Image deconvolution methods based on fourier transform phase and bounded energy

Abstract

We developed deconvolution algorithms based on Fourier transform phase and bounded energy. Deconvolution is a major area of study in image processing applications. In general, restoration of original images from noisy filtered observation images is an ill-posed problem. We use Fourier transform phase as a constraint in developed image recovery methods. The Fourier phase information is robust to noise, which makes it suitable as a frequency domain constraint. One of our focus is microscopy images where the blur is caused by slight disturbances of the focus. Because of the symmetrical optical parameters, it may be assumed that the Point Spread Function (PSF) is symmetrical. This symmetry of PSF results in zero phase distortion in the Fourier transform coefficients of the original image. Since the convolution leads to multiplication in Fourier domain, we assume that the Fourier phase of some of the frequencies of observed image around the origin represents the Fourier phase of the original image in the same set of frequencies. Therefore the Fourier transform phases of the original image can be estimated from the phase of the observed image and this information can be used as a Fourier domain constraint. In order to complete the algorithm, we also use a Total Variation (TV) reduction based regularization in spatial domain. We embed the proposed Fourier phase relation and spatial domain regularization as additional constraints in well-known blind Ayers-Dainty deconvolution method. Another problem we focused on is the restoration of highly blurry Magnetic Particle Imaging (MPI) applications. In this study we developed a standalone iterative algorithm. The algorithm again relies on the symmetry property of the MPI PSF. The phase estimates of the true image are obtained from the observed image. In this case we employ an `1 projection based regularization algorithm. The `1 projection reduces the small coefficients to zero which is suitable for MPI application because the contrast between foreground and background is sufficiently large by nature. Finally, a more general restoration algorithm is developed for deconvolution of non-symmetrical filters. The algorithm uses the known Fourier phase properties of the PSF in order to estimate the Fourier transform phase of the original image. We also update the estimated Fourier transform magnitudes iteratively using the knowledge of observed image and the PSF. A TV reduction based regularization method completes the algorithm in spatial domain. Simulations and experimental results show that the proposed algorithm outperforms the Wiener filter. We also conclude that the addition of estimate of Fourier transform phase is useful in any deconvolution method.