Unformatted text preview: Weighted Neighbor Likelihood vs. EMMA
Weighted neighbor likelihood and EMMA are both smoothly di erentiable functions that can
be used to align signals when the imaging function is unknown. Qualitatively the EMMA
estimate of joint entropy seems better. Joint entropy seems to have a wider basin in these
synthetic experiments.
If weighted neighbor likelihood and EMMA are so similar, why is there a di erence?
Recall that weighted neighbor likelihood measures the conditional entropy of the image given
the model. It does this under the assumption that the conditional distribution of the image
is Gaussian. Weighted neighbor likelihood use the data around a point to estimate the mean
of a Gaussian. The log likelihood of that point is then proportional to the squared di erence
from this mean. In general log likelihood calculations are very sensitive to outliers. Outliers
are points that are, because of noise or measurement error, perturbed and land far from
where they should have. Recall that the log likelihood of a sample is the sum of the log
likelihoods of each point in the sample. As a result a single outlier can ruin a sample that
would otherwise have had a high likelihood.
A more reasonable measure might introduce a bound on the penalty for a single point.
Once a single point moved beyond a certain distance from the local mean the cost would
no longer increase. Calculating likelihood in this way is closely related to the concept of a
robust statistic
EMMA on the other hand does not assume that the conditional distribution of the image
given the model is Gaussian. Instead it approximates the density nonparametrically. EMMA
can handle situations where there are multiple peaks in the conditional distribution. While
there is a likelihood penalty if a group of points are perturbed away from the local mean,
it is not a function of the distance from the mean. Once these points move outside the
e ective range of the smoothing function there is no additional penalty. EMMA's robust
nature prevents it from getting swamped by a few outliers in the joint distribution. This
gives it a greater ability to deal with the distributions that arise from misalignments between
model and image.
In the next sections we will describe a number of other situations where EMMA is better
than weighted neighbor likelihood.
93 Paul A. Viola CHAPTER 4. MATCHING AND ALIGNMENT 4.2.1 Nonfunctional Signals
Up until this point our analysis of alignment has assumed that there exists an imaging
function that relates the model and the image. For at least two classes of problems there will
be no imaging function at all. The rst arises from a common situation in computer vision:
occlusion. The second arises when the model does not contain all of the information required
to predict the image. In both cases no single function, regardless of exogenous variables, can
be used to predict the image from the model.
Figure 4.14 shows a graph of our original pair of signals, except that vx has now been
corrupted by an occlusion. Occlusion proves to be particularly bothersome for the alignment
techniques we have proposed. For example, the basic assumption behind normalized cost
has been violated; the occluded signal is not a linearly transformed version of the model.
In addition, a quick glance at the joint space shows that the assumption behind weighted
neighbor likelihood has also been violated see Figure 4.15; even when the signals are aligned,
there is no longer any function that relates ux and vx. Figure 4.16 show a graph of
weighted neighbor likelihood versus translation. The global minimum no longer coincides
with the correct translation.
In some cases EMMA can be used to align partially occluded signals. Joint entropy does
not su er from the strong assumption that the signals are functionally related. Though part
of the signal may be corrupted, the remaining parts retain their low entropy relationship.
Figure 4.17 show a plot of joint entropy for the occluded pair of signals.
The simplest example of nonfunctional signals often arises when the model and the image
are swapped. Whenever the function between the model and the image is nonmonotonic,
the relationship between the image and the model is nonfunctional. The nonmonotonically
related signals shown in Figure 4.2 are an example. Figure 4.18 shows the joint space of the
swapped signals and a weighted neighbor function approximation. The function t to this
joint space is a terrible approximation of the data. The quality of the function approximation
points out an important limitation of weighted neighbor likelihood. While normalized cost
is a symmetric comparison metric, weighted neighbor likelihood is not. It may seem at rst
that this is an unimportant distinction. It is not. Symmetric measures allow us to match
images to models as well as models to images. This can be critical when it is not possible to
construct a detailed model.
94 4.2. WEIGHTED NEIGHBOR LIKELIHOOD VS. EMMA AITR 1548 u(x)
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Position 400 Figure 4.14: Graph of ux and vx where vx has been perturbed b...
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 Spring '10
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 The Land, Probability distribution, Probability theory, probability density function, Mutual Information, Paul A. Viola

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