Algorithms for Blind Deconvolution

The function and for this particular example are shown in Figures 6.13(f) and (g). Suppose we apply a smoothing operator S to (6.3.14). Assuming the smoothing operating S is linear, we have

	(6.4.15)

Since both and are smooth functions, smoothing will not affect significantly. From Figure 6.20(g), however, smoothing will reduce significantly. Based of this observation, (6.4.15) reduces to,

	(6.4.16)

From (6.4.16)

	(6.4.17)

Equation (6.4.17) is the basis for estimating . The numerator term can be determined from the blurred image . The denominator term is approximately the same for similar classes of images. Differences in the image details affect , but they do not appear to significantly affect . Based on this observation,

	(6.4.18)

where is obtained from an original undergraded image that is similar in content to . From (6.4.17) and (6.4.18)

	(6.4.19)

Despite various assumptions and heuristics used in derivation of (6.4.19), a reasonable estimate of can be obtained from (6.4.19). It is possible to derive an expression for the phase in manner analogous to the derivation of in (6.4.19). This approach has not yet been successful, however, partly because an image's details affect the phase function, and the phase of one image appears to contain little information about the phase of another image, even when the two images have similar contents. In the absence of the good method to estimate is assumed to be zero. Once is estimated, inverse filtering or its variation, discussed in Section 6.4.1, can be used to reduce blurring.

Another blind deconvolotion method can be developed by assuming that the effective size (region of support) of is much smaller than . In this method, the degraded image is segmented into subimages by using nonoverlaping rectangular windows. The window size is chosen such that it is much larger than the effective size of but smaller than size . The subimage is then assumed to be

	(6.4.20)

where is the original image segmented by the same window used to obtain .