To reiterate their system nds an estimated correction

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Unformatted text preview: ch 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 non-uniformity 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 Levenberg-Marquardt. 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 multi-class 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 non-smooth 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 model-based 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 low-frequency 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 AI-TR 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 low-frequency 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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