Our analysis to degenerate solutions of motion transparency are based
upon a singular value decomposition (SVD) of image measurements[
1
]. Consider the vector
of measurements that are collected into an
matrix
. The singular value decomposition is given by
where the columns of
are orthonormal and
is a diagonal square-root matrix. Let
be the
matrix of rank
formed from the
largest singular values, and singular vectors of
. Then
is the rank
least-squares approximation of
in that it minimises:
for all matrices
of rank
or less [
6
]. Accordingly, we partition
as:
The matrix of residuals and signal is:
From equation ( 2 ), there are three combinations of second order derivatives that yield coherent motion. They are:
In our coherent model, we have combined the first two to permit a
least-squares estimate of image velocity from the SVD of
. So that we have
n
= 6 and
. Using the first two constraints from equation (
16
) we consider the model:
where
is a residual. We now minimise this model according to:
where
and
Q
is again a residual after including
.
for a single motion can be viewed as a Lagrangian based upon the phase
velocity. Also,
. From this, the model's parameters are obtained from:
where
k
refers to the number of measurements taken and
the unbiased estimate of the variance
. Note, a small value of
Q
, requires that the third row and column of the matrix
: a null vector. This will occur when the regression matrix
is of full rank, the variation
is large by comparision to the variation of the measurements, and the
measured temporal derivatives are small in magnitude.
An estimate of the generalised covariance of the measurements is given by:
The model with the smallest generalised variance implies one whose
stability is highest. Equally, we can consider the model which takes
into account the most variance. To do this, we define the confidence
measure
taken from the geometric mean of the eigenvalues to the regression
matrix
:
where
and
C
is a constant. Equation (
21
) may be regarded as a test statistic that the matrix
has a simple form like
[
6
], and
is large. We have used this measure to determine our model certainty
estimates because it reflects the stability of a model's parameters
posed in terms of the total variation of the image measurements.
The 1-d (phase) velocity
and the confidence measure used to obtain the curves shown in figure
3
was taken from the model:
whose parameters were determined from the first eigenvector to the SVD
of the matrix
[
1
].
Degeneracies of rank 3 or 4 to a 2-fold transparent motion model are ambiguous (ie to cases of three or four sinusoids) and not considered here. Feasible solutions may, however, be obtained using linear optimisation and the constraints given by equations (6)-(8). A unique single-flow vector may be detected if the system is determined to be of rank 3. Here one is required to collect the three constraints from equation ( 16 ) from their respective eigenvector elements of the 3 largest eigenvalues to the matrix A into a single least-squares system.
Adrian F Clark
