Gait Analysis and Human Motion Tracking - Intelligent Video Event Analysis and Understanding - page 44

A i indicates individual fundamental matrices F i determined by the i th subsampling
group. Second, the coefficients A i are associated with the population covariates by

A i = v i b + γ , (4) i

×

where v i is an 8 8 population design matrix, b is an 8-vector, γ i is a vector of

random effects independent of ε

i . An optimal solution for Eq. (3) leads to the min-

imisation of

i .
To obtain a solution for A i , the two-stage strategy is combined with a moment
estimator, based on the optimisation technique proposed by Demidenko and Stukel
[9]. The first stage refers to Eq. (3), while the second stage contributes to Eq. (4). The
moment estimator is used to compute the variance between the measurements and
the estimates against the number of the remaining samples. Omitting intermediate
steps that can be found in their paper, we have:

ε

A i =( ∑ i Z i H − 1 Z i ) − 1

∑ i

Z i H − 1 C A i ,

(5)

i i

where A i is an unbiased solution to the fundamental matrix. C = Z = I (identity ma-
trix), and A i and H can be computed using the numerical technique introduced in
[42]. The proposed linear mixed-effects model is not robust to the existence of out-
liers that perturb the optimization by introducing ambiguities. To make the proposed
strategy robust, we investigate the squared residuals of the Euclidean distances from
image points to their corresponding epipolar lines in accordance with the two-stage
estimator [39].
A confidence interval, , is used to determine the probability of inliers (valid

correspondences) or outliers (rogue points). This confidence interval is constructed
as a m −

μ

t m s ,where a m is the sample mean, s is the standard deviation,
and t m depends on the degrees of freedom, which is equal to one less than the size
of a random subsample, i.e. eight, and the level of confidence. To determine t m ,the
t distribution is used. The t distribution is an infinite mixture of Gaussians, and is
often employed as a robust measure to “correct" the distorted estimation created by
a few extreme measurements.

t m s ≤ μ ≤ a m +

1

∑ i ( d i − a m )

2

μ ≈ a m ± 2 . 896 × , (6)

M − 1

Assuming there are i correspondences across images, the confidence interval of the
distribution is determined by Eq. (6), where a m =

d M , M is the number of the image
correspondences, and d i are the residuals computed by applying the fundamental
matrix estimated in the last subsection to the overall image correspondences, and
2.896 is based on a significance level of 0.01 and the 8 degrees of freedom (with
reference to the upper critical values of the student's t distribution). Examples of
the epipolar geometry estimation can be found in Fig. 2. Once the fundamental
matrix has been obtained, the essential matrix, E , can be obtained using the known