Instead we will examine the distribution of the

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Unformatted text preview: of the MRI is very similar to the other entropy manipulation applications we have described. Once again we sample points from the image, x, where each point now has a value, vx, and a current estimate for the bias eld, bx. Proceeding as we have before, we approximate the entropy of the bias compensated image, cx = vx , bx see Equation 3.22. The bias eld is adjusted to minimize hcx by taking steps in the direction of the derivative as approximated in Equation 3.24. In this case the parameters over which we are minimizing entropy are the coe cients of the bias eld polynomial. The derivatives of the bias eld, since they are polynomial, are easy to compute. The rst experiment is identical to a synthetic experiment proposed by Wells et al. A binary checkerboard is used as a prototypical example of a two class image. Half of the pixels belong to the black class, the other half to the white class. The pixel entropy of a checkerboard is very low. The checkerboard is then corrupted by a large unknown bias eld see Figures 6.1 and 6.2. The corruption is so large that any cluster structure in the data has disappeared. This is apparent both from the distribution of pixels and a thresholding of the corrupted image see Figures 6.3 and 6.2. Entropy minimization comes very close to exactly compensating for the bias eld. Figure 6.2 shows the corrected image. Its distribution is also shown in Figure 6.3. The image is not perfectly corrected because we use a di erent bias eld representation than Wells et al. One of the goals of the work by Wells et al. is to correctly classify white versus grey matter in the brain see Bezdek et al., 1993 for a comprehensive overview of MRI segmentation. They show that classi cation is much easier if the bias eld of a scan is known beforehand. 139 Paul A. Viola CHAPTER 6. OTHER APPLICATIONS OF EMMA Figure 6.1: The original checkerboard image and the bias eld that corrupts it. Figure 6.2: Left: the corrupted checkerboard. Center: a thresholded version of the corrupted image. Right: the corrected checkerboard. We have performed a number of experiments where EMMA is used to nd the correction eld. While Wells et al.'s system is designed to give a tissue classi cation for the corrected scan, EMMA based correction does not provide a classi cation. Instead we will examine the distribution of the corrected scan and attempt to determine if the scan has been corrected in a way that would make classi cation of white and grey matter easier. Figure 6.4 shows a slice of MRI data taken from a brain. Figure 6.5 shows the histogram of the scan before and after correction. In the histogram of the original scan white and grey matter tissue classes are confounded into a single peak ranging from about 0.4 to 0.7. The histogram of the corrected scan shows much better separation between these two classes. We can highlight the pixels in this distribution by mapping all the pixels below the rst peak to black, all the pixels above the second peak to white, and linearly scaling between white matter appears darker than grey matter in this MRI scan. Figure 6.6 shows the original and corrected scans in this manner. Notice that the inhomogeneity in the original image becomes immediately apparent. The lower left hand portion of the original scan is dark. Since the corrected scan does not show this inhomogeneity, the white and grey matter of the corrected scan are distinct. 140 6.1. BIAS COMPENSATION AI-TR 1548 18000 Corrupted Corrected 16000 14000 12000 10000 8000 6000 4000 2000 0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Figure 6.3: The pixel density of the corrupted checkerboard and the compensated checkerboard. Figure 6.4: A slice from an MRI scan of a head. A Second Experiment The procedure for bias eld correction has been repeated for a number of di erent scans. Figures 6.7, 6.8 and 6.9 show the results of an experiment performed on a coronal slice of an MRI scan. 141 Paul A. Viola CHAPTER 6. OTHER APPLICATIONS OF EMMA Corrupted Corrected 1200 1000 800 600 400 200 0 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Figure 6.5: The distribution of pixel value in the MRI scan before and after correction. Experimental Details All of the experiments use a smoothing function variance of 0.01. The learning rate was -0.003. The sample size was 30. Every run was concluded after 8000 parameter updates. The pmin value was 0.01. The checkerboard image is 256 by 256 pixels. The checkerboard experiment uses a bias eld which is a 20 by 20 pixel low resolution image. This 20 by 20 image is then bilinearly interpolated to create a continuous bias o set for each pixel in the checkerboard. The 400 parameters in this bias eld are the most ever simultaneously approximated by an EMMA based technique. The MRI scans varied in size from 100 square to 256 square. The bias eld for the MRI experiments was a 3rd order polynomial in x and y location. 6.2 Alignment of Line Drawings Jones and Poggio have constructed a system that automatically analyzes hand drawn faces Jones and Poggio, 19...
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