## The Inverse Convolution Problem

Convolution is a determinate process: you start with an image, convolve it with some point-spread function, and get a single well-defined result. Deconvolution is not determinate. In principle, the image that you capture at the telescope could have been created from many different original images. Consider a one-dimensional example; this one-dimensional image has just been captured:

s(x) = [... 000122222221000...]-You happen to know that the exact convolution kernel is:

which is a basic Gaussian-like blur. Recovering the original should be easy, right? Wrong. Here are two candidates for the original image:

If you calculate their convolutions with k, you will find that both reproduce the central string of 2's in s(x) quite nicely. The only differences are the values at the ends of the 2's, and those small differences tell you nothing about the middle of the image. Like it or not, it is impossible to distinguish which of these two very different candidates produced ^(x). You can readily imagine many other candidate originals, also indistinguishable after convolution.

Let's look at the example again, but this time with an image that has a small amount of noise. The image s(x) + n contains random noise:

[... 0.1 -0.1 0.1 0.9 1.9 2.2 2.1 2.0 1.9 2.0 2.1 1.1 0 -0.1 0 ...] •

Adding noise means that you cannot "trust" any value in the image to better than some standard deviation. This uncertainty increases the number of candidate originals astronomically, because it is no longer necessary that they match exactly; any match within the noise envelope is a candidate solution.

Now consider the convolution kernel, which we have assumed that you know perfectly. With a real image, where would it come from? Astronomers are extremely lucky because celestial images are littered with point sources—stars— each of which is a record of the point-spread function. So you can measure the point-spread function from an image, like this image of a one-dimensional star:

Unfortunately, even the best point-spread function is slightly noisy. Thus, any realistic scheme for deconvolution should be able to determine the original image from a noisy detected image and a noisy point-spread function.

Mathematicians call deconvolution an ill-posed problem. The problem is ill-posed because it doesn't have a unique solution. Furthermore, in images that are noisy, it is possible that no original image can produce the image you have captured. Finally, the solution is divergent—that is, tiny errors in the detected image can produce large errors during an attempt to restore the original.

The key to image restoration lies in the physical limitations that real images place on the purely mathematical solutions. Rather than seek a unique original image, we look for a statistically plausible original. And rather than search among all possible mathematical solutions, we constrain the solution to a realistic original image by insisting, for example, that it conform to reality by having no areas of negative brightness. In the end, the key to deconvolution lies in shifting the goal from a mathematically rigorous solution to finding ways of searching for a best approximation of the original image. 