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Derivation of Kanade-Lucas-Tomasi Tracking Equation

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Derivation of Kanade-Lucas-Tomasi  Tracking Equation Powered By Docstoc
					   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:
                                              d          d
                      ǫ=             [J(x +     ) − I(x − )]2 w(x) dx,                (1)
                                 W            2          2

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 + d ) − I(x − d )].
                                                                   2           2
    Now the Taylor series expansion of J about a point a = [ax , ay ]T , truncated
to the linear term, is:
                                                 ∂J                 ∂J
                 J(ξ) ≈ J(a) + (ξx − ax )           (a) + (ξy − ay ) (a),
                                                 ∂x                 ∂y
where ξ = [ξx , ξy ]T .
                                                          d
   Following the derivation in [3], we let x +            2   = ξ and x = a to get:
                               d            dx ∂J       dy ∂J
                      J(x +      ) ≈ J(x) +       (x) +       (x).
                               2             2 ∂x       2 ∂y
Similarly,
                               d            dx ∂I       dy ∂I
                       I(x −     ) ≈ I(x) −       (x) −       (x).
                               2             2 ∂x       2 ∂y
   Therefore,

  ∂ǫ                             d          d ∂J(x + d ) ∂I(x − d )
                                                     2          2
        =    2          [J(x +     ) − I(x − )][        −           ]w(x) dx,
  ∂d               W             2          2    ∂d         ∂d
        ≈             [J(x) − I(x) + gT d]g(x)w(x) dx,
                  W

where
                                      ∂    I+J       ∂    I+J    T
                            g=        ∂x    2        ∂y    2         .
   To find the displacement d, we set the derivative to zero:
                 ∂ǫ
                    =          [J(x) − I(x) + gT (x)d]g(x)w(x) dx = 0.
                 ∂d        W


                                                 1
Rearranging terms, we get

            [J(x) − I(x)]g(x)w(x) dx     = −          gT (x)dg(x)w(x) dx,
        W                                         W

                                         = −           g(x)gT (x)w(x) dx d.
                                                   W

In other words, we must solve the equation

                                       Zd = e,                                (2)

where Z is the following 2 × 2 matrix:

                         Z=            g(x)gT (x)w(x) dx
                                   W

and e is the following 2 × 1 vector:

                      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.




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