Blind Image Deconvolution using Approximate Greatest Common Divisor and Approximate Polynomial Factorisation

Images play a significant and important role in diverse areas of everyday modern life. Examples of the areas where the use of images is routine include medicine, forensic investigations, engineering applications and astronomical science. The procedures and methods that depend on image processing would benefit considerably from images that are free of blur. Most images are unfortunately affected by noise and blur that result from the practical limitations of image sourcing systems. The blurring and noise effects render the image less useful. An efficient method for image restoration is hence important for many applications.
Restoration of true images from blurred images is the inverse of the naturally occurring problem of true image convolution through a blurring function. The deconvolution of images from blurred images is a non-trivial task. One challenge is that the computation of the mathematical function that represents the blurring process, which is known as the point spread function (PSF), is an ill-posed problem, i.e. an infinite number of solutions are possible for given inexact data. The blind image deconvolution (BID) problem is the central subject of this thesis. There are a number of approaches for solving the BID problem, including statistical methods and linear algebraic methods. The approach adopted in this research study for solving this problem falls within the class of linear algebraic methods. Polynomial linear algebra offers a way of computing the PSF size and its components without requiring any prior knowledge about the true image and the blurring PSF.
This research study has developed a BID method for image restoration based on the approximate greatest common divisor (AGCD) algorithms, specifically, the approximate polynomial factorization (APF) algorithm of two polynomials. The developed method uses the Sylvester resultant matrix algorithm in the computation of the AGCD and the QR decomposition for computing the degree of the AGCD. It is shown that the AGCD is equal to the PSF and the deblurred image can be computed from the coprime polynomials.
In practice, the PSF can be spatially variant or invariant. PSF spatial invariance means that the blurred image pixels are the convolution of the true image pixels and the same PSF. Some of the PSF bivariate functions, in particular, separable functions, can be further simplified as the multiplication of two univariate polynomials. This research study is focused on the invariant separable and non-separable PSF cases.
The performance of state-of-the-art image restoration methods varies in terms of computational speed and accuracy. In addition, most of these methods require prior knowledge about the true image and the blurring function, which in a significant number of applications is an impractical requirement. The development of image restoration methods that require no prior knowledge about the true image and the blurring functions is hence desirable. Previous attempts at developing BID methods resulted in methods that have a robust performance against noise perturbations; however, their good performance is limited to blurring functions of small size. In addition, even for blurring functions of small size, these methods require the size of the blurring functions to be known and an estimate of the noise level to be present in the blurred image.
The developed method has better performance than all the other state-of-the-art methods, in particular, it determines the correct size and coefficients of the PSF and then uses it to recover the original image. It does not require any prior knowledge about the PSF, which is a prerequisite for all the other methods.