6.4.2 Algorithms for Blind Deconvolution

If the blurring function is not accurately know, must be estimated prior to inverse filtering. Since we are attempting to deconvolve without detailed knowledge of this is called the blind deconvolution problem. If we know nothing about or , it is not possible to solve the blind deconbolution problem. The problem is analogous to determining two numbers from their sun when nothing is known about either of the two numbers. To solve the blind deconblution problem, therefore, something must be known about , or both. Blind deconvoltion algorithms differ depending on what is assumed known and how that knowledge is exploited.

Suppose and are finite extent sequence with nonfactorable * z-transforms and .

Then we can recover within translation and a scale factor from , using a polynomial factorization algorithm. Specifically, , the z-transforms of , is given by . Since we assume that and are finite-extent sequences, is a finite-order 2-D polynomial in and . In addition, we assume that and are nonfactorable, and therefore the only nontrivial factors of are and . Polynomial factorization algorithms that determine nontrivial factors of may be used in determining or within translation and a scale factor. Unfortunately, this approach to solving the blind deconvolution problem has serious practical difficulties. Polynomial factorization algorithms developed to data are very expensive computationally. In addition, the algorithms are extremely sensitive to any deviation from the assumption that or . In practice, the convolutional model of is not exact due to the presence of some background noise or due to approximations made in the modeling process.

One practical blind deconvolution algorithm is based on the assumption that is a smooth function. This assumption is approximately valid in some applications. When an image is blurred by a thin circular lens, the modulation transfer function is a fairly smooth circularly symmetric lowpass filter. When a long-exposure image is blurred by atmospheric turbulence, the blurring function b(x, y) and its Fourier transform are approximately Gaussian-shaped. When an image is blurred by horizontal motion, is a sine function and is smooth except in the regions where crosses zero.

To estimate under the assumption that is smooth, we first note that

	(6.4.12)

Examples of and are shown in Figures 6.13(a), 6.13(b), and 6.13(c). The function can be viewed as a sum of two components, a smooth function, denoted by , and a rapidly varying function, denoted by .

	(6.4.13)

The functions and for in Figure 6.13(a) are shown in Figure 6.13(a) and 6.13(e). From

	(6.4.14)