Regularization of the image division approach to blind deconvolution

Abstract

Randomly inhomogeneous media, such as a turbulent atmosphere, degrade images taken by optical systems. This imposes strong limitations on the resolution achieved by optical systems. The quest for increasing the angular resolution of terrestrial telescopes is still open. This work is a small contribution in that quest. A problem of blind deconvolution arises when one attempts to restore a short-exposure image that has been degraded by random atmospheric turbulence. The image division method attacks this problem by using two short-exposure images of the same object and taking the ratio of their respective Fourier transforms. The result is the quotient of the unknowns transfer functions. The latter are expressed as Fourier series in corresponding point-spread functions. Cross multiplying the division equation gives a system of linear equations with the point-spread functions as unknowns. It is found that the system of linear equations, resulting from the implementation of the image division method, has a multiplicity of solutions. Moreover such system of equations is poorly conditioned. This brings the necessity of a regularization approach. This dissertation describes the development and implementation of a regularization algorithm for the image division method. Using this regularization algorithm the blind deconvolution problem is posed as a constrained least-squares problem. A least-squares solution is found by computing a QR factorization of the system matrix. The Householder transformation method is used to find this factorization. The QR decomposition transforms the problem into an upper-triangular system of equations which is solved by backsubstitution. Prior partial knowledge about the point-spread functions and the object (such as finite support and positivity) is used to impose constrains on the solution, solving the multiplicity-solutions problem. The regularization algorithm is tested with simulated and real data. Good quality reconstructions are obtained from the implementation of the regularized image division method on computer simulated atmospheric degraded images corrupted with up to 5% of additive Gaussian noise, or corrupted with Poisson noise with 100 or more photons as the average number of photons per pixel. It also yields good results when tested with real infrared short-exposure images.