Two-Dimensional Blind Deconvolution Using a Robust GCD Approach

In this paper we examine the applicability of the previously proposed Greatest Common Divisor GCD method to blind image deconvolution In this method the desired image is approximated as the GCD of the two dimensional polynomials corresponding to the z transforms of two or more distorted and noisy versions of the same scene assuming that the distortion lters are FIR and relatively co prime We justify the break down of two dimensional GCD into one dimensional Sylvester type GCD algorithms which lowers the com putational complexity while maintaining the noise ro bustness A way of determining the support size of the true image is also described We also provide a solu tion to deblurring using the GCD method when only one blurred image is available Experimental results are shown using both synthetically blurred images and real motion blurred pictures Introduction Blind image deconvolution is the process of identi fying both the true image and the blurring function from the degraded image using partial information about the imaging system Although this has been shown to be possible by making use of the irreducible property of D polynomial factors a small amount of additive noise can lead to large deviations and a unique solution may not be found In existing methods dealing with such problems reliability is often traded with high computational complexity A novel approach has been proposed in that identi es the true image from two blurred versions of the same scene Assuming that the distortion l ters are FIR the z transform of the images and lters can be written as two dimensional polynomials The problem is then transformed into estimating the GCD This research work was supported by the O ce of Naval Research under contract N J P of D polynomials corresponding to the z transforms of the blurred images and a robust interpolative D GCD method is introduced based on a D Sylvester type GCD algorithm In the following section we will rst brie y describe the method and then intro duce an extension to determine the order of the GCD polynomial when the support size of the true image is unknown A justi cation is provided to the process of D interpolation of the D GCD using Discrete Fourier Transform points on the unit circle Section shows how to apply the GCD method on a single blurred image with the example of a real motion blurred picture The D GCD Approach with Two Blurred Images Problem Formulation Let f m n and f m n represent two distorted versions of a true image p m n in presence of noise Thus f m n p m n d m n n m n f m n p m n d m n n m n