91:
20:
1496:
1325:
1771:
While the
Anscombe transform is appropriate for pure Poisson data, in many applications the data presents also an additive Gaussian component. These cases are treated by a Generalized Anscombe transform and its asymptotically unbiased or exact unbiased inverses.
1761:
1506:
There are many other possible variance-stabilizing transformations for the
Poisson distribution. Bar-Lev and Enis report a family of such transformations which includes the Anscombe transform. Another member of the family is the Freeman-Tukey transformation
713:
1340:
1052:
75:. The Anscombe transform is widely used in photon-limited imaging (astronomy, X-ray) where images naturally follow the Poisson law. The Anscombe transform is usually used to pre-process the data in order to make the
1147:
1158:
190:
548:
1564:
412:
766:
866:
1628:
1614:
901:
599:
290:
822:
248:
1491:{\displaystyle y\mapsto {\frac {1}{4}}y^{2}-{\frac {1}{8}}+{\frac {1}{4}}{\sqrt {\frac {3}{2}}}y^{-1}-{\frac {11}{8}}y^{-2}+{\frac {5}{8}}{\sqrt {\frac {3}{2}}}y^{-3}.}
210:
112:
490:
1079:
969:
949:
929:
786:
594:
574:
464:
444:
41:
322:
1152:
mitigates the issue of bias, but this is not the case in photon-limited imaging, for which the exact unbiased inverse given by the implicit mapping
980:
1923:
Mäkitalo, M.; Foi, A. (2011), "A closed-form approximation of the exact unbiased inverse of the
Anscombe variance-stabilizing transformation",
1320:{\displaystyle \operatorname {E} \left=2\sum _{x=0}^{+\infty }\left({\sqrt {x+{\tfrac {3}{8}}}}\cdot {\frac {m^{x}e^{-m}}{x!}}\right)\mapsto m}
1845:
Bar-Lev, S. K.; Enis, P. (1988), "On the classical choice of variance stabilizing transformations and an application for a
Poisson variate",
1091:
553:
aims at transforming the data so that the variance is set approximately 1 for large enough mean; for mean zero, the variance is still zero.
117:
1619:
which, while it is not quite so good at stabilizing the variance, has the advantage of being more easily understood. Indeed, from the
2135:
498:
1513:
327:
2034:
Mäkitalo, M.; Foi, A. (2013), "Optimal inversion of the generalized
Anscombe transformation for Poisson-Gaussian noise",
1781:
1570:
718:
60:
2084:
Starck, J.-L.; Murtagh, F. (2001), "Astronomical image and signal processing: looking at noise, information and scale",
1867:
Mäkitalo, M.; Foi, A. (2011), "Optimal inversion of the
Anscombe transformation in low-count Poisson image denoising",
2018:
827:
1756:{\displaystyle V\approx \left({\frac {d(2{\sqrt {m}})}{dm}}\right)^{2}V=\left({\frac {1}{\sqrt {m}}}\right)^{2}m=1}
87:
are used; the final estimate is then obtained by applying an inverse
Anscombe transformation to the denoised data.
1579:
2125:
84:
2130:
708:{\displaystyle 2{\sqrt {m+{\tfrac {3}{8}}}}-{\tfrac {1}{4\,m^{1/2}}}+O\left({\tfrac {1}{m^{3/2}}}\right)}
1786:
871:
253:
951:), its inverse transform is also needed in order to return the variance-stabilized and denoised data
1885:
794:
215:
1880:
1331:
195:
2089:
2039:
1928:
1872:
1058:
423:
114:
is the mean of the
Anscombe-transformed Poisson distribution, normalized by subtracting by
68:
97:
8:
469:
72:
2093:
2043:
1932:
1876:
23:
Standard deviation of the transformed
Poisson random variable as a function of the mean
2105:
2063:
2007:
1988:
1952:
1906:
1825:
1064:
954:
934:
914:
911:
When the
Anscombe transform is used in denoising (i.e. when the goal is to obtain from
771:
579:
559:
449:
429:
76:
26:
295:
2067:
2055:
2014:
1944:
1898:
2109:
1910:
2097:
2047:
1978:
1956:
1936:
1890:
1850:
1817:
972:
56:
80:
64:
1821:
1809:
1983:
1854:
1812:(1948), "The transformation of Poisson, binomial and negative-binomial data",
1047:{\displaystyle A^{-1}:y\mapsto \left({\frac {y}{2}}\right)^{2}-{\frac {3}{8}}}
90:
2119:
2051:
1940:
1894:
19:
2059:
1948:
1902:
1620:
1992:
1970:
1829:
1142:{\displaystyle y\mapsto \left({\frac {y}{2}}\right)^{2}-{\frac {1}{8}}}
1082:
48:
2101:
1973:(1950), "Transformations related to the angular and the square root",
185:{\displaystyle 2{\sqrt {m+{\tfrac {3}{8}}}}-{\tfrac {1}{4\,m^{1/2}}}}
212:
is its standard deviation (estimated empirically). We notice that
1571:
primitive of the reciprocal of the standard deviation of the data
903:, which is exactly the reason why this value was picked.
824:, the expression for the variance has an additional term
543:{\displaystyle A:x\mapsto 2{\sqrt {x+{\tfrac {3}{8}}}}\,}
1559:{\displaystyle A:x\mapsto {\sqrt {x+1}}+{\sqrt {x}}.\,}
1253:
1185:
1085:. Sometimes using the asymptotically unbiased inverse
882:
835:
736:
675:
632:
615:
526:
150:
133:
1631:
1582:
1516:
1343:
1161:
1094:
1067:
983:
957:
937:
917:
874:
830:
797:
774:
721:
602:
582:
562:
501:
472:
452:
432:
330:
298:
256:
218:
198:
120:
100:
29:
2004:
768:. This approximation gets more accurate for larger
407:{\displaystyle \mu =O(m^{-3/2}),\sigma =1+O(m^{-2})}
1081:, because the forward square-root transform is not
761:{\displaystyle 1+O\left({\tfrac {1}{m^{2}}}\right)}
2006:
1816:, vol. 35, no. 3–4, , pp. 246–254,
1755:
1608:
1558:
1490:
1319:
1141:
1073:
1046:
963:
943:
923:
895:
860:
816:
780:
760:
707:
588:
568:
542:
484:
458:
438:
406:
316:
284:
242:
204:
184:
106:
35:
2117:
1927:, vol. 20, no. 9, pp. 2697–2698,
1334:approximation of this exact unbiased inverse is
2005:Starck, J.L.; Murtagh, F.; Bijaoui, A. (1998).
1977:, vol. 21, no. 4, pp. 607–611,
1849:, vol. 75, no. 4, pp. 803–804,
1804:
1802:
861:{\displaystyle {\frac {{\tfrac {3}{8}}-c}{m}}}
324:over the period, giving empirical support for
2083:
2038:, vol. 22, no. 1, pp. 91–103,
1871:, vol. 20, no. 1, pp. 99–109,
1569:A simplified transformation, obtained as the
2088:, vol. 18, no. 2, pp. 30–40,
1968:
2033:
1922:
1866:
1844:
1799:
1982:
1884:
1609:{\displaystyle A:x\mapsto 2{\sqrt {x}}\,}
1605:
1555:
641:
596:) to approximately Gaussian data of mean
539:
159:
83:algorithms designed for the framework of
1840:
1838:
1808:
89:
71:into one with an approximately standard
18:
791:For a transformed variable of the form
2118:
2036:IEEE Transactions on Image Processing
1975:The Annals of Mathematical Statistics
1925:IEEE Transactions on Image Processing
1869:IEEE Transactions on Image Processing
1835:
788:, as can be also seen in the figure.
971:to the original range. Applying the
1782:Variance-stabilizing transformation
61:variance-stabilizing transformation
13:
2077:
2009:Image Processing and Data Analysis
1234:
1162:
94:Anscombe transform animated. Here
14:
2147:
1766:
896:{\displaystyle c={\tfrac {3}{8}}}
2086:Signal Processing Magazine, IEEE
292:remains roughly in the range of
285:{\displaystyle m^{2}(\sigma -1)}
2136:Statistical data transformation
1501:
2027:
2013:. Cambridge University Press.
1998:
1962:
1916:
1860:
1711:
1705:
1678:
1665:
1648:
1635:
1592:
1526:
1347:
1311:
1098:
1003:
817:{\displaystyle 2{\sqrt {x+c}}}
556:It transforms Poissonian data
511:
401:
385:
364:
340:
311:
299:
279:
267:
79:approximately constant. Then
1:
1792:
1057:usually introduces undesired
417:
85:additive white Gaussian noise
1061:to the estimate of the mean
906:
7:
1775:
868:; it is reduced to zero at
243:{\displaystyle m^{3/2}\mu }
10:
2152:
1822:10.1093/biomet/35.3-4.246
492:. The Anscombe transform
2052:10.1109/TIP.2012.2202675
1941:10.1109/TIP.2011.2121085
1895:10.1109/TIP.2010.2056693
1984:10.1214/aoms/1177729756
1855:10.1093/biomet/75.4.803
715:and standard deviation
205:{\displaystyle \sigma }
1787:Box–Cox transformation
1757:
1610:
1560:
1492:
1321:
1238:
1143:
1075:
1048:
965:
945:
925:
897:
862:
818:
782:
762:
709:
590:
570:
544:
486:
460:
440:
414:
408:
318:
286:
244:
206:
186:
108:
44:
37:
1758:
1611:
1561:
1493:
1322:
1215:
1144:
1076:
1049:
966:
946:
926:
898:
863:
819:
783:
763:
710:
591:
571:
545:
487:
466:are not independent:
461:
441:
409:
319:
287:
245:
207:
187:
109:
93:
73:Gaussian distribution
38:
22:
2126:Poisson distribution
1629:
1580:
1514:
1341:
1159:
1092:
1065:
981:
955:
935:
915:
872:
828:
795:
772:
719:
600:
580:
560:
499:
470:
450:
430:
424:Poisson distribution
328:
296:
254:
216:
196:
118:
107:{\displaystyle \mu }
98:
69:Poisson distribution
27:
2131:Normal distribution
2094:2001ISPM...18...30S
2044:2013ITIP...22...91M
1933:2011ITIP...20.2697M
1877:2011ITIP...20...99M
485:{\displaystyle m=v}
16:Statistical concept
1753:
1606:
1556:
1488:
1330:should be used. A
1317:
1262:
1194:
1139:
1071:
1044:
961:
941:
921:
893:
891:
858:
844:
814:
778:
758:
752:
705:
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662:
624:
586:
566:
540:
535:
482:
456:
436:
415:
404:
314:
282:
240:
202:
182:
180:
142:
104:
77:standard deviation
63:that transforms a
53:Anscombe transform
45:
33:
2102:10.1109/79.916319
1732:
1731:
1690:
1676:
1646:
1603:
1550:
1540:
1470:
1469:
1458:
1432:
1406:
1405:
1394:
1381:
1358:
1304:
1264:
1261:
1196:
1193:
1137:
1114:
1074:{\displaystyle m}
1042:
1019:
973:algebraic inverse
964:{\displaystyle y}
944:{\displaystyle m}
924:{\displaystyle x}
890:
856:
843:
812:
781:{\displaystyle m}
751:
698:
661:
626:
623:
589:{\displaystyle m}
569:{\displaystyle x}
537:
534:
459:{\displaystyle v}
439:{\displaystyle m}
179:
144:
141:
36:{\displaystyle m}
2143:
2112:
2071:
2070:
2031:
2025:
2024:
2012:
2002:
1996:
1995:
1986:
1969:Freeman, M. F.;
1966:
1960:
1959:
1920:
1914:
1913:
1888:
1864:
1858:
1857:
1842:
1833:
1832:
1806:
1762:
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1759:
1754:
1743:
1742:
1737:
1733:
1727:
1723:
1701:
1700:
1695:
1691:
1689:
1681:
1677:
1672:
1660:
1647:
1642:
1615:
1613:
1612:
1607:
1604:
1599:
1565:
1563:
1562:
1557:
1551:
1546:
1541:
1530:
1497:
1495:
1494:
1489:
1484:
1483:
1471:
1462:
1461:
1459:
1451:
1446:
1445:
1433:
1425:
1420:
1419:
1407:
1398:
1397:
1395:
1387:
1382:
1374:
1369:
1368:
1359:
1351:
1326:
1324:
1323:
1318:
1310:
1306:
1305:
1303:
1295:
1294:
1293:
1281:
1280:
1270:
1265:
1263:
1254:
1245:
1237:
1229:
1208:
1204:
1197:
1195:
1186:
1177:
1148:
1146:
1145:
1140:
1138:
1130:
1125:
1124:
1119:
1115:
1107:
1080:
1078:
1077:
1072:
1053:
1051:
1050:
1045:
1043:
1035:
1030:
1029:
1024:
1020:
1012:
996:
995:
970:
968:
967:
962:
950:
948:
947:
942:
930:
928:
927:
922:
902:
900:
899:
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883:
867:
865:
864:
859:
857:
852:
845:
836:
832:
823:
821:
820:
815:
813:
802:
787:
785:
784:
779:
767:
765:
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749:
737:
714:
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676:
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654:
633:
627:
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616:
607:
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593:
592:
587:
575:
573:
572:
567:
549:
547:
546:
541:
538:
536:
527:
518:
491:
489:
488:
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465:
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457:
445:
443:
442:
437:
413:
411:
410:
405:
400:
399:
363:
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323:
321:
320:
317:{\displaystyle }
315:
291:
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134:
125:
113:
111:
110:
105:
57:Francis Anscombe
42:
40:
39:
34:
2151:
2150:
2146:
2145:
2144:
2142:
2141:
2140:
2116:
2115:
2080:
2078:Further reading
2075:
2074:
2032:
2028:
2021:
2003:
1999:
1967:
1963:
1921:
1917:
1886:10.1.1.219.6735
1865:
1861:
1843:
1836:
1810:Anscombe, F. J.
1807:
1800:
1795:
1778:
1769:
1738:
1722:
1718:
1717:
1696:
1682:
1671:
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1659:
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1654:
1641:
1630:
1627:
1626:
1598:
1581:
1578:
1577:
1545:
1529:
1515:
1512:
1511:
1504:
1476:
1472:
1460:
1450:
1438:
1434:
1424:
1412:
1408:
1396:
1386:
1373:
1364:
1360:
1350:
1342:
1339:
1338:
1296:
1286:
1282:
1276:
1272:
1271:
1269:
1252:
1244:
1243:
1239:
1230:
1219:
1184:
1176:
1172:
1168:
1160:
1157:
1156:
1129:
1120:
1106:
1102:
1101:
1093:
1090:
1089:
1066:
1063:
1062:
1034:
1025:
1011:
1007:
1006:
988:
984:
982:
979:
978:
956:
953:
952:
936:
933:
932:
931:an estimate of
916:
913:
912:
909:
881:
873:
870:
869:
834:
833:
831:
829:
826:
825:
801:
796:
793:
792:
773:
770:
769:
745:
741:
735:
731:
720:
717:
716:
688:
684:
680:
674:
670:
650:
646:
642:
637:
631:
614:
606:
601:
598:
597:
581:
578:
577:
561:
558:
557:
525:
517:
500:
497:
496:
471:
468:
467:
451:
448:
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392:
388:
354:
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343:
329:
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297:
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293:
261:
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227:
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213:
197:
194:
193:
168:
164:
160:
155:
149:
132:
124:
119:
116:
115:
99:
96:
95:
65:random variable
28:
25:
24:
17:
12:
11:
5:
2149:
2139:
2138:
2133:
2128:
2114:
2113:
2079:
2076:
2073:
2072:
2026:
2019:
1997:
1961:
1915:
1859:
1834:
1797:
1796:
1794:
1791:
1790:
1789:
1784:
1777:
1774:
1768:
1767:Generalization
1765:
1752:
1749:
1746:
1741:
1736:
1730:
1726:
1721:
1716:
1713:
1710:
1707:
1704:
1699:
1694:
1688:
1685:
1680:
1675:
1670:
1667:
1664:
1658:
1653:
1650:
1645:
1640:
1637:
1634:
1617:
1616:
1602:
1597:
1594:
1591:
1588:
1585:
1567:
1566:
1554:
1549:
1544:
1539:
1536:
1533:
1528:
1525:
1522:
1519:
1503:
1500:
1499:
1498:
1487:
1482:
1479:
1475:
1468:
1465:
1457:
1454:
1449:
1444:
1441:
1437:
1431:
1428:
1423:
1418:
1415:
1411:
1404:
1401:
1393:
1390:
1385:
1380:
1377:
1372:
1367:
1363:
1357:
1354:
1349:
1346:
1328:
1327:
1316:
1313:
1309:
1302:
1299:
1292:
1289:
1285:
1279:
1275:
1268:
1260:
1257:
1251:
1248:
1242:
1236:
1233:
1228:
1225:
1222:
1218:
1214:
1211:
1207:
1203:
1200:
1192:
1189:
1183:
1180:
1175:
1171:
1167:
1164:
1150:
1149:
1136:
1133:
1128:
1123:
1118:
1113:
1110:
1105:
1100:
1097:
1070:
1055:
1054:
1041:
1038:
1033:
1028:
1023:
1018:
1015:
1010:
1005:
1002:
999:
994:
991:
987:
960:
940:
920:
908:
905:
889:
886:
880:
877:
855:
851:
848:
842:
839:
811:
808:
805:
800:
777:
756:
748:
744:
740:
734:
730:
727:
724:
703:
695:
691:
687:
683:
679:
673:
669:
666:
657:
653:
649:
645:
640:
636:
630:
622:
619:
613:
610:
605:
585:
565:
551:
550:
533:
530:
524:
521:
516:
513:
510:
507:
504:
481:
478:
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455:
435:
419:
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403:
398:
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387:
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333:
313:
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307:
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281:
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234:
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175:
171:
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163:
158:
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148:
140:
137:
131:
128:
123:
103:
55:, named after
32:
15:
9:
6:
4:
3:
2:
2148:
2137:
2134:
2132:
2129:
2127:
2124:
2123:
2121:
2111:
2107:
2103:
2099:
2095:
2091:
2087:
2082:
2081:
2069:
2065:
2061:
2057:
2053:
2049:
2045:
2041:
2037:
2030:
2022:
2020:9780521599146
2016:
2011:
2010:
2001:
1994:
1990:
1985:
1980:
1976:
1972:
1965:
1958:
1954:
1950:
1946:
1942:
1938:
1934:
1930:
1926:
1919:
1912:
1908:
1904:
1900:
1896:
1892:
1887:
1882:
1878:
1874:
1870:
1863:
1856:
1852:
1848:
1841:
1839:
1831:
1827:
1823:
1819:
1815:
1811:
1805:
1803:
1798:
1788:
1785:
1783:
1780:
1779:
1773:
1764:
1750:
1747:
1744:
1739:
1734:
1728:
1724:
1719:
1714:
1708:
1702:
1697:
1692:
1686:
1683:
1673:
1668:
1662:
1656:
1651:
1643:
1638:
1632:
1624:
1622:
1600:
1595:
1589:
1586:
1583:
1576:
1575:
1574:
1572:
1552:
1547:
1542:
1537:
1534:
1531:
1523:
1520:
1517:
1510:
1509:
1508:
1485:
1480:
1477:
1473:
1466:
1463:
1455:
1452:
1447:
1442:
1439:
1435:
1429:
1426:
1421:
1416:
1413:
1409:
1402:
1399:
1391:
1388:
1383:
1378:
1375:
1370:
1365:
1361:
1355:
1352:
1344:
1337:
1336:
1335:
1333:
1314:
1307:
1300:
1297:
1290:
1287:
1283:
1277:
1273:
1266:
1258:
1255:
1249:
1246:
1240:
1231:
1226:
1223:
1220:
1216:
1212:
1209:
1205:
1201:
1198:
1190:
1187:
1181:
1178:
1173:
1169:
1165:
1155:
1154:
1153:
1134:
1131:
1126:
1121:
1116:
1111:
1108:
1103:
1095:
1088:
1087:
1086:
1084:
1068:
1060:
1039:
1036:
1031:
1026:
1021:
1016:
1013:
1008:
1000:
997:
992:
989:
985:
977:
976:
975:
974:
958:
938:
918:
904:
887:
884:
878:
875:
853:
849:
846:
840:
837:
809:
806:
803:
798:
789:
775:
754:
746:
742:
738:
732:
728:
725:
722:
701:
693:
689:
685:
681:
677:
671:
667:
664:
655:
651:
647:
643:
638:
634:
628:
620:
617:
611:
608:
603:
583:
563:
554:
531:
528:
522:
519:
514:
508:
505:
502:
495:
494:
493:
479:
476:
473:
453:
446:and variance
433:
425:
396:
393:
389:
382:
379:
376:
373:
370:
367:
359:
355:
351:
348:
344:
337:
334:
331:
308:
305:
302:
276:
273:
270:
262:
258:
237:
232:
228:
224:
220:
199:
173:
169:
165:
161:
156:
152:
146:
138:
135:
129:
126:
121:
101:
92:
88:
86:
82:
78:
74:
70:
66:
62:
58:
54:
50:
30:
21:
2085:
2035:
2029:
2008:
2000:
1974:
1971:Tukey, J. W.
1964:
1924:
1918:
1868:
1862:
1846:
1813:
1770:
1625:
1621:delta method
1618:
1568:
1505:
1502:Alternatives
1329:
1151:
1056:
910:
790:
555:
552:
421:
52:
46:
1332:closed-form
576:(with mean
2120:Categories
1847:Biometrika
1814:Biometrika
1793:References
418:Definition
49:statistics
2068:206724566
1881:CiteSeerX
1652:≈
1593:↦
1527:↦
1478:−
1440:−
1422:−
1414:−
1371:−
1348:↦
1312:↦
1288:−
1267:⋅
1235:∞
1217:∑
1199:∣
1166:
1127:−
1099:↦
1032:−
1004:↦
990:−
907:Inversion
847:−
629:−
512:↦
426:the mean
394:−
371:σ
349:−
332:μ
274:−
271:σ
238:μ
200:σ
147:−
102:μ
81:denoising
2110:13210703
2060:22692910
1949:21356615
1911:10229455
1903:20615809
1776:See also
422:For the
2090:Bibcode
2040:Bibcode
1993:2236611
1957:7937596
1929:Bibcode
1873:Bibcode
1830:2332343
67:with a
59:, is a
2108:
2066:
2058:
2017:
1991:
1955:
1947:
1909:
1901:
1883:
1828:
1083:linear
192:, and
51:, the
2106:S2CID
2064:S2CID
1989:JSTOR
1953:S2CID
1907:S2CID
1826:JSTOR
1573:, is
2056:PMID
2015:ISBN
1945:PMID
1899:PMID
1059:bias
250:and
2098:doi
2048:doi
1979:doi
1937:doi
1891:doi
1851:doi
1818:doi
47:In
2122::
2104:,
2096:,
2062:,
2054:,
2046:,
1987:,
1951:,
1943:,
1935:,
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1889:,
1879:,
1837:^
1824:,
1801:^
1763:.
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1427:11
309:10
2100::
2092::
2050::
2042::
2023:.
1981::
1939::
1931::
1893::
1875::
1853::
1820::
1751:1
1748:=
1745:m
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1735:)
1729:m
1725:1
1720:(
1715:=
1712:]
1709:x
1706:[
1703:V
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1663:d
1657:(
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1524:x
1521::
1518:A
1486:.
1481:3
1474:y
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1464:3
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1430:8
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1403:2
1400:3
1392:4
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1366:2
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1308:)
1301:!
1298:x
1291:m
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1224:=
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1210:=
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1202:m
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1009:(
1001:y
998::
993:1
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879:=
876:c
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850:c
841:8
838:3
810:c
807:+
804:x
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747:2
743:m
739:1
733:(
729:O
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690:/
686:3
682:m
678:1
672:(
668:O
665:+
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652:/
648:1
644:m
639:4
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621:8
618:3
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604:2
584:m
564:x
532:8
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515:2
509:x
506::
503:A
480:v
477:=
474:m
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397:2
390:m
386:(
383:O
380:+
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360:2
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341:(
338:O
335:=
312:]
306:,
303:0
300:[
280:)
277:1
268:(
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233:2
229:/
225:3
221:m
174:2
170:/
166:1
162:m
157:4
153:1
139:8
136:3
130:+
127:m
122:2
43:.
31:m
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