots are locations where samples need not be taken. A single cell containing one red dot is shown shaded. The area of the cell is The corresponding reduction in sampling density is shown in Figure 4 where the red dots
956:
In the previous example, the squares in Figure 3 can be made arbitrarily small and increased in number so that, asymptotically, all of the area equal to zero can be covered. Thus, the sampling density can be reduced to the support of the spectrum, i.e., to the area where the spectrum is not
241:
603: domain results in spectrum replication in the Fourier domain. If the uniform sampling density were lower, the replications would overlap and an attempt at reconstruction of the original function would result in image
88:
19:
The
Papoulis-Marks-Cheung approach is a theorem in multidimensional Shannon sampling theory that shows that the sampling density of a two-dimensional bandlimited function can be reduced to the
960:
The
Papoulis-Marks-Cheung approach can straightforwardly be generalized to higher dimensions. Also, replication geometry need not be rectangular but can be any shape that will tile the entire
788:
different two-dimensional signals. All of the samples for the signals corresponding to the light green areas are zero and do not have to be considered. The area of the two green squares is
1444:
Cheung, Kwang F. (Dec 6, 2012). "A Multidimensional
Extension of Papoulis' Generalized Sampling Expansion with Application in Minimum Sampling Density". In Marks, Robert J. II (ed.).
556:
To the right in Figure 1 is pictured a rectangular replication of the half circle which occurs when the two-dimensional function is sampled at spatial locations shown in Figure 2.
748:
identical squares. Note that two of these squares lie totally in an area where the spectral replication is identically zero. These squares are shaded light green. Think of each of
834:
886:
611:
to achieve this is equal to the area of the rectangular lattice cell of the spectrum replication. The corresponding area of the rectangle used in the replication is equal to
459:
493:
726:
689:
551:
531:
83:
990:
601:
295:
268:
946:
995:
A more detailed mathematical description of the
Papoulis-Marks-Cheung approach is available in the original paper by Marks and Cheung and their derivative work.
918:
786:
766:
746:
854:
669:
649:
629:
375:
355:
335:
315:
461:, is zero outside the half circle. Inside the circle, the spectrum' is arbitrary but is well behaved. The half-circle, with unit radius, has an area of
412:
The
Papoulis-Marks-Cheung approach is best explained by example. Consider from Figure 1 the half circle shown on the right half plane. A signal's spectrum,
651:
samples per unit area. The
Papoulis-Marks-Cheung approach says that this sampling density can be reduced to the area of the half circle, namely from
836:. Since the samples corresponding to these squares do not have to be considered (they are all zero), the overall sampling density is reduced from
1303:
Herley, Cormac; Wong, Ping Wah; Member, Senior (1999). "Minimum rate sampling and reconstruction of signals with arbitrary frequency support".
1475:
1453:
1387:
631:(cycles per unit length) squared. As confirmed by Figure 2, the sampling density required to achieve the spectral replication is therefore
236:{\displaystyle F(u_{x},y_{y})=\iint \limits _{x,y}f(x,y){\rm {e}}^{-i2\pi (xu_{x}+yu_{y})}\operatorname {d} \!x\operatorname {d} \!y}
1420:
1349:
1279:
1234:
900:
are locations where samples need not be taken. A single cell containing one red dot is shown shaded. The area of the cell is
920:
units. The sampling density is therefore, as also seen from the areas of two green squares in Figure 3, reduced by
44:
568:
891:
694:
559:
396:
1312:
1219:[Proceedings] ICASSP 91: 1991 International Conference on Acoustics, Speech, and Signal Processing
48:
401:
20:
791:
35:
and Kwang Fai Cheung. The approach has been called "elegant," "remarkably" closed, and "interesting."
1179:
1129:
859:
895:
Figure 4: Reduction in sampling density: The red dots are locations where samples need not be taken.
415:
1317:
464:
28:
1285:
1240:
1056:
1028:
563:
Figure 2: This sampling geometry gives rise to the spectral replication on the right in Figure 1.
32:
705:
388:
In deriving their result, Marks and Cheung relied on
Papoulis' generalized sampling expansion.
1449:
1426:
1416:
1393:
1383:
1355:
1345:
1275:
1230:
1195:
1145:
1117:
1098:
1048:
24:
1167:
1322:
1267:
1222:
1187:
1137:
1090:
1038:
674:
536:
501:
53:
963:
574:
273:
246:
923:
1183:
1133:
903:
771:
751:
731:
839:
654:
634:
614:
404:. Outside of the half circle, the spectrum is identically zero. (Right) Nonoverlapping
360:
340:
320:
300:
1469:
1244:
1289:
498:
According to the
Papoulis-Marks-Cheung approach, the sampling density for the image
1060:
608:
1260:"Spectrum-blind minimum-rate sampling and reconstruction of 2-D multiband signals"
553:
samples per unit area. The
Papoulis-Marks-Cheung approach informs how to do this.
1226:
1078:
1259:
1214:
1397:
1271:
1199:
1149:
1102:
1094:
1052:
1043:
1016:
1359:
27:
of the function. Applying a multidimensional generalization of a theorem by
1430:
1191:
1141:
380:
Prelee and
Neuhoff describe the Papoulis-Marks-Cheung approach as follows.
604:
405:
1215:"FIR filtering of images on a lattice with periodically deleted samples"
571:
that shows that the sampling of a two-dimensional signal in the spatial
1326:
1264:
Proceedings of 3rd IEEE International Conference on Image Processing
377:
are lengths, spatial frequency has units of cycles per unit length.
1033:
698:
Figure 3: The half-circle spectrum support divided into 32 squares.
408:
rectangular replication of the support of the spectrum on the left.
890:
693:
558:
395:
1118:"Multidimensional-signal sample dependency at Nyquist densities"
702:
To see how this reduction happens, consider Figure 3 where the
1415:. Russell M. Mersereau. Englewood Cliffs, NJ: Prentice-Hall.
1168:"Imaging sampling below the Nyquist density without aliasing"
1446:
Advanced Topics in Shannon Sampling and Interpolation Theory
1017:"Multidimensional Manhattan Sampling and Reconstruction"
966:
926:
906:
862:
842:
794:
774:
754:
734:
708:
677:
657:
637:
617:
577:
539:
504:
467:
418:
363:
343:
323:
303:
276:
249:
91:
56:
1380:
Handbook of Fourier analysis & its applications
297: are the spatial frequencies corresponding to
1015:Prelee, Matthew A.; Neuhoff, David L. (May 2016).
984:
940:
912:
880:
848:
828:
780:
760:
740:
720:
683:
663:
643:
623:
595:
545:
525:
487:
453:
369:
349:
329:
309:
289:
262:
235:
77:
992: plane such as parallelograms and hexagons.
229:
222:
1166:Cheung, Kwan F.; Marks, Robert J. (1990-01-01).
382:
8:
1344:. Englewood Cliffs, N.J: Prentice Hall PTR.
1413:Multidimensional digital signal processing
1448:. Springer Science & Business Media.
1316:
1083:IEEE Transactions on Circuits and Systems
1042:
1032:
965:
930:
925:
905:
861:
841:
812:
798:
793:
773:
753:
733:
728:rectangular lattice cell is divided into
707:
676:
656:
636:
616:
576:
567:This replication is a consequence of the
538:
503:
471:
466:
442:
429:
417:
362:
342:
322:
302:
281:
275:
254:
248:
208:
192:
169:
163:
162:
131:
115:
102:
90:
55:
1266:. Vol. 1. pp. 701–704 vol.1.
1021:IEEE Transactions on Information Theory
1004:
1213:Gardos, T.R.; Mersereau, R.M. (1991).
400:Figure 1: (Left) Half circle spectral
31:, the approach was first proposed by
7:
1373:
1371:
1369:
1161:
1159:
1072:
1070:
1010:
1008:
1382:. Oxford: Oxford University Press.
495:(cycles per unit length) squared.
226:
219:
164:
14:
569:multidimensional sampling theorem
45:two-dimensional Fourier transform
1258:Bresler, Y.; Feng, Ping (1996).
1079:"Generalized sampling expansion"
829:{\displaystyle 2/32=1/16=0.0625}
1116:Marks, Robert J. (1986-02-01).
881:{\displaystyle 2-0.0625=1.9375}
1077:Papoulis, A. (November 1977).
979:
967:
590:
578:
520:
508:
454:{\displaystyle F(u_{x},u_{y})}
448:
422:
214:
182:
158:
146:
121:
95:
72:
60:
47:, or frequency spectrum, of a
1:
488:{\displaystyle \pi /2=1.5708}
1476:Theorems in Fourier analysis
1378:Marks, Robert J. II (2009).
1221:. pp. 2873–2876 vol.4.
1492:
1305:IEEE Trans. Inform. Theory
1227:10.1109/ICASSP.1991.151002
16:Theorem in sampling theory
856:samples per unit area to
721:{\displaystyle 1\times 2}
1411:Dudgeon, Dan E. (1984).
1272:10.1109/ICIP.1996.559595
1095:10.1109/TCS.1977.1084284
1044:10.1109/TIT.2016.2542081
888:samples per unit area.
691:samples per unit area.
533: can be reduced to
1342:Time-frequency analysis
948:samples per unit area.
1192:10.1364/JOSAA.7.000092
1142:10.1364/JOSAA.3.000268
986:
942:
914:
896:
882:
850:
830:
782:
768:squares as spectra of
762:
742:
722:
699:
685:
684:{\displaystyle 1.5708}
665:
645:
625:
597:
564:
547:
546:{\displaystyle 1.5708}
527:
526:{\displaystyle f(x,y)}
489:
455:
409:
386:
371:
351:
331:
311:
291:
264:
237:
79:
78:{\displaystyle f(x,y)}
987:
985:{\displaystyle (x,y)}
943:
915:
894:
883:
851:
831:
783:
763:
743:
723:
697:
686:
666:
646:
626:
598:
596:{\displaystyle (x,y)}
562:
548:
528:
490:
456:
399:
372:
352:
332:
312:
292:
290:{\displaystyle u_{x}}
265:
263:{\displaystyle u_{x}}
238:
80:
1340:Cohen, Leon (1995).
964:
941:{\displaystyle 1/16}
924:
904:
860:
840:
792:
772:
752:
732:
706:
675:
655:
635:
615:
575:
537:
502:
465:
416:
361:
341:
321:
301:
274:
247:
89:
54:
1184:1990JOSAA...7...92C
1134:1986JOSAA...3..268M
29:Athanasios Papoulis
982:
957:identically zero.
938:
913:{\displaystyle 16}
910:
897:
878:
846:
826:
781:{\displaystyle 32}
778:
761:{\displaystyle 32}
758:
741:{\displaystyle 32}
738:
718:
700:
681:
661:
641:
621:
593:
565:
543:
523:
485:
451:
410:
367:
347:
327:
307:
287:
260:
233:
142:
75:
33:Robert J. Marks II
1455:978-1-4613-9757-1
1389:978-0-19-533592-7
1327:10.1109/18.771158
849:{\displaystyle 2}
664:{\displaystyle 2}
644:{\displaystyle 2}
624:{\displaystyle 2}
370:{\displaystyle y}
350:{\displaystyle x}
330:{\displaystyle y}
310:{\displaystyle x}
127:
25:Fourier transform
1483:
1460:
1459:
1441:
1435:
1434:
1408:
1402:
1401:
1375:
1364:
1363:
1337:
1331:
1330:
1320:
1311:(5): 1555–1564.
1300:
1294:
1293:
1255:
1249:
1248:
1210:
1204:
1203:
1163:
1154:
1153:
1113:
1107:
1106:
1074:
1065:
1064:
1046:
1036:
1027:(5): 2772–2787.
1012:
991:
989:
988:
983:
947:
945:
944:
939:
934:
919:
917:
916:
911:
887:
885:
884:
879:
855:
853:
852:
847:
835:
833:
832:
827:
816:
802:
787:
785:
784:
779:
767:
765:
764:
759:
747:
745:
744:
739:
727:
725:
724:
719:
690:
688:
687:
682:
670:
668:
667:
662:
650:
648:
647:
642:
630:
628:
627:
622:
609:sampling density
602:
600:
599:
594:
552:
550:
549:
544:
532:
530:
529:
524:
494:
492:
491:
486:
475:
460:
458:
457:
452:
447:
446:
434:
433:
376:
374:
373:
368:
356:
354:
353:
348:
336:
334:
333:
328:
316:
314:
313:
308:
296:
294:
293:
288:
286:
285:
269:
267:
266:
261:
259:
258:
242:
240:
239:
234:
218:
217:
213:
212:
197:
196:
168:
167:
141:
120:
119:
107:
106:
84:
82:
81:
76:
1491:
1490:
1486:
1485:
1484:
1482:
1481:
1480:
1466:
1465:
1464:
1463:
1456:
1443:
1442:
1438:
1423:
1410:
1409:
1405:
1390:
1377:
1376:
1367:
1352:
1339:
1338:
1334:
1302:
1301:
1297:
1282:
1257:
1256:
1252:
1237:
1212:
1211:
1207:
1165:
1164:
1157:
1115:
1114:
1110:
1089:(11): 652–654.
1076:
1075:
1068:
1014:
1013:
1006:
1001:
962:
961:
954:
922:
921:
902:
901:
858:
857:
838:
837:
790:
789:
770:
769:
750:
749:
730:
729:
704:
703:
673:
672:
653:
652:
633:
632:
613:
612:
573:
572:
535:
534:
500:
499:
463:
462:
438:
425:
414:
413:
394:
359:
358:
339:
338:
319:
318:
299:
298:
277:
272:
271:
250:
245:
244:
204:
188:
161:
111:
98:
87:
86:
52:
51:
41:
17:
12:
11:
5:
1489:
1487:
1479:
1478:
1468:
1467:
1462:
1461:
1454:
1436:
1421:
1403:
1388:
1365:
1350:
1332:
1318:10.1.1.83.8638
1295:
1280:
1250:
1235:
1205:
1155:
1128:(2): 268–273.
1108:
1066:
1003:
1002:
1000:
997:
981:
978:
975:
972:
969:
953:
950:
937:
933:
929:
909:
877:
874:
871:
868:
865:
845:
825:
822:
819:
815:
811:
808:
805:
801:
797:
777:
757:
737:
717:
714:
711:
680:
660:
640:
620:
592:
589:
586:
583:
580:
542:
522:
519:
516:
513:
510:
507:
484:
481:
478:
474:
470:
450:
445:
441:
437:
432:
428:
424:
421:
393:
390:
366:
346:
326:
306:
284:
280:
257:
253:
232:
228:
225:
221:
216:
211:
207:
203:
200:
195:
191:
187:
184:
181:
178:
175:
172:
166:
160:
157:
154:
151:
148:
145:
140:
137:
134:
130:
126:
123:
118:
114:
110:
105:
101:
97:
94:
74:
71:
68:
65:
62:
59:
40:
37:
15:
13:
10:
9:
6:
4:
3:
2:
1488:
1477:
1474:
1473:
1471:
1457:
1451:
1447:
1440:
1437:
1432:
1428:
1424:
1422:0-13-604959-1
1418:
1414:
1407:
1404:
1399:
1395:
1391:
1385:
1381:
1374:
1372:
1370:
1366:
1361:
1357:
1353:
1351:0-13-594532-1
1347:
1343:
1336:
1333:
1328:
1324:
1319:
1314:
1310:
1306:
1299:
1296:
1291:
1287:
1283:
1281:0-7803-3259-8
1277:
1273:
1269:
1265:
1261:
1254:
1251:
1246:
1242:
1238:
1236:0-7803-0003-3
1232:
1228:
1224:
1220:
1216:
1209:
1206:
1201:
1197:
1193:
1189:
1185:
1181:
1178:(1): 92–105.
1177:
1173:
1169:
1162:
1160:
1156:
1151:
1147:
1143:
1139:
1135:
1131:
1127:
1123:
1119:
1112:
1109:
1104:
1100:
1096:
1092:
1088:
1084:
1080:
1073:
1071:
1067:
1062:
1058:
1054:
1050:
1045:
1040:
1035:
1030:
1026:
1022:
1018:
1011:
1009:
1005:
998:
996:
993:
976:
973:
970:
958:
951:
949:
935:
931:
927:
907:
893:
889:
875:
872:
869:
866:
863:
843:
823:
820:
817:
813:
809:
806:
803:
799:
795:
775:
755:
735:
715:
712:
709:
696:
692:
678:
658:
638:
618:
610:
606:
587:
584:
581:
570:
561:
557:
554:
540:
517:
514:
511:
505:
496:
482:
479:
476:
472:
468:
443:
439:
435:
430:
426:
419:
407:
403:
398:
391:
389:
385:
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378:
364:
344:
324:
304:
282:
278:
255:
251:
230:
223:
209:
205:
201:
198:
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189:
185:
179:
176:
173:
170:
155:
152:
149:
143:
138:
135:
132:
128:
124:
116:
112:
108:
103:
99:
92:
69:
66:
63:
57:
50:
46:
38:
36:
34:
30:
26:
22:
1445:
1439:
1412:
1406:
1379:
1341:
1335:
1308:
1304:
1298:
1263:
1253:
1218:
1208:
1175:
1171:
1125:
1121:
1111:
1086:
1082:
1024:
1020:
994:
959:
955:
898:
701:
566:
555:
497:
411:
387:
383:
379:
42:
18:
392:Explanation
39:The Theorem
1034:1502.01694
999:References
406:nonaliased
357: and
317: and
270: and
1398:227191901
1313:CiteSeerX
1245:121408946
1200:1520-8532
1150:1520-8532
1103:0098-4094
1053:0018-9448
952:Extension
867:−
713:×
469:π
180:π
171:−
129:∬
85: is
1470:Category
1360:31516509
1290:44441155
605:aliasing
49:function
1431:9282699
1180:Bibcode
1130:Bibcode
1061:3199882
402:support
337:. When
23:of the
21:support
1452:
1429:
1419:
1396:
1386:
1358:
1348:
1315:
1288:
1278:
1243:
1233:
1198:
1172:JOSA A
1148:
1122:JOSA A
1101:
1059:
1051:
876:1.9375
870:0.0625
824:0.0625
679:1.5708
607:. The
541:1.5708
483:1.5708
243:where
1286:S2CID
1241:S2CID
1057:S2CID
1029:arXiv
1450:ISBN
1427:OCLC
1417:ISBN
1394:OCLC
1384:ISBN
1356:OCLC
1346:ISBN
1276:ISBN
1231:ISBN
1196:ISSN
1146:ISSN
1099:ISSN
1049:ISSN
43:The
1323:doi
1268:doi
1223:doi
1188:doi
1138:doi
1091:doi
1039:doi
671:to
1472::
1425:.
1392:.
1368:^
1354:.
1321:.
1309:45
1307:.
1284:.
1274:.
1262:.
1239:.
1229:.
1217:.
1194:.
1186:.
1174:.
1170:.
1158:^
1144:.
1136:.
1124:.
1120:.
1097:.
1087:24
1085:.
1081:.
1069:^
1055:.
1047:.
1037:.
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1292:.
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844:2
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619:2
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144:f
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117:y
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109:,
104:x
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93:F
73:)
70:y
67:,
64:x
61:(
58:f
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