In the deblurring process, one applies a deblurring filter denoted as h(x,y) in the position- coordinate domain or H(ρ) in the spatial-frequency domain. The problem is to choose H(ρ) to make the resulting image approximate f(x,y) with acceptably high accuracy. In the present algorithm, the deblurring filter is given by

H(ρ)=W(ρ)/G(ρ)

Figure 2. The P-Deblurring Function is the product of (1) the inverse of a Gaussian blurring function and (2) a power window described in the text. In the absence of the power window, the inverse Gaussian would cause magnification of noise at high spatial frequencies.

W(ρ)≡exp(−αρ^m )

and the exponent m is chosen in the adaptation process. In the case of a Gaussian blurring function (see Figure 2), the P-deblurring function is given by

P(ρ)=exp[−αρ^m+(ρ²⁄2σ²)]

where σ is the standard deviation of the Gaussian function. Hence, the problem of choosing H(ρ) becomes the problem of choosing appropriate values of α, m, and σ in the adaptation process.

A complete characterization of the effect of the power window and a complete description of the adaptation process would greatly exceed the space available for this article. It must suffice to summarize as follows: The power window preserves most of the spatial-frequency components in its pass band while its attenuation increases rapidly but smoothly with increasing spatial frequency in a transition band, which lies between a peak at the outer edge (ρ_p)of the pass band and a transition point (ρ_n) defined as the spatial frequency above which the noise signal exceeds the input image signal. The smoothness of the transition helps to limit a ripple effect that can be especially pronounced in the outputs of prior, more-abrupt-edged deblurring filters. In the deblurring process, the energies of the signal and noise components of the image are estimated in order to estimate the pass band and transition band of the filter and, thereby, to provide guidance for choosing the parameters α, m, and σ. The main criteria applied in selecting the parameters are expressed by the following equations:

P(ρ)>1 for ρ<ρ_n
P(ρ)=1 for ρ=ρ_n
P(ρ)<1 for ρ>ρ_n

The adaptively designed deblurring filter is able to deblur the image by a desired amount based on the estimated or known blurring function, while reducing noise in the output image. The effectiveness of the adaptive deblurring algorithm has been demonstrated in human-perception experiments involving comparisons of blurred noisy images with their P-deblurred counterparts.

This work was done by S. Susan Young, Ronald G. Driggers, Brian P. Teaney, and Eddie L. Jacobs of the Army Research Laboratory.
ARL-0027

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