Blind image deconvolution via fast approximate GCD

The problem of blind image deconvolution can be solved by computing approximate greatest common divisors (GCD) of polynomials. The bivariate polynomials corresponding to the z-transforms of several blurred images have an approximate GCD corresponding to the z-transform of the original image. Since blurring functions as cofactors have very low degree in general, this GCD will be of high degree. On the other hand, if we only have one blurred image and want to identify the original scene, the blurred image can be partitioned such that each part completely contains the blurring function, hence the blurring function becomes the GCD which is of low degree. Therefore, we design a specialized algorithm for computing GCDs of polynomials to recover true images in two different cases. The new algorithm is based on the fast GCD algorithm for univariate polynomials and the Fast Fourier Transform (FFT) algorithm. The complexity of our specialized algorithm for identifying both the true image and the blurring functions from blurred images of size n × n is 0(n ² log(n)) in the case of blurring functions of very low degree. The algorithm has been implemented in Maple and can extract true images of hundreds by hundreds pixel images from blurred images in a few seconds.