Gait Analysis and Human Motion Tracking - Intelligent Video Event Analysis and Understanding - page 43

or between corresponding points in the two images,

p 2 T F p 1 = 0 , (2)

where the positions of points p 1 and p 2 are defined in image coordinates. We solve
the equation, x T a

=

0, where a is a 9-vector including the entries of the fundamental

matrix F ,and x T

9 parameter matrix whose elements come from those ele-
ments of p 1 and p 2 . Solutions for F are well understood [36], but in our case outliers
may be caused by incorrect image correspondences. In general, errors embedded in
the sum of the squared residuals of Euclidean distance from image points to their
corresponding epipolar lines are non-Gaussian , as illustrated in the images and his-
togram of Fig. 1. Therefore we employ a two-stage, generalised least squares model
[42]. This is an extended version of linear mixed-effects models that have been used
for the analysis of balanced or unbalanced grouped data, such as longitudinal data,
repeated measures, and multilevel data [24, 16].

is a 1 ×

(a) Image 1 (b) Image 2 (c) Histogram of distance

Fig. 1 An example demonstrating the distribution of the squared residuals of Euclidean dis-
tance: (a) and (b) a pair of real images with the corner features superimposed, and (c) the
frequency (point numbers) versus Euclidean distances (pixels), where the width of each bin
is 1 pixel.

For simplicity, the general form of the Gauss-Markov linear model is considered.
Let Y i be a 9-vector with index i (each element of Y i is -1):

Y i = X i A i + ε i , i = 1 , 2 , ..., N , (3)

where X i is a 9 8 parameter matrix (9 random samples from the corresponding

points and 8 unknown elements of the F matrix), A i is an 8-vector including the
unknown elements of the F matrix (the ninth is 1 given a normalised F matrix), and
ε i is an independent error. The dimension of X i is designed such that the computa-
tion of the F matrix becomes more efficient and robust [40]. For variable index i ,

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