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5.1 Active contours 243

form
1 T 2
E shape = b diag(λ 0 ...λ M−1 ) b = b /2λ m . (5.17)
m
2
m
Alternatively, the range of allowable b m values can be limited to some range, e.g., |b m |≤
√
3 λ m (Cootes, Cooper, Taylor et al. 1995). Alternative approaches for deriving a set of
shape vectors are reviewed by Isard and Blake (1998).
√ √
Varying the individual shape parameters b m over the range −2 λ m ≤ 2 λ m can give
a good indication of the expected variation in appearance, as shown in Figure 5.4d. Another
example, this time related to face contours, is shown in Figure 5.5a.
In order to align a point distribution model with an image, each control point searches
in a direction normal to the contour to find the most likely corresponding image edge point
(Figure 5.5b). These individual measurements can be combined with priors on the shape
parameters (and, if desired, position, scale, and orientation parameters) to estimate a new set
of parameters. The resulting Active Shape Model (ASM) can be iteratively minimized to fit
images to non-rigidly deforming objects such as medical images or body parts such as hands
(Cootes, Cooper, Taylor et al. 1995). The ASM can also be combined with a PCA analysis of
the underlying gray-level distribution to create an Active Appearance Model (AAM) (Cootes,
Edwards, and Taylor 2001), which we discuss in more detail in Section 14.2.2.

5.1.2 Dynamic snakes and CONDENSATION

In many applications of active contours, the object of interest is being tracked from frame
to frame as it deforms and evolves. In this case, it makes sense to use estimates from the
previous frame to predict and constrain the new estimates.
One way to do this is to use Kalman filtering, which results in a formulation called Kalman
snakes (Terzopoulos and Szeliski 1992; Blake, Curwen, and Zisserman 1993). The Kalman
filter is based on a linear dynamic model of shape parameter evolution,
x t = Ax t−1 + w t , (5.18)

where x t and x t−1 are the current and previous state variables, A is the linear transition
matrix, and w is a noise (perturbation) vector, which is often modeled as a Gaussian (Gelb
1974). The matrices A and the noise covariance can be learned ahead of time by observing
typical sequences of the object being tracked (Blake and Isard 1998).
The qualitative behavior of the Kalman filter can be seen in Figure 5.6a. The linear dy-
namic model causes a deterministic change (drift) in the previous estimate, while the process
noise (perturbation) causes a stochastic diffusion that increases the system entropy (lack of
certainty). New measurements from the current frame restore some of the certainty (peaked-
ness) in the updated estimate.
In many situations, however, such as when tracking in clutter, a better estimate for the
contour can be obtained if we remove the assumptions that the distribution are Gaussian,
which is what the Kalman filter requires. In this case, a general multi-modal distribution is
propagated, as shown in Figure 5.6b. In order to model such multi-modal distributions, Isard
and Blake (1998) introduced the use of particle filtering to the computer vision community. 5

5 Alternatives to modeling multi-modal distributions include mixtures of Gaussians (Bishop 2006) and multiple
hypothesis tracking (Bar-Shalom and Fortmann 1988; Cham and Rehg 1999).