Experimentation and Results

Two-Dimensional Tests

Our first experiment with thin object restoration was with thin lines in two dimensions. Figure 4, below, shows a two-dimensional test pattern with different amounts of added noise. Lines in the test pattern have pixels of value 1 while the background is 0. The added noise in the examples has standard deviation sigma = 0.1 and 0.4. Below the noisy images are the restorations, thresholded to have values of 0 and 1 only. To produce those images, we applied the restoration algorithm iteratively, using three iterations. Alpha, the weighted probability of having a blank, is 0.5 and the dictionary was the one shown in Figure 2. As you can see, the restoration is successful.

Figure 4: Two-dimensional tests of thin-line binary restoration. The top two images show the noisy input pattern of 1s and 0s. The noise on the left image is 0.1 and the right is 0.4. The corresponding restorations below used three iterations of binary restoration with alpha = 0.5 and the dictionary of Figure 2.

Figure 5 shows the effect changing the probability of a blank has on the restoration. On the left alpha is 0.2. As a result, much of the image that should be blank gets restored as spurious line segments. When alpha is 0.8, right image, the restoration assumes things are blank when it should not, and much of the lines in the test pattern are missing. At present we rely on experimentation to find the appropriate value for alpha.

Figure 5: The effect of alpha. If alpha is too small (0.2), left, then spurious line segments appear in the restoration. If alpha is too large (0.8), right, then line segments that should be restored are not.

We can restore lines from even noisier images if we use a larger neighborhood. Figure 6 shows the restoration of a line from an image with sigma = 0.7. The restoration requires an 11 by 11 neighborhood. The test pattern contains only a straight line segment. With such a large neighborhood it is not possible to restore arcs such as those in Figure 4 since the curvature of the arc must be small with respect to the size of the neighborhood.

Figure 6: Binary restoration with higher noise levels. The image on top contains a line segment of 1s on a background of 0s. The noise level is sigma = 0.7. On the bottom is the restoration using an 11 by 11 neighborhood, alpha = 0.8 on the left and alpha = 0.98 on the right.

Three-Dimensional Tests

To test three-dimensional restoration we look at a thin sheet of 1s situated in a volume of 0s. Figure 7 shows a 3D plot of the positions of the 1s in the volume in the upper left-hand corner. Beside that is a plot of all voxels with values above a threshold of 0.5 when the noise is sigma = 0.5. The presence of the thin sheet is completely obscured. Using a dictionary for thin sheets in a 3-cubed neighborhood and alpha = 0.7, we get the restoration shown in the lower left-hand corner in only one iteration. Using a larger neighborhood we can restore noisier data. The lower right-hand corner of Figure 7 shows the restoration for the same thin sheet with sigma = 0.7 obtained using a 5-cubed neighborhood.

Figure 7: Example restorations of a thin sheet in three dimensions. Upper left shows points in the sheet with no noise while upper right shows the points above a threshold of 0.5 for noise of sigma = 0.5 added. On the bottom are restored images for sigma = 0.5 (3x3x3 neighborhood) and sigma = 0.7 (5x5x5 neighborhood). Click on the plots to view mpeg animations.

A common source of three-dimensional volumetric data is computed tomography (CT). CT produces a three-dimensional map of the density of matter in the CT scanner's field of view. While densities in nature are not normally binary, density data can be very much like binary data if there is some dense material sitting amongst some light matter. The data are even more like binary data if the CT scanner is sensitive to some unique property found only in one type of material. In this case, we can apply binary restoration to CT images.

Although noise in a CT image is normally distributed, it is not uniform throughout the image. Because the contents of a CT scanner attenuate the signal, noise levels vary with position and contents. Nevertheless, we consider the noise in a CT image to be uniform and apply the restoration as before. The results are shown in Figure 8. On the left is shown the restoration of a synthetic CT scan of a thin sheet. Densities in the scan are scaled so that the sheet has a density of one. The noise level is approximately sigma = 0.5. We used a 5-cubed neighborhood, alpha = 0.7, and one iteration of the restoration algorithm. The image on the right shows a similar restoration for sigma = 0.65, approximately.

Figure 8: Restorations of computed tomography data. The image on the left shows a restoration for sigma = 0.5, approximately, and is done with a 5x5x5 neighborhood and alpha = 0.7. Only one iteration is required. On the right is a similar restoration, but the noise level was higher, sigma = 0.65, approximately. Click on the plots to view mpeg animations.

As we might expect, the partial volume effects and non-uniformity of noise have reduced the quality of the restoration from that obtained for the more ideal data of Figure 7. Still, the restoration is high-quality and excellent for detection purposes. It would be easy to determine automatically that a thin sheet is present in the restored image.

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