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problems and their solutions. In the rst part of the chapter we will show how EMMA can
be used to correct images that have been have been corrupted" by a slowly varying bias
eld. Examples include: MRI corruption that arises from nonuniformity in magnetic eld,
and lightness correction in visual images. The second part of the chapter is devoted to an
application of stochastic gradient descent outside of entropy manipulation. Jones and Poggio
have presented a system that aligns line drawings of faces with novel line drawings Jones
and Poggio, 1995. Their published work uses a complex second order gradient descent
technique known as LevenbergMarquardt. We will show that similar if not better results
can be obtained with stochastic gradient descent. The resulting algorithm operates roughly
30 times as fast as the original. 6.1 Bias Compensation
A magnetic resonance image MRI is a 2 or 3 dimensional image that records the density
of tissues inside the body. In the head, as in other parts of the body, there are a number
of distinct tissue classes including: bone, water, white matter, grey matter, and fat. In
principle the distribution of pixel values in an MRI scan should be clustered, with one
137 Paul A. Viola CHAPTER 6. OTHER APPLICATIONS OF EMMA cluster for each tissue class. In reality MRI signals are corrupted by a bias eld, an additive
or multiplicative o set that varies slowly in space. The bias eld results from unavoidable
variations in magnetic eld see Wells III et al., 1994 for an overview of this problem. The
bias eld makes constructing automatic tissue classi ers di cult.
Wells et al. have built a system for bias correction around the assumption that an uncorrupted MRI scan will have a particular distribution of pixel values. This distribution
will have a peak for each type of tissue. Using an explicit physical model of MRI image
formation they construct a prior model for this distribution as a mixture of Gaussians, with
one Gaussian for each tissue type. The model can then be used to compute the likelihood of
an MRI. Corrupted MRI's will be unlikely because the bias eld blurs together the clusters.
Wells et al. use maximum likelihood to select the correction eld the inverse of the bias
eld that makes a corrupted MRI most likely.
To reiterate, their system nds an estimated correction eld that when applied to the data
makes it look like a particular type of clustered multiclass data. Applying the correction eld
sharpens up the classes and makes automatic classi cation easy. As in the learning problems
encountered in previous chapters, some assumption about the nature of the correction eld
is necessary to condition the problem. If we have prior knowledge that the bias eld varies
slowly across space, the correction eld should also vary slowly. Wells et al. assume that the
bias eld is smooth. To encourage smoothness they introduce a probabilistic prior in which
smooth elds are more likely than nonsmooth ones.
The main disadvantage of their MRI correction system is that it requires a fairly accurate
model of the tissue distribution. These models can be di cult to construct. Furthermore,
since the model includes estimates for the relative proportions of the tissue types, a di erent
model is required for each region of the body.
Using entropy we can proceed in a much less modelbased way. Since Wells et al.'s
technique has proven to be quite e ective, we can safely assume that the pixel values of an
uncorrupted MRI image are clustered into distinct classes. Such a distribution should have
low entropy. Corruption from the bias eld blurs together the clusters. The bias eld acts
like noise, adding entropy to the pixel distribution. This insight is the central idea behind
our approach. We attempt to nd the lowfrequency correction eld that when applied to
the image, makes the pixel distribution have a lower entropy. The resulting bias corrected"
image will have a tighter clustering than the original distribution.
138 6.1. BIAS COMPENSATION AITR 1548 Insuring the smoothness of the correction eld can be tricky. Wells et al. estimate a dense
correction eld, with one estimate for every pixel in the MRI. They insure smoothness by
periodically, at every iteration, smoothing the correction eld estimates. Another approach
would be to represent the bias eld as smooth in the rst place. This can be done parametrically by representing the correction eld as a smooth parameterized function. Or it can
be done using a lowfrequency correction image, with say 1 pixel for every 10 in the image.
Another approach, one which is guaranteed to be better when it is possible, is to represent
the correction eld with an explicit physical model. In the case of MRI, physics can be used
to show that the bias eld should be a low order polynomial of location M. Tincher and Williams, 1993. We take this approach for correcting MRI scans, representing the correction
eld as a third degree polynomial in the x and y coordinates of the scan.
The code that minimizes the entropy...
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