To begin with we need to be able to represent shape deformation of an
object, which we assume is a non-rigid contour (B-spline). This is done
using the concept of a shape-space. A shape-space is a linear mapping of
a ``shape-space vector''
to a spline-vector
,
where
is called a shape matrix. The elements of
act as weights on the columns of
.
is a constant offset, for example, a mean shape.
As an example, a planar affine shape space is described by,
where
are the
x
template coordinates of control points, chosen with the centroid at the
origin; similarly for
. Here the first two columns of
represent horizontal and vertical translation. The third and fourth
columns represent scaling (width and height respectively). The last two
columns deal with rotation. Rather than using Equation
1
or Equation
2
it is possible to apply a principal component analysis [
10
] to the data to determine the size of space that could be used to
represent the motion (or shape deformation) of the object. The advantage
is that the resulting
matrix is finely tuned to the deformations of the object of interest, in
our case the left ventricle. The disadvantage is that interpretation of
the resulting
matrix is less clear. We return to this point in Section
3
.
To track an object, a shape model is propagated over time using stochastic differential equations.
Tracker dynamics can be described by a second order autoregressive model which can be written in discrete form as,
A Kalman filter framework [ 5 ] is used to iteratively update the tracking algorithm using a prediction-update strategy. The prediction step updates the motion based on the model of the tracker dynamics. This prediction is then corrected in the update step using information provided by the measurement process. In the original tracker implementation measurements are made along the normals to the present estimate of the contour to save computational expense. Features are detected by applying a one-dimensional gradient operator along the sampled normals and selecting the strongest response as the most probable feature.
and
govern the behaviour of the tracking algorithm and can either be set by
specifying `reasonable' default dynamics or learnt from extended
training sequences. In practice, choosing a set of good default dynamics
is time-consuming and problematic and training is necessary. Suppose
that we are given a training sequence of data. We can estimate
(or equivalently
) by noting that the covariance of the data set is
. The procedure for finding the coefficient matrices for
and
is a little more complicated [
2
].
Briefly, first a principal component analysis is applied to the data to
estimate
. This is done in order to restrict the state space to a low-dimensional
subspace during training to avoid overfitting. Training data is
collected by tracking an ultrasound sequence using a tracker with good
default dynamics. The learning exercise is then to estimate the
coefficients
and
from this training sequence of spline contours. The discrete-time system
parameters are estimated via Maximum Likelihood estimation (MLE).
Assuming that the noise is Gaussian, it is straightforward to set up and
maximise the likelihood function.
Gary Jacob
