1. The Image Deblurring Problem
Society for Industrial and Applied Mathematics eBooks · 2006
You cannot depend on your eyes when your imagination is out of focus. — Mark Twain When we use a camera, we want the recorded image to be a faithful representation of the scene that we see—but every image is more or less blurry. Thus, image deblurring is fundamental in making pictures sharp and useful. A digital image is composed of picture elements called pixels. Each pixel is assigned an intensity, meant to characterize the color of a small rectangular segment of the scene. A small image typically has around 2562 = 65536 pixels while a high-resolution image often has 5 to 10 million pixels. Some blurring always arises in the recording of a digital image, because it is unavoidable that scene information “spills over” to neighboring pixels. For example, the optical system in a camera lens may be out of focus, so that the incoming light is smeared out. The same problem arises, for example, in astronomical imaging where the incoming light in the telescope has been slightly bent by turbulence in the atmosphere. In these and similar situations, the inevitable result is that we record a blurred image. In image deblurring, we seek to recover the original, sharp image by using a mathematical model of the blurring process. The key issue is that some information on the lost details is indeed present in the blurred image—but this information is “hidden” and can only be recovered if we know the details of the blurring process. Unfortunately there is no hope that we can recover the original image exactly! This is due to various unavoidable errors in the recorded image. The most important errors are fluctuations in the recording process and approximation errors when representing the image with a limited number of digits. The influence of this noise puts a limit on the size of the details that we can hope to recover in the reconstructed image, and the limit depends on both the noise and the blurring process.