On average the coe cients of the nal transformation

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Unformatted text preview: t procedure. A typical initial alignment appears in Figure 5.19. Notice that this image is a scaled, sheared, rotated and translated version of the original. The nal alignment is displayed as a checkerboard. Here every other 20x20 pixel block is taken either from the model image or aligned target image. Notice that the boundary of the brain in the two images is in close agreement. We represent transformation space as a 6 element a ne matrix that is used to project two dimensional points from the image into the model. This scheme can represent any combination of scaling, shearing, rotation and translation. The remaining algorithmic details are summarized in Table 5.5. In order to determine the reliability and precision of the alignment procedure, 50 random126 5.2. MEDICAL REGISTRATION EXPERIMENTS AI-TR 1548 1. De ne the model and image u and v: u is the model image, v is the target image. 2. Sampling x: Sample the pixels of the model image uniformly. 3. Transformation space T : The space of a ne transformations mapping pixels locations from the model into pixel locations in the image. 4. De nition of dvy=dy: This is the intensity gradient. 5. Distance metric: Euclidean distance. 6. Variance, : Assuming diagonal covariance matrices, three different variance are necessary: two for the joint entropy and one for the image entropy. The variances were 0.1 in all cases. 7. Minimum probability, pmin : 0.01. 8. Number of samples: One sample of 20 using cross-validation. 9. Update rate, : 0.02 for 500 steps and then 0.005 for 500 steps. Table 5.5: Summary of MRI alignment experiments. ized alignments were performed. The initial transformations were randomly selected, having a translation of up to 35 pixels this is about one third of the width of the head, a rotation of up to 30 degrees, and a scaling of up to 20. The correct alignment was obtained in 100 of the experiments. After alignment the a ne transformations had an average translation error of 0.1 pixels. The remaining a ne parameters represent a mixing of rotation, scale and shearing. They are somewhat more di cult to interpret. Given that the correct transformation is the identity matrix we can evaluate the nal matrices by measuring the di erence from the identity. On average the coe cients of the nal transformation were in error by 0:02. These experiments demonstrate that EMMA alignment is both precise and reliable. These two MRI images are fairly similar. Good alignment could probably have been obtained with a normalized correlation metric. Normalized correlation assumes, at least locally, that one signal is a scaled and o set version of the other. Our technique makes no such assumption. In fact, it will work across a wide variety of non-linear transformations. More di cult alignment problems are easily simulated. In Figure 5.20 we show the model image after a non-monotonic non-linear function has been applied. Recall that initially the image lies in the range 0,1 . We subtract 0.5, square the result, and renormalize to 0,1 . This operation is shown at the right of the gure. After applying this non-linear transformation the two images are anti-correlated; no variant of correlation can correctly align them. EMMA 127 Paul A. Viola CHAPTER 5. ALIGNMENT EXPERIMENTS Figure 5.18: MR Images alignment performance, however, is not e ected. 5.2.1 Three Dimensional MR CT Alignment The medical alignment procedure described above can be extended to volumetric data. In the resulting system both the model and image are 3D arrays and a full three dimensional aligning transformation is estimated. Recently a number of three dimensional CT MRI alignments have been performed. Because these results are preliminary, many of the experimental details are still in ux. Our description here will be necessarily brief. A more complete description can be found in Wells III and Viola, 1995. The scans used were obtained from the same patient at di erent times4. Display of these three dimensional scans, and their alignment, is a di cult problem. Though the entire scan cannot be shown, some feeling for the data can obtained be displaying the three central slices. The central slices are perpendicular planes that pass trough the centroid of the data. Figure 5.21 shows the three central slices of the CT scan. Figure 5.22 shows the initial The images and the standard transformations were provided as part of the project, Evaluation of Retrospective Image Registration", National Institutes of Health, Project Number 1 R01 NS33926-01, Principal Investigator, J. Michael Fitzpatrick, Vanderbilt University, Nashville, TN. 4 128 5.2. MEDICAL REGISTRATION EXPERIMENTS AI-TR 1548 Figure 5.19: Initial Pose and Display of Result alignment of the CT MR pair as a checkerboard. When the signals involved are very di erent, the checkerboard representation can be somewhat confusing. Figure 5.23 shows the nal alignment as the composition of the MR data with the intensity edges computed from the CT data. Notice the close agreement between the skull in both scans. 129 Paul A. Viola CHAPTER 5. ALIGNMENT EXPERIMENTS 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 Figure 5.20: Transformed Model...
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This note was uploaded on 02/10/2010 for the course TBE 2300 taught by Professor Cudeback during the Spring '10 term at Webber.

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