Derivation of Kanade-Lucas-Tomasi Tracking
Equation
Stan Birchfield
January 20, A. D. 1997
Carlo Tomasi [1] has recently proposed the following symmetric definition
for the dissimilarity between two windows, one in image I and one in image J:
Z Z
d
d
ǫ=
[J(x + ) − I(x − )]2 w(x) dx,
(1)
2
2
W
where x = [x, y]T , the displacement d = [dx , dy ]T , and the weighting function
w(x) is usually set to the constant 1. Equation (1) is identical to the equation
given in [2] except that the current version has been made symmetric with
respect to both images by replacing [J(x) − I(x − d)] with [J(x + d2 ) − I(x − d2 )].
Now the Taylor series expansion of J about a point a = [ax , ay ]T , truncated
to the linear term, is:
J(ξ) ≈ J(a) + (ξx − ax )
∂J
∂J
(a) + (ξy − ay ) (a),
∂x
∂y
where ξ = [ξx , ξy ]T .
Following the derivation in [3], we let x +
d
2
= ξ and x = a to get:
J(x +
d
dx ∂J
dy ∂J
) ≈ J(x) +
(x) +
(x).
2
2 ∂x
2 ∂y
I(x −
d
dx ∂I
dy ∂I
) ≈ I(x) −
(x) −
(x).
2
2 ∂x
2 ∂y
Similarly,
Therefore,
Z Z
d
d ∂J(x + d2 ) ∂I(x − d2 )
∂ǫ
[J(x + ) − I(x − )][
= 2
−
]w(x) dx,
∂d
2
2
∂d
∂d
Z Z W
≈
[J(x) − I(x) + gT d]g(x)w(x) dx,
W
where
g=
∂
∂x
I+J
2
∂
∂y
I+J
2
T
.
To find the displacement d, we set the derivative to zero:
Z Z
∂ǫ
=
[J(x) − I(x) + gT (x)d]g(x)w(x) dx = 0.
∂d
W
1
Rearranging terms, we get
Z Z
[J(x) − I(x)]g(x)w(x) dx
W
Z Z
gT (x)dg(x)w(x) dx,
Z Z
T
= −
g(x)g (x)w(x) dx d.
= −
W
W
In other words, we must solve the equation
Zd = e,
(2)
where Z is the following 2 × 2 matrix:
Z Z
g(x)gT (x)w(x) dx
Z=
W
and e is the following 2 × 1 vector:
Z Z
e=
[I(x) − J(x)]g(x)w(x) dx.
W
References
[1] C. Tomasi. Personal correspondence, May 1996.
[2] C. Tomasi and T. Kanade. Detection and Tracking of Point Features.
Carnegie Mellon University Technical Report CMU-CS-91-132, April 1991.
[3] J. Shi and C. Tomasi. Good features to track. In CVPR, pages 593-600,
1994.
2