2485:
820:
177:
satisfies a similar relationship, the essential matrix is a metric object pertaining to calibrated cameras, while the fundamental matrix describes the correspondence in more general and fundamental terms of projective geometry. This is captured mathematically by the relationship between a fundamental
346:
The fundamental matrix is a relationship between any two images of the same scene that constrains where the projection of points from the scene can occur in both images. Given the projection of a scene point into one of the images the corresponding point in the other image is constrained to a line,
934:
595:
138:
Being of rank two and determined only up to scale, the fundamental matrix can be estimated given at least seven point correspondences. Its seven parameters represent the only geometric information about cameras that can be obtained through point correspondences alone.
999:). Therefore, there are multiple projection centers for one image scene and the epipolar line is formed as an epipolar curve. However, in special conditions such as small image tiles, the satellite images could be rectified using the fundamental matrix.
285:
698:
828:
509:
501:
428:
687:
133:
703:
514:
639:
968:
336:
452:
309:
220:
198:
46:
1534:
815:{\displaystyle {\begin{aligned}{\textbf {P}}_{0}&={\textbf {P}}{\textbf {H}}^{-1}\\{\textbf {P}}_{0}'&={\textbf {P}}'{\textbf {H}}^{-1}\end{aligned}}}
228:
383:
to world points with the help of camera matrices derived directly from this fundamental matrix. The scene composed of these world points is within a
2143:
929:{\displaystyle {\textbf {P}}_{0}{\textbf {X}}_{0}={\textbf {P}}{\textbf {H}}^{-1}{\textbf {H}}{\textbf {X}}={\textbf {P}}{\textbf {X}}=\mathbf {x} }
1326:, Reinhard Koch and Luc van Gool (1999). "Self-Calibration and Metric Reconstruction in spite of Varying and Unknown Intrinsic Camera Parameters".
590:{\displaystyle {\begin{aligned}\mathbf {x} &={\textbf {P}}{\textbf {X}}\\\mathbf {x'} &={\textbf {P}}'{\textbf {X}}\end{aligned}}}
1079:
2357:
1576:
1502:
644:
2448:
457:
398:
1430:
1311:
1221:
1182:
2521:
2133:
1388:
Philip H. S. Torr and A. Zisserman (2000). "MLESAC: A New Robust
Estimator with Application to Estimating Image Geometry".
1254:
Nurollah Tatar (2019). "Stereo rectification of pushbroom satellite images by robustly estimating the fundamental matrix".
90:
1496:
1357:
Philip H. S. Torr (1997). "The
Development and Comparison of Robust Methods for Estimating the Fundamental Matrix".
1077:"Novel Approach to Epipolar Resampling of HRSI and Satellite Stereo Imagery-based Georeferencing of Aerial Images"
2168:
1142:
Q.T. Luong and
Olivier D. Faugeras (1996). "The Fundamental Matrix: Theory, Algorithms, and Stability Analysis".
1063:
1715:
1016:
607:
170:
1499:) fundamental matrix estimation from matched point pairs and various objective functions (Manolis Lourakis).
1932:
1569:
1121:
Olivier D. Faugeras; Q.T. Luong; Steven
Maybank (1992). "Camera self-calibration: Theory and experiments".
601:
2007:
1542:
166:
939:
2163:
1685:
2267:
2138:
2052:
1195:
1100:
Olivier D. Faugeras (1992). "What can be seen in three dimensions with an uncalibrated stereo rig?".
384:
1402:
1110:
2372:
2262:
1970:
1650:
1484:
433:
347:
helping the search, and allowing for the detection of wrong correspondences. The relation between
314:
292:
203:
181:
29:
2407:
2336:
2218:
2078:
1675:
1562:
61:
1524:
2277:
1860:
1665:
1397:
1105:
376:
1529:
2223:
1960:
1810:
1805:
1640:
1615:
1610:
1043:
1008:
2417:
1775:
1605:
1585:
348:
49:
1442:
Zhengyou Zhang (1998). "Determining the epipolar geometry and its uncertainty: A review".
8:
2438:
2412:
1990:
1795:
1785:
1076:
84:′ on the other image must lie. That means, for all pairs of corresponding points holds
2489:
2443:
2433:
2387:
2382:
2311:
2247:
2113:
1850:
1845:
1780:
1770:
1635:
1459:
1374:
1343:
1159:
161:
The above relation which defines the fundamental matrix was published in 1992 by both
2500:
2484:
2287:
2282:
2272:
2252:
2213:
2208:
2037:
2032:
2017:
2012:
2003:
1998:
1945:
1840:
1790:
1735:
1705:
1700:
1680:
1670:
1630:
1492:
1426:
1307:
1299:
1295:
1217:
1178:
1028:
988:
57:
1378:
2495:
2463:
2392:
2331:
2326:
2306:
2242:
2148:
2118:
2103:
2083:
2022:
1975:
1950:
1940:
1911:
1830:
1825:
1800:
1730:
1710:
1620:
1600:
1525:
Epipolar
Geometry and the Fundamental Matrix (chapter from Hartley & Zisserman)
1463:
1451:
1407:
1366:
1335:
1263:
1242:
1163:
1151:
1126:
1033:
996:
992:
974:
174:
162:
151:
2088:
1347:
1267:
280:{\displaystyle \mathbf {E} =({\mathbf {K} '})^{\top }\;\mathbf {F} \;\mathbf {K} }
2193:
2128:
2108:
2093:
2073:
2057:
1955:
1886:
1876:
1835:
1720:
1690:
1083:
1038:
1012:
991:
in images taken with perspective cameras appears as straight lines. However, in
20:
1530:
Determining the epipolar geometry and its uncertainty: A review (Zhengyou Zhang)
1289:
2453:
2397:
2377:
2362:
2321:
2198:
2158:
2123:
2047:
1986:
1965:
1906:
1896:
1881:
1815:
1760:
1750:
1745:
1655:
1477:
1323:
1291:
1455:
1370:
1339:
1322:
2515:
2458:
2316:
2257:
2188:
2178:
2173:
2098:
2027:
1901:
1891:
1820:
1740:
1725:
1660:
1131:
987:
The fundamental matrix expresses the epipolar geometry in stereo images. The
979:
The fundamental matrix can also be derived using the coplanarity condition.
155:
77:
2341:
2298:
2203:
1916:
1855:
1765:
1645:
1507:
1411:
1120:
2183:
2153:
1921:
1755:
1625:
53:
146:
in his influential PhD thesis. It is sometimes also referred to as the "
2234:
1695:
1155:
1246:
1233:
Richard I. Hartley (1997). "In
Defense of the Eight-Point Algorithm".
2468:
2042:
1278:
338:
being the intrinsic calibration matrices of the two images involved.
1211:
2402:
143:
1548:
1387:
1196:"Estimation of relative camera positions for uncalibrated cameras"
1141:
995:, the image is formed during the sensor movement along its orbit (
1554:
1280:
Matrice fondamentale et auto-calibration en vision par ordinateur
975:
Derivation of the fundamental matrix using coplanarity condition
1172:
1235:
IEEE Transactions on
Pattern Analysis and Machine Intelligence
1538:
1488:
351:, which the fundamental matrix represents, is referred to as
682:{\displaystyle {\textbf {X}}_{0}={\textbf {H}}{\textbf {X}}}
1512:
1503:
Structure and Motion
Toolkit in MATLAB (Philip H. S. Torr)
1423:
Epipolar geometry in Stereo, Motion and Object
Recognition
1420:
1481:
496:{\displaystyle \left({\textbf {P}},{\textbf {P}}'\right)}
423:{\displaystyle \mathbf {x} \leftrightarrow \mathbf {x'} }
379:. Additionally, these corresponding image points may be
1508:
Fundamental Matrix
Estimation Toolbox (Joaquim Salvi)
1203:
Proceedings of European Conference on Computer Vision
1123:
Proceedings of European Conference on Computer Vision
1102:
Proceedings of European Conference on Computer Vision
942:
831:
701:
647:
610:
512:
460:
436:
401:
375:
The fundamental matrix can be determined by a set of
317:
295:
231:
206:
184:
128:{\displaystyle \mathbf {x} '^{\top }\mathbf {Fx} =0.}
93:
32:
370:
1099:
72:′, of corresponding points in a stereo image pair,
1232:
1193:
962:
928:
814:
681:
633:
589:
495:
446:
422:
330:
303:
279:
214:
192:
127:
40:
1356:
2513:
158:relating points in distinct coordinate systems.
1551:Video demonstrating laws of epipolar geometry.
1441:
1253:
1570:
1212:Richard Hartley and Andrew Zisserman (2003).
142:The term "fundamental matrix" was coined by
2144:Fundamental (linear differential equation)
1577:
1563:
1276:
271:
265:
1401:
1283:. PhD Thesis, University of Paris, Orsay.
1214:Multiple View Geometry in Computer Vision
1130:
1109:
1064:Multiple View Geometry in Computer Vision
634:{\displaystyle {\textbf {H}}_{4\times 4}}
1444:International Journal of Computer Vision
1359:International Journal of Computer Vision
1328:International Journal of Computer Vision
1173:Olivier Faugeras and Q.T. Luong (2001).
1144:International Journal of Computer Vision
395:Say that the image point correspondence
2449:Matrix representation of conic sections
1390:Computer Vision and Image Understanding
1256:International Journal of Remote Sensing
982:
200:and its corresponding essential matrix
2514:
1086:, 2011, pp. 22–29 accessed 2011-08-05.
1062:Richard Hartley and Andrew Zisserman "
52:which relates corresponding points in
1558:
600:Say we transform space by a general
1537:(originally by Sylvain Bougnoux of
1513:The Epipolar Geometry Toolbox (EGT)
1421:Gang Xu and Zhengyou Zhang (1996).
970:still get us the same image points.
946:
913:
906:
896:
889:
873:
865:
849:
835:
794:
782:
758:
737:
729:
709:
674:
667:
651:
614:
578:
567:
538:
531:
479:
468:
439:
80:) on which the corresponding point
13:
1584:
1535:Visualization of epipolar geometry
963:{\displaystyle {\textbf {P}}_{0}'}
260:
105:
14:
2533:
1518:
371:Projective reconstruction theorem
2483:
922:
549:
518:
412:
403:
320:
297:
273:
267:
246:
233:
208:
186:
115:
112:
96:
34:
2351:Used in science and engineering
1175:The Geometry of Multiple Images
341:
1594:Explicitly constrained entries
1425:. Kluwer Academic Publishers.
1216:. Cambridge University Press.
1069:
1056:
692:The cameras then transform as
407:
256:
240:
171:H. Christopher Longuet-Higgins
1:
2368:Fundamental (computer vision)
1497:Levenberg–Marquardt algorithm
1268:10.1080/01431161.2019.1624862
1093:
1007:The fundamental matrix is of
1002:
447:{\displaystyle {\textbf {X}}}
430:derives from the world point
331:{\displaystyle \mathbf {K} '}
62:homogeneous image coordinates
1471:
361:discrete matching constraint
304:{\displaystyle \mathbf {K} }
215:{\displaystyle \mathbf {E} }
193:{\displaystyle \mathbf {F} }
41:{\displaystyle \mathbf {F} }
7:
2522:Geometry in computer vision
2134:Duplication and elimination
1933:eigenvalues or eigenvectors
1549:The Fundamental Matrix Song
1495:, non-linear (based on the
1304:An Invitation to 3-D Vision
1194:Richard I. Hartley (1992).
1022:
10:
2538:
2067:With specific applications
1696:Discrete Fourier Transform
454:under the camera matrices
16:Matrix in computer version
2477:
2426:
2358:Cabibbo–Kobayashi–Maskawa
2350:
2296:
2232:
2066:
1985:Satisfying conditions on
1984:
1930:
1869:
1593:
385:projective transformation
1132:10.1007/3-540-55426-2_37
1049:
390:
1716:Generalized permutation
1456:10.1023/A:1007941100561
1371:10.1023/A:1007927408552
1340:10.1023/A:1008109111715
150:". As a tensor it is a
2490:Mathematics portal
1412:10.1006/cviu.1999.0832
964:
930:
816:
683:
635:
591:
497:
448:
424:
332:
305:
281:
216:
194:
129:
42:
1044:Eight-point algorithm
965:
931:
817:
684:
636:
592:
498:
449:
425:
377:point correspondences
333:
306:
282:
217:
195:
130:
76:describes a line (an
43:
983:For satellite images
940:
829:
699:
645:
608:
510:
458:
434:
399:
349:corresponding points
315:
293:
229:
204:
182:
91:
30:
2439:Linear independence
1686:Diagonally dominant
1541:Robotvis, requires
1277:Q.T. Luong (1992).
1066:" 2003, pp. 266–267
959:
771:
387:of the true scene.
357:matching constraint
353:epipolar constraint
2444:Matrix exponential
2434:Jordan normal form
2268:Fisher information
2139:Euclidean distance
2053:Totally unimodular
1156:10.1007/BF00127818
1082:2012-03-31 at the
960:
943:
936:and likewise with
926:
812:
810:
755:
679:
631:
587:
585:
493:
444:
420:
365:incidence relation
328:
301:
277:
212:
190:
125:
38:
25:fundamental matrix
2509:
2508:
2501:Category:Matrices
2373:Fuzzy associative
2263:Doubly stochastic
1971:Positive-definite
1651:Block tridiagonal
1432:978-0-7923-4199-4
1313:978-0-387-00893-6
1300:S. Shankar Sastry
1247:10.1109/34.601246
1223:978-0-521-54051-3
1184:978-0-262-06220-6
1029:Epipolar geometry
989:epipolar geometry
948:
915:
908:
898:
891:
875:
867:
851:
837:
796:
784:
760:
739:
731:
711:
676:
669:
653:
616:
580:
569:
540:
533:
481:
470:
441:
58:epipolar geometry
2529:
2496:List of matrices
2488:
2487:
2464:Row echelon form
2408:State transition
2337:Seidel adjacency
2219:Totally positive
2079:Alternating sign
1676:Complex Hadamard
1579:
1572:
1565:
1556:
1555:
1467:
1436:
1415:
1405:
1382:
1351:
1317:
1284:
1271:
1250:
1227:
1206:
1200:
1188:
1167:
1136:
1134:
1115:
1113:
1087:
1073:
1067:
1060:
1034:Essential matrix
997:pushbroom sensor
993:satellite images
969:
967:
966:
961:
955:
950:
949:
935:
933:
932:
927:
925:
917:
916:
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852:
845:
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175:essential matrix
163:Olivier Faugeras
154:in that it is a
152:two-point tensor
134:
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118:
110:
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99:
47:
45:
44:
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37:
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2505:
2482:
2473:
2422:
2346:
2292:
2228:
2062:
1980:
1926:
1865:
1666:Centrosymmetric
1589:
1583:
1521:
1474:
1433:
1403:10.1.1.110.5740
1314:
1224:
1198:
1185:
1111:10.1.1.462.4708
1096:
1091:
1090:
1084:Wayback Machine
1074:
1070:
1061:
1057:
1052:
1039:Trifocal tensor
1025:
1005:
985:
977:
951:
945:
944:
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938:
937:
921:
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905:
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619:
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207:
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185:
183:
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167:Richard Hartley
111:
104:
100:
95:
94:
92:
89:
88:
33:
31:
28:
27:
21:computer vision
17:
12:
11:
5:
2535:
2525:
2524:
2507:
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2504:
2503:
2498:
2493:
2478:
2475:
2474:
2472:
2471:
2466:
2461:
2456:
2454:Perfect matrix
2451:
2446:
2441:
2436:
2430:
2428:
2424:
2423:
2421:
2420:
2415:
2410:
2405:
2400:
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2314:
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2275:
2270:
2265:
2260:
2255:
2250:
2245:
2239:
2237:
2230:
2229:
2227:
2226:
2224:Transformation
2221:
2216:
2211:
2206:
2201:
2196:
2191:
2186:
2181:
2176:
2171:
2166:
2161:
2156:
2151:
2146:
2141:
2136:
2131:
2126:
2121:
2116:
2111:
2106:
2101:
2096:
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2068:
2064:
2063:
2061:
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2055:
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2035:
2030:
2025:
2020:
2015:
2010:
2001:
1995:
1993:
1982:
1981:
1979:
1978:
1973:
1968:
1963:
1961:Diagonalizable
1958:
1953:
1948:
1943:
1937:
1935:
1931:Conditions on
1928:
1927:
1925:
1924:
1919:
1914:
1909:
1904:
1899:
1894:
1889:
1884:
1879:
1873:
1871:
1867:
1866:
1864:
1863:
1858:
1853:
1848:
1843:
1838:
1833:
1828:
1823:
1818:
1813:
1811:Skew-symmetric
1808:
1806:Skew-Hermitian
1803:
1798:
1793:
1788:
1783:
1778:
1773:
1768:
1763:
1758:
1753:
1748:
1743:
1738:
1733:
1728:
1723:
1718:
1713:
1708:
1703:
1698:
1693:
1688:
1683:
1678:
1673:
1668:
1663:
1658:
1653:
1648:
1643:
1641:Block-diagonal
1638:
1633:
1628:
1623:
1618:
1616:Anti-symmetric
1613:
1611:Anti-Hermitian
1608:
1603:
1597:
1595:
1591:
1590:
1582:
1581:
1574:
1567:
1559:
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1520:
1519:External links
1517:
1516:
1515:
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1500:
1473:
1470:
1469:
1468:
1450:(2): 161–195.
1438:
1437:
1431:
1417:
1416:
1396:(1): 138–156.
1384:
1383:
1365:(3): 271–300.
1353:
1352:
1324:Marc Pollefeys
1319:
1318:
1312:
1292:Stefano Soatto
1286:
1285:
1273:
1272:
1251:
1241:(6): 580–593.
1229:
1228:
1222:
1208:
1207:
1190:
1189:
1183:
1169:
1168:
1138:
1137:
1117:
1116:
1095:
1092:
1089:
1088:
1068:
1054:
1053:
1051:
1048:
1047:
1046:
1041:
1036:
1031:
1024:
1021:
1004:
1001:
984:
981:
976:
973:
972:
971:
958:
954:
924:
920:
903:
884:
881:
862:
857:
843:
823:
822:
805:
802:
789:
778:
775:
773:
770:
766:
754:
753:
748:
745:
726:
723:
721:
717:
705:
704:
664:
659:
628:
625:
622:
598:
597:
574:
563:
560:
558:
554:
551:
546:
545:
528:
525:
523:
520:
516:
515:
491:
486:
475:
464:
417:
414:
409:
405:
392:
389:
372:
369:
343:
340:
326:
322:
299:
288:
287:
275:
269:
262:
258:
252:
248:
242:
239:
235:
210:
188:
148:bifocal tensor
136:
135:
124:
121:
117:
114:
107:
103:
98:
36:
15:
9:
6:
4:
3:
2:
2534:
2523:
2520:
2519:
2517:
2502:
2499:
2497:
2494:
2492:
2491:
2486:
2480:
2479:
2476:
2470:
2467:
2465:
2462:
2460:
2459:Pseudoinverse
2457:
2455:
2452:
2450:
2447:
2445:
2442:
2440:
2437:
2435:
2432:
2431:
2429:
2427:Related terms
2425:
2419:
2418:Z (chemistry)
2416:
2414:
2411:
2409:
2406:
2404:
2401:
2399:
2396:
2394:
2391:
2389:
2386:
2384:
2381:
2379:
2376:
2374:
2371:
2369:
2366:
2364:
2361:
2359:
2356:
2355:
2353:
2349:
2343:
2340:
2338:
2335:
2333:
2330:
2328:
2325:
2323:
2320:
2318:
2315:
2313:
2310:
2308:
2305:
2304:
2302:
2300:
2295:
2289:
2286:
2284:
2281:
2279:
2276:
2274:
2271:
2269:
2266:
2264:
2261:
2259:
2256:
2254:
2251:
2249:
2246:
2244:
2241:
2240:
2238:
2236:
2231:
2225:
2222:
2220:
2217:
2215:
2212:
2210:
2207:
2205:
2202:
2200:
2197:
2195:
2192:
2190:
2187:
2185:
2182:
2180:
2177:
2175:
2172:
2170:
2167:
2165:
2162:
2160:
2157:
2155:
2152:
2150:
2147:
2145:
2142:
2140:
2137:
2135:
2132:
2130:
2127:
2125:
2122:
2120:
2117:
2115:
2112:
2110:
2107:
2105:
2102:
2100:
2097:
2095:
2092:
2090:
2087:
2085:
2082:
2080:
2077:
2075:
2072:
2071:
2069:
2065:
2059:
2056:
2054:
2051:
2049:
2046:
2044:
2041:
2039:
2036:
2034:
2031:
2029:
2026:
2024:
2021:
2019:
2016:
2014:
2011:
2009:
2005:
2002:
2000:
1997:
1996:
1994:
1992:
1988:
1983:
1977:
1974:
1972:
1969:
1967:
1964:
1962:
1959:
1957:
1954:
1952:
1949:
1947:
1944:
1942:
1939:
1938:
1936:
1934:
1929:
1923:
1920:
1918:
1915:
1913:
1910:
1908:
1905:
1903:
1900:
1898:
1895:
1893:
1890:
1888:
1885:
1883:
1880:
1878:
1875:
1874:
1872:
1868:
1862:
1859:
1857:
1854:
1852:
1849:
1847:
1844:
1842:
1839:
1837:
1834:
1832:
1829:
1827:
1824:
1822:
1819:
1817:
1814:
1812:
1809:
1807:
1804:
1802:
1799:
1797:
1794:
1792:
1789:
1787:
1784:
1782:
1779:
1777:
1776:Pentadiagonal
1774:
1772:
1769:
1767:
1764:
1762:
1759:
1757:
1754:
1752:
1749:
1747:
1744:
1742:
1739:
1737:
1734:
1732:
1729:
1727:
1724:
1722:
1719:
1717:
1714:
1712:
1709:
1707:
1704:
1702:
1699:
1697:
1694:
1692:
1689:
1687:
1684:
1682:
1679:
1677:
1674:
1672:
1669:
1667:
1664:
1662:
1659:
1657:
1654:
1652:
1649:
1647:
1644:
1642:
1639:
1637:
1634:
1632:
1629:
1627:
1624:
1622:
1619:
1617:
1614:
1612:
1609:
1607:
1606:Anti-diagonal
1604:
1602:
1599:
1598:
1596:
1592:
1587:
1580:
1575:
1573:
1568:
1566:
1561:
1560:
1557:
1550:
1547:
1544:
1540:
1536:
1533:
1531:
1528:
1526:
1523:
1522:
1514:
1511:
1509:
1506:
1504:
1501:
1498:
1494:
1490:
1486:
1483:
1479:
1476:
1475:
1465:
1461:
1457:
1453:
1449:
1445:
1440:
1439:
1434:
1428:
1424:
1419:
1418:
1413:
1409:
1404:
1399:
1395:
1391:
1386:
1385:
1380:
1376:
1372:
1368:
1364:
1360:
1355:
1354:
1349:
1345:
1341:
1337:
1333:
1329:
1325:
1321:
1320:
1315:
1309:
1305:
1301:
1297:
1293:
1288:
1287:
1282:
1281:
1275:
1274:
1269:
1265:
1261:
1257:
1252:
1248:
1244:
1240:
1236:
1231:
1230:
1225:
1219:
1215:
1210:
1209:
1204:
1197:
1192:
1191:
1186:
1180:
1177:. MIT Press.
1176:
1171:
1170:
1165:
1161:
1157:
1153:
1149:
1145:
1140:
1139:
1133:
1128:
1124:
1119:
1118:
1112:
1107:
1103:
1098:
1097:
1085:
1081:
1078:
1072:
1065:
1059:
1055:
1045:
1042:
1040:
1037:
1035:
1032:
1030:
1027:
1026:
1020:
1018:
1014:
1010:
1000:
998:
994:
990:
980:
956:
952:
918:
901:
882:
879:
860:
855:
841:
825:
824:
803:
800:
787:
776:
774:
768:
764:
746:
743:
724:
722:
715:
695:
694:
693:
690:
662:
657:
626:
623:
620:
603:
572:
561:
559:
552:
526:
524:
506:
505:
504:
489:
484:
473:
462:
415:
388:
386:
382:
378:
368:
366:
362:
358:
354:
350:
339:
324:
250:
237:
225:
224:
223:
176:
172:
168:
164:
159:
157:
156:bilinear form
153:
149:
145:
140:
122:
119:
101:
87:
86:
85:
83:
79:
78:epipolar line
75:
71:
67:
63:
59:
55:
54:stereo images
51:
26:
22:
2481:
2413:Substitution
2367:
2299:graph theory
1796:Quaternionic
1786:Persymmetric
1491:library for
1447:
1443:
1422:
1393:
1389:
1362:
1358:
1331:
1327:
1306:. Springer.
1303:
1296:Jana Košecká
1279:
1262:(20): 1–19.
1259:
1255:
1238:
1234:
1213:
1202:
1174:
1150:(1): 43–75.
1147:
1143:
1122:
1101:
1075:Jaehong Oh.
1071:
1058:
1015:defines the
1006:
986:
978:
691:
599:
394:
381:triangulated
380:
374:
364:
360:
356:
352:
345:
342:Introduction
289:
160:
147:
141:
137:
81:
73:
69:
65:
24:
18:
2388:Hamiltonian
2312:Biadjacency
2248:Correlation
2164:Householder
2114:Commutation
1851:Vandermonde
1846:Tridiagonal
1781:Permutation
1771:Nonnegative
1756:Matrix unit
1636:Bisymmetric
1334:(1): 7–25.
222:, which is
169:. Although
2288:Transition
2283:Stochastic
2253:Covariance
2235:statistics
2214:Symplectic
2209:Similarity
2038:Unimodular
2033:Orthogonal
2018:Involutory
2013:Invertible
2008:Projection
2004:Idempotent
1946:Convergent
1841:Triangular
1791:Polynomial
1736:Hessenberg
1706:Equivalent
1701:Elementary
1681:Copositive
1671:Conference
1631:Bidiagonal
1094:References
1003:Properties
641:such that
602:homography
2469:Wronskian
2393:Irregular
2383:Gell-Mann
2332:Laplacian
2327:Incidence
2307:Adjacency
2278:Precision
2243:Centering
2149:Generator
2119:Confusion
2104:Circulant
2084:Augmented
2043:Unipotent
2023:Nilpotent
1999:Congruent
1976:Stieltjes
1951:Defective
1941:Companion
1912:Redheffer
1831:Symmetric
1826:Sylvester
1801:Signature
1731:Hermitian
1711:Frobenius
1621:Arrowhead
1601:Alternant
1472:Toolboxes
1398:CiteSeerX
1106:CiteSeerX
880:−
801:−
744:−
624:×
408:↔
261:⊤
106:⊤
48:is a 3Ă—3
2516:Category
2297:Used in
2233:Used in
2194:Rotation
2169:Jacobian
2129:Distance
2109:Cofactor
2094:Carleman
2074:Adjugate
2058:Weighing
1991:inverses
1987:products
1956:Definite
1887:Identity
1877:Exchange
1870:Constant
1836:Toeplitz
1721:Hadamard
1691:Diagonal
1379:12031059
1302:(2004).
1080:Archived
1023:See also
957:′
788:′
769:′
573:′
553:′
485:′
416:′
325:′
251:′
144:QT Luong
102:′
2398:Overlap
2363:Density
2322:Edmonds
2199:Seifert
2159:Hessian
2124:Coxeter
2048:Unitary
1966:Hurwitz
1897:Of ones
1882:Hilbert
1816:Skyline
1761:Metzler
1751:Logical
1746:Integer
1656:Boolean
1588:classes
1478:fundest
1464:3190498
1290:Yi Ma;
1164:2582003
1017:epipole
1011:2. Its
604:matrix
178:matrix
60:, with
2317:Degree
2258:Design
2189:Random
2179:Payoff
2174:Moment
2099:Cartan
2089:BĂ©zout
2028:Normal
1902:Pascal
1892:Lehmer
1821:Sparse
1741:Hollow
1726:Hankel
1661:Cauchy
1586:Matrix
1493:robust
1462:
1429:
1400:
1377:
1348:306722
1346:
1310:
1220:
1181:
1162:
1108:
1013:kernel
56:. In
50:matrix
23:, the
2378:Gamma
2342:Tutte
2204:Shear
1917:Shift
1907:Pauli
1856:Walsh
1766:Moore
1646:Block
1539:INRIA
1480:is a
1460:S2CID
1375:S2CID
1344:S2CID
1199:(PDF)
1160:S2CID
1050:Notes
391:Proof
363:, or
2184:Pick
2154:Gram
1922:Zero
1626:Band
1543:Java
1427:ISBN
1308:ISBN
1218:ISBN
1179:ISBN
1009:rank
311:and
165:and
68:and
2273:Hat
2006:or
1989:or
1489:C++
1482:GPL
1452:doi
1408:doi
1367:doi
1336:doi
1264:doi
1243:doi
1152:doi
1127:doi
503:as
19:In
2518::
1458:.
1448:27
1446:.
1406:.
1394:78
1392:.
1373:.
1363:24
1361:.
1342:.
1332:32
1330:.
1298:;
1294:;
1260:40
1258:.
1239:19
1237:.
1201:.
1158:.
1148:17
1146:.
1125:.
1104:.
1019:.
689:.
367:.
359:,
355:,
173:'
123:0.
74:Fx
64:,
2403:S
1861:Z
1578:e
1571:t
1564:v
1545:)
1487:/
1485:C
1466:.
1454::
1435:.
1414:.
1410::
1381:.
1369::
1350:.
1338::
1316:.
1270:.
1266::
1249:.
1245::
1226:.
1205:.
1187:.
1166:.
1154::
1135:.
1129::
1114:.
953:0
947:P
923:x
919:=
914:X
907:P
902:=
897:X
890:H
883:1
874:H
866:P
861:=
856:0
850:X
842:0
836:P
804:1
795:H
783:P
777:=
765:0
759:P
747:1
738:H
730:P
725:=
716:0
710:P
675:X
668:H
663:=
658:0
652:X
627:4
621:4
615:H
579:X
568:P
562:=
550:x
539:X
532:P
527:=
519:x
490:)
480:P
474:,
469:P
463:(
440:X
413:x
404:x
321:K
298:K
274:K
268:F
257:)
247:K
241:(
238:=
234:E
209:E
187:F
120:=
116:x
113:F
97:x
82:x
70:x
66:x
35:F
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