Image Deconvolution Based On Fourier-Total Variation Regularization

Image deconvolution is one of the most common techniques for image restoration problem, which has been used in many fields for science and technology, such as space exploration, medical imaging, remote sensing, historical and cultural research, etc. We present an interactive iterative regularization algorithm based on shrinkage operator in Fourier domain and total variation model for image deconvolution. The preliminary experimental results show the effectiveness of the algorithm.Generally, we assume that the observed blurred image is the result of the original clear image convolutes with a point spread function, while image deconvolution is to obtain an estimate of the original clear image from the observed image and the point spread function. Because image will be mixed with random noise inevitably in the process of collection, transmission, transform, etc. which leads to the ill-posedness of the deconvolution problem. Usually, a regularization method is used to change this ill-posed problem into a well-posed problem, namely, using the prior knowledge of the practical problem to add more constraints to the problem such that the solution of the problem will depends on the observed data continuously and be meaningful in practical application.Among general linear transformations, the Fourier transform captures the maximum colored noise energy using a fixed number of coefficients. Thus a shrinkage operator in Fourier domain, such as Tikhonov regularization operator, is used firstly to make an initial estimate of the observed blurred image in our proposed algorithm, which can significantly reduce the blurring of the image by choosing appropriate parameters, but left more colored noise. One of the most successful models in image denoising is the total variation regularization model that is consequently used to regularize the above initial estimate from Tikhonov regularization, which can reduce the colored noise. Then, we use Tikhonov regularization with the above result again to find back some of the lost image information in the process of denoising. The experimental results show that using several times of the interactive iteration of Tikhonov Regularization and Total Variation Regularization can have the satisfied deconvolution results.On the one hand, Tikhonov regularization can be directly calculated individually in the Fourier domain, which results in the initial estimate fast. On the other hand, the total variation denoising model with the L1 norm can be solved quickly by using the split Bregman iteration algorithm. So, even if using several interactive iterations of the two regularization methods, we can also have relatively fast processing speed.At last, we give two possible improvements for the image deconvolution algorithm based on Fourier Regularization and Total Variation Regularization. Firstly, we can use the LTI Wiener deconvolution operator to instead of Tikhonov regularization operator, which reduce the Mean-Squeared Error between the real image and the estimate image. Secondly, using the total variation model with adaptive fidelity term can preserve the details of the image such as edges and textures better, because the distribution of the colored noise corresponds to the spatial location of the image.