. '/? = 1.6 /= 15 •

Figure 19.3 Results with van Cittert image estimation depend on the radius of the Gaussian point-spread function, R, and the number of iterations, I. A radius of 1.0 pixel is slightly too small, while 1.6 is slightly too large. The optimum number of iterations depends to some extent on personal preference.

Figure 19.3 Results with van Cittert image estimation depend on the radius of the Gaussian point-spread function, R, and the number of iterations, I. A radius of 1.0 pixel is slightly too small, while 1.6 is slightly too large. The optimum number of iterations depends to some extent on personal preference.

it will also hold for an image that is similar to the original, but not exactly the same—that is, for similar inputs, we expect similar outputs. So, unless the CCD image has been terribly degraded, it is a fairly close approximation of the original one.

What happens if we simply decide that we'll compute a new version of the original image by treating the CCD image as if it were the original?

To accomplish this, we convolve the CCD image with the point-spread function, which generates a blurred version of it. We subtract this blurred image from the original CCD image, and obtain a difference image. Because the two initial images are similar, the difference image should contain small pixel values with both positive and negative signs. If we sum the difference image with that from the CCD, we will obtain a new version of the original image that is different from the one that we started with. So far so good.

Figure 19.4 In this case study, watch the lunar crater Davy as it changes from the original image through 99 iterations. The best stopping point is subjective—a balance between enhancement and noise amplification—but most observers would probably agree that it's somewhere between 30 and 60 iterations.

Figure 19.4 In this case study, watch the lunar crater Davy as it changes from the original image through 99 iterations. The best stopping point is subjective—a balance between enhancement and noise amplification—but most observers would probably agree that it's somewhere between 30 and 60 iterations.

The big conceptual leap is to feed the new version of the original image back into the same equation as a new old version, and to repeat this process over and over. If the process is well-behaved, at each iteration we should find an increasingly better approximation of the original image.

In other words, using the degraded CCD image to "seed" the process, we compute a new estimate of the original, and use that to compute a better estimate, and that to compute a still better estimate. When we satisfy the condition that s{x, y) - e(x, y) = 0, we will have recovered the original image.

This iterative method is so simple that anyone can try it using standard image-processing tools to blur, add, and subtract images. Figure 19.2 shows three successive approximations. Each new approximation appears sharper, and we see gratifyingly enough, that the amplitudes in the difference image become smaller with each iteration.

Of course there is no guarantee that a series of approximations will converge until e(x, y) disappears. For one thing, we have ignored the noise which does have a finite value; and for another, we cannot be sure that successive difference images will not become larger and larger instead of smaller and smaller. In fact, the van Cittert and Richardson-Lucy deconvolution methods exist because we can ignore small amounts of noise and because their iterations do converge.

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