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and may be solved for each pixel in the image. However, since the solution depends on the neighboring values of the flow field, it must be repeated once the neighbors have been updated. The following iterative scheme is derived using
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708:{\displaystyle {\frac {\partial L}{\partial v}}-{\frac {\partial }{\partial x}}{\frac {\partial L}{\partial v_{x}}}-{\frac {\partial }{\partial y}}{\frac {\partial L}{\partial v_{y}}}=0}
574:{\displaystyle {\frac {\partial L}{\partial u}}-{\frac {\partial }{\partial x}}{\frac {\partial L}{\partial u_{x}}}-{\frac {\partial }{\partial y}}{\frac {\partial L}{\partial u_{y}}}=0}
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The Horn-Schunck algorithm assumes smoothness in the flow over the whole image. Thus, it tries to minimize distortions in flow and prefers solutions which show more smoothness.
948:
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Advantages of the Horn–Schunck algorithm include that it yields a high density of flow vectors, i.e. the flow information missing in inner parts of homogeneous objects is
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1881:{\displaystyle v^{k+1}={\overline {v}}^{k}-{\frac {I_{y}(I_{x}{\overline {u}}^{k}+I_{y}{\overline {v}}^{k}+I_{t})}{\alpha ^{2}+I_{x}^{2}+I_{y}^{2}}}}
1684:{\displaystyle u^{k+1}={\overline {u}}^{k}-{\frac {I_{x}(I_{x}{\overline {u}}^{k}+I_{y}{\overline {v}}^{k}+I_{t})}{\alpha ^{2}+I_{x}^{2}+I_{y}^{2}}}}
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calculated in a neighborhood around the pixel at location (x,y). Using this notation the above equation system may be written
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1022:{\displaystyle \Delta ={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}}
1033:. In practice the Laplacian is approximated numerically using finite differences, and may be written
1919:
from the motion boundaries. On the negative side, it is more sensitive to noise than local methods.
313:
are the derivatives of the image intensity values along the x, y and time dimensions respectively,
53:
which is then sought to be minimized. This function is given for two-dimensional image streams as:
50:
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1907:, applied to the large, sparse system arising when solving for all pixels simultaneously.
1437:{\displaystyle I_{x}I_{y}u+(I_{y}^{2}+\alpha ^{2})v=\alpha ^{2}{\overline {v}}-I_{y}I_{t}}
1310:{\displaystyle (I_{x}^{2}+\alpha ^{2})u+I_{x}I_{y}v=\alpha ^{2}{\overline {u}}-I_{x}I_{t}}
8:
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lead to a smoother flow. This functional can be minimized by solving the associated
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935:{\displaystyle I_{y}(I_{x}u+I_{y}v+I_{t})-\alpha ^{2}\Delta v=0}
834:{\displaystyle I_{x}(I_{x}u+I_{y}v+I_{t})-\alpha ^{2}\Delta u=0}
1119:{\displaystyle \Delta u(x,y)=({\overline {u}}(x,y)-u(x,y))}
24:
is a global method which introduces a global constraint of
1940:
B.K.P. Horn and B.G. Schunck, "Determining optical flow."
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denotes the next iteration, which is to be calculated and
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where subscripts again denote partial differentiation and
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is the last calculated result. This is in essence a
222:{\displaystyle E=\iint \left{{\rm {d}}x{\rm {d}}y}}
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443:multi-dimensional Euler–Lagrange equations
49:The flow is formulated as a global energy
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1161:{\displaystyle {\overline {u}}(x,y)}
390:{\displaystyle {\vec {V}}=^{\top }}
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1944:, vol 17, pp 185–203, 1981.
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38:for further description).
1974:Motion in computer vision
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1948:available on MIT server.
1942:Artificial Intelligence
1903:method, similar to the
434:{\displaystyle \alpha }
414:{\displaystyle \alpha }
1891:where the superscript
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1959:OpenCV implementation
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42:Mathematical details
1929:Lucas–Kanade method
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1447:which is linear in
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18:Horn–Schunck method
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1901:Matrix splitting
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85:
81:
73:
69:
66:
63:
56:
55:
54:
52:
47:
39:
37:
33:
32:
28:to solve the
27:
23:
19:
1941:
1916:
1914:
1896:
1892:
1890:
1446:
1029:denotes the
944:
717:
445:. These are
398:
231:
48:
45:
36:Optical Flow
29:
25:
22:optical flow
17:
15:
1946:Manuscript
1935:References
1911:Properties
51:functional
26:smoothness
1917:filled in
1831:α
1801:¯
1771:¯
1736:−
1725:¯
1634:α
1604:¯
1574:¯
1539:−
1528:¯
1412:−
1407:¯
1393:α
1374:α
1285:−
1280:¯
1266:α
1221:α
1139:¯
1093:−
1073:¯
1041:Δ
1004:∂
994:∂
972:∂
962:∂
953:Δ
921:Δ
912:α
908:−
820:Δ
811:α
807:−
684:∂
676:∂
664:∂
660:∂
655:−
639:∂
631:∂
619:∂
615:∂
610:−
601:∂
593:∂
550:∂
542:∂
530:∂
526:∂
521:−
505:∂
497:∂
485:∂
481:∂
476:−
467:∂
459:∂
429:α
409:α
383:⊤
327:→
181:‖
174:∇
171:‖
159:‖
152:∇
149:‖
137:α
70:∬
1968:Category
1923:See also
1126:where
718:where
232:where
34:(see
1467:and
286:and
16:The
1893:k+1
399:for
1970::
1492::
259:,
1897:k
1871:2
1866:y
1862:I
1858:+
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1848:x
1844:I
1840:+
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1820:t
1816:I
1812:+
1807:k
1798:v
1790:y
1786:I
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1777:k
1768:u
1760:x
1756:I
1752:(
1747:y
1743:I
1731:k
1722:v
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1711:1
1708:+
1705:k
1701:v
1674:2
1669:y
1665:I
1661:+
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1651:x
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1623:t
1619:I
1615:+
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1601:v
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1550:x
1546:I
1534:k
1525:u
1519:=
1514:1
1511:+
1508:k
1504:u
1475:v
1455:u
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1420:y
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1386:v
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1378:2
1370:+
1365:2
1360:y
1356:I
1352:(
1349:+
1346:u
1341:y
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1331:x
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1303:t
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1259:v
1254:y
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1225:2
1217:+
1212:2
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1108:y
1105:,
1102:x
1099:(
1096:u
1090:)
1087:y
1084:,
1081:x
1078:(
1070:u
1065:(
1062:=
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1050:x
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1012:2
1008:y
998:2
988:+
980:2
976:x
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956:=
930:0
927:=
924:v
916:2
905:)
900:t
896:I
892:+
889:v
884:y
880:I
876:+
873:u
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864:I
860:(
855:y
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829:0
826:=
823:u
815:2
804:)
799:t
795:I
791:+
788:v
783:y
779:I
775:+
772:u
767:x
763:I
759:(
754:x
750:I
726:L
703:0
700:=
692:y
688:v
679:L
667:y
647:x
643:v
634:L
622:x
604:v
596:L
569:0
566:=
558:y
554:u
545:L
533:y
513:x
509:u
500:L
488:x
470:u
462:L
379:]
375:)
372:y
369:,
366:x
363:(
360:v
357:,
354:)
351:y
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345:x
342:(
339:u
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333:=
324:V
299:t
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272:y
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245:x
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216:y
211:d
206:x
201:d
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190:)
185:2
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146:(
141:2
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128:2
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118:t
114:I
110:+
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102:y
98:I
94:+
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86:x
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