Image Estimation by Iteration

Successful deconvolution methods work using a constrained iterative process of image estimation and correction. Rather than attempt to solve the inverse convo lution problem in a single step, image estimation relies on a series of small steps-successive approximations—that eventually converge on a best estimate of the original image.

Let's consider how successive approximation works. We begin by defining the original ideal image and the captured image:

• s(x, y): the degraded noisy image captured by a CCD camera;

• o(x, y): the (unknown) original undegraded noise-free image;

• k: the point-spread function, known from star images; and

Recall from Equation 19.1 that the relationship between the two images, the point-spread function, and image noise is:

that is, the image that the CCD records is the convolution product of the point-spread function with an undegraded image, with detector noise added.

We also define another image, e(x,y), which represents our best estimate of the convolved-but-noiseless image that fell on the detector; namely:

This image is the convolution of the original one with the point-spread function. Of course, we don't know what e(x,y) is, but we can estimate the point-spread function from star images in y). If we knew what the original was, we could compute e(x, y) by convolving the original with the point-spread function; but, of course, the original image is what we're trying to recover.

This sounds like variation on the chicken-and-egg problem. In the world of mathematics, it is sometimes possible to solve such problems.

We begin by asking what condition(s) must be satisfied to recover the original. Suppose that we subtract e from both sides of Equation 19.1 to obtain:

This says that the difference between these image is the noise. If we set the condition—one that we know cannot be met—that the noise is zero, or at least too small to matter, then:

This equation states that if we knew what e(x, y) was, there would be no difference (except for a negligible amount of noise) between the two images. This will remain true if we add the original image, o(x, y), to both sides of the relationship:

For the sake of clarity, we will designate the original image on the left side of the equation as o'(x, y), and we replace e(x, y) with its definition:

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