28:
1155:
692:
1085:
751:, effects of spatially selective excitation, signal detection coil properties, motion etc.) or nonzero phase due to just physical reasons (such as the different chemical shift of fat and water in gradient echo techniques).
322:
250:
1014:
950:
823:
861:
907:
500:
468:
176:
144:
404:
579:
348:
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is prepared to be proportional simply to a contrast-weighted proton density and thus is a real quantity. In such a case, the signal at two opposite locations in
509:-space has the same number of rows and columns as the final image and is filled with raw data during the scan, usually one line per TR (Repetition Time).
1025:
256:
184:
747:). However, these techniques are approximate due to phase errors in the MRI data which can rarely be completely controlled (due to imperfect
743:) or in the FE (Frequency Encode) direction, allowing for lower sampling frequencies and/or shorter echo times (such a technique is known as
69:
1178:
Twieg D (1983). "The k-trajectory formulation of the NMR imaging process with applications in analysis and synthesis of imaging methods".
955:
1260:
528:
1153:, Richard S. Likes, "Moving Gradient Zeugmatography", issued 1981-12-22, assigned to General Electric Company
918:
430:
of this encoded signal results in a representation of the spin density distribution in two dimensions. Thus position (
791:
777:
is present in MRI data acquisition, but is absent in NMR data acquisition. As a result of this difference, the NMR
44:
828:
94:-space is full (at the end of the scan) the data are mathematically processed to produce a final image. Thus
731:-space information is somewhat redundant then, and an image can be reconstructed using only one half of the
866:
555:-space; in this way, different contrast images can be acquired without the need of running full scans.
519:
in the sample at a specific time point after an excitation. Conventional qualitative interpretation of
473:
441:
149:
117:
1150:
380:
1187:
687:{\displaystyle S(-k_{\mathrm {FE} },-k_{\mathrm {PE} })=S^{*}(k_{\mathrm {FE} },k_{\mathrm {PE} })\,}
90:, usually a matrix, in which data from digitized MR signals are stored during data acquisition. When
724:
58:
551:-space is acquired, and subsequent scans are performed for acquiring just the central part of the
735:-space, either in the PE (Phase Encode) direction saving scan time (such a technique is known as
540:
411:
512:
An MR image is a complex-valued map of the spatial distribution of the transverse magnetization
1203:
532:
427:
330:
65:
of the image measured. It was introduced in 1979 by Likes and in 1983 by
Ljunggren and Twieg.
62:
1109:
1195:
748:
520:
53:
353:
27:
765:
in all aspects, both being used for raw data storage. The only difference between the MRI
700:
1191:
535:
information of the image, whereas high spatial frequencies (outer peripheral regions of
82:, i.e. an accurately timed sequence of radiofrequency and gradient pulses. In practice,
1254:
78:
1080:{\displaystyle \omega _{\mathrm {r} }=\omega _{\mathrm {0} }+{\bar {\gamma }}rG}
317:{\displaystyle k_{\mathrm {PE} }={\bar {\gamma }}n\Delta G_{\mathrm {PE} }\tau }
111:
17:
245:{\displaystyle k_{\mathrm {FE} }={\bar {\gamma }}G_{\mathrm {FE} }m\Delta t}
1207:
76:-space during an MR measurement in a premeditated scheme controlled by a
1199:
1009:{\displaystyle (\omega _{\mathrm {r} }t)/(\omega _{\mathrm {r} }t)}
35:-space is conjugate symmetric: the imaginary component at opposite
1222:
Principles of nuclear magnetic resonance in one and two dimensions
543:. This is the basis for advanced scanning techniques, such as the
26:
759:
350:
is the sampling time (the reciprocal of sampling frequency),
523:
asserts that low spatial frequencies (near the center of
178:
for "frequency encoding" (FE) and "phase encoding" (PE):
1168:
Ljunggren S. Journal of
Magnetic Resonance 1983; 54:338.
422:
is the sample number in the PE direction (also known as
1112:
1028:
958:
921:
869:
831:
794:
703:
582:
476:
444:
383:
356:
333:
259:
187:
152:
120:
945:{\displaystyle {\text{Spin-Echo}}=M_{\mathrm {0} }}
1118:
1079:
1008:
944:
901:
855:
817:
715:
686:
494:
462:
398:
362:
342:
316:
244:
170:
138:
539:-space) contain the information determining the
1220:Ernst RR, Bodenhausen G and Wokaun A (1987),
818:{\displaystyle {\text{FID}}=M_{\mathrm {0} }}
418:is the sample number in the FE direction and
8:
785:signal take different mathematical forms:
1111:
1060:
1059:
1049:
1048:
1034:
1033:
1027:
993:
992:
980:
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966:
957:
935:
934:
922:
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889:
888:
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868:
856:{\displaystyle (\omega _{\mathrm {0} }t)}
840:
839:
830:
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795:
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683:
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223:
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126:
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119:
39:-space coordinates has the opposite sign.
1142:
547:acquisition, in which a first complete
502:) constitute a Fourier transform pair.
902:{\displaystyle (-t/T_{\mathrm {2} })}
7:
1090:Due to the presence of the gradient
31:For a real image, the corresponding
558:A nice symmetry property exists in
1035:
994:
968:
674:
671:
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622:
619:
601:
598:
562:-space if the image magnetization
486:
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451:
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25:
495:{\displaystyle k_{\mathrm {PE} }}
463:{\displaystyle k_{\mathrm {FE} }}
171:{\displaystyle k_{\mathrm {PE} }}
139:{\displaystyle k_{\mathrm {FE} }}
109:It can be formulated by defining
1106:) is encoded onto the frequency
399:{\displaystyle {\bar {\gamma }}}
72:, complex values are sampled in
1103:spatial frequency information k
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985:
977:
959:
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832:
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644:
628:
586:
390:
284:
212:
61:) is obtained as the 2D or 3D
1:
1238:MRI, From picture to proton.
1126:, and at the same time the
86:-space often refers to the
1277:
1261:Magnetic resonance imaging
1224:, Oxford University Press.
45:magnetic resonance imaging
438:) and spatial frequency (
57:(a mathematical space of
343:{\displaystyle \Delta t}
1119:{\displaystyle \omega }
1120:
1081:
1010:
946:
903:
857:
819:
717:
688:
496:
464:
400:
364:
344:
318:
246:
172:
140:
40:
1151:US patent 4307343
1121:
1097:spatial information r
1082:
1011:
947:
904:
858:
820:
718:
689:
497:
465:
401:
365:
363:{\displaystyle \tau }
345:
319:
247:
173:
141:
88:temporary image space
30:
1243:Hashemi Ray, et al.
1236:McRobbie D., et al.
1110:
1026:
956:
919:
867:
829:
792:
716:{\displaystyle ^{*}}
701:
580:
527:-space) contain the
474:
442:
381:
354:
331:
257:
185:
150:
118:
1192:1983MedPh..10..610T
781:signal and the MRI
773:is that a gradient
725:complex conjugation
370:is the duration of
59:spatial frequencies
1116:
1077:
1006:
942:
899:
853:
815:
713:
684:
492:
460:
412:gyromagnetic ratio
396:
360:
340:
314:
242:
168:
136:
41:
1068:
925:
798:
749:static field shim
428:Fourier Transform
393:
287:
215:
63:Fourier transform
16:(Redirected from
1268:
1225:
1218:
1212:
1211:
1200:10.1118/1.595331
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697:where the star (
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541:image resolution
521:Fourier Analysis
501:
499:
498:
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491:
490:
489:
469:
467:
466:
461:
459:
458:
457:
426:). Then, the 2D-
424:partition number
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349:
347:
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323:
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217:
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177:
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54:reciprocal space
21:
1276:
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1269:
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1250:
1245:MRI, The Basics
1233:
1231:Further reading
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1219:
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1180:Medical Physics
1177:
1176:
1172:
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962:
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864:
835:
827:
826:
803:
790:
789:
704:
699:
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665:
647:
634:
613:
592:
578:
577:
568:
529:signal to noise
518:
477:
472:
471:
445:
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379:
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352:
351:
329:
328:
296:
260:
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153:
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23:
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15:
12:
11:
5:
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1248:
1241:
1232:
1229:
1227:
1226:
1213:
1170:
1161:
1141:
1139:
1136:
1130:is renamed as
1115:
1088:
1087:
1076:
1073:
1067:
1064:
1058:
1052:
1047:
1043:
1037:
1032:
1017:
1016:
1005:
1002:
996:
991:
987:
983:
979:
976:
970:
965:
961:
938:
933:
929:
910:
909:
898:
892:
887:
882:
878:
875:
872:
852:
849:
843:
838:
834:
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758:is related to
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221:
214:
211:
205:
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164:
161:
156:
132:
129:
124:
104:reconstruction
79:pulse sequence
24:
14:
13:
10:
9:
6:
4:
3:
2:
1273:
1262:
1259:
1258:
1256:
1246:
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1234:
1230:
1223:
1217:
1214:
1209:
1205:
1201:
1197:
1193:
1189:
1186:(5): 610–21.
1185:
1181:
1174:
1171:
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1152:
1146:
1143:
1137:
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1021:
1020:
1000:
989:
981:
974:
963:
936:
931:
927:
915:
914:
913:
890:
885:
880:
876:
873:
847:
841:
836:
809:
804:
800:
788:
787:
786:
784:
780:
776:
772:
768:
764:
761:
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750:
746:
742:
738:
734:
730:
726:
708:
705:
666:
662:
648:
639:
635:
631:
614:
610:
607:
593:
589:
583:
576:
575:
574:
572:
565:
561:
556:
554:
550:
546:
542:
538:
534:
530:
526:
522:
515:
510:
508:
503:
478:
446:
437:
433:
429:
425:
421:
417:
413:
409:
387:
373:
357:
337:
311:
297:
290:
281:
275:
261:
253:
239:
233:
219:
209:
203:
189:
181:
180:
179:
154:
122:
114:
113:
107:
105:
101:
98:-space holds
97:
93:
89:
85:
81:
80:
75:
71:
66:
64:
60:
56:
55:
50:
46:
38:
34:
29:
19:
18:K-space (MRI)
1244:
1237:
1221:
1216:
1183:
1179:
1173:
1164:
1145:
1131:
1127:
1102:
1101:
1096:
1095:
1091:
1089:
1018:
911:
782:
778:
774:
770:
769:and the NMR
766:
762:
755:
753:
744:
740:
737:half Fourier
736:
732:
728:
696:
570:
563:
559:
557:
552:
548:
544:
536:
524:
513:
511:
506:
504:
435:
431:
423:
419:
415:
407:
371:
326:
112:wave vectors
110:
108:
103:
102:data before
99:
95:
91:
87:
83:
77:
73:
67:
52:
48:
42:
36:
32:
1128:time-domain
771:time domain
763:time-domain
573:-space is:
505:Typically,
70:MRI physics
47:(MRI), the
1247:2ED. 2004.
1138:References
723:) denotes
51:-space or
1114:ω
1100:(not the
1066:¯
1063:γ
1046:ω
1031:ω
990:ω
964:ω
924:Spin-Echo
874:−
837:ω
783:spin-echo
745:half echo
741:half scan
709:∗
640:∗
611:−
590:−
410:) is the
408:gamma bar
391:¯
388:γ
358:τ
335:Δ
312:τ
294:Δ
285:¯
282:γ
237:Δ
213:¯
210:γ
1255:Category
533:contrast
1208:6646065
1188:Bibcode
1132:k-space
767:k-space
756:k-space
727:. Thus
545:keyhole
1206:
1157:
1094:, the
1019:where
327:where
912:and
1240:2003
1204:PMID
754:MRI
531:and
146:and
1196:doi
952:sin
863:exp
825:cos
797:FID
779:FID
760:NMR
739:or
100:raw
68:In
43:In
1257::
1202:.
1194:.
1184:10
1182:.
1134:.
567:xy
517:xy
470:,
414:,
377:,
375:PE
106:.
1210:.
1198::
1190::
1092:G
1075:G
1072:r
1057:+
1051:0
1042:=
1036:r
1004:)
1001:t
995:r
986:(
982:/
978:)
975:t
969:r
960:(
937:0
932:M
928:=
897:)
891:2
886:T
881:/
877:t
871:(
851:)
848:t
842:0
833:(
810:0
805:M
801:=
775:G
733:k
729:k
681:)
675:E
672:P
667:k
663:,
657:E
654:F
649:k
645:(
636:S
632:=
629:)
623:E
620:P
615:k
608:,
602:E
599:F
594:k
587:(
584:S
571:k
564:M
560:k
553:k
549:k
537:k
525:k
514:M
507:k
487:E
484:P
479:k
455:E
452:F
447:k
436:y
434:,
432:x
420:n
416:m
406:(
372:G
338:t
306:E
303:P
298:G
291:n
276:=
270:E
267:P
262:k
240:t
234:m
228:E
225:F
220:G
204:=
198:E
195:F
190:k
163:E
160:P
155:k
131:E
128:F
123:k
96:k
92:k
84:k
74:k
49:k
37:k
33:k
20:)
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