Hyper-Laplacian priors combined with rotating pupils for image restoration in sparse aperture systems

A hyper-Laplacian can model the heavy-tailed distribution of gradients in natural scenes well, which have proven effective priors for deconvolution and denoising. However, because of missing point spread function (PSF) information in the two-dimensional spatial domain of optical sparse aperture (OSA) systems, a hyper-Laplacian prior of single exposure cannot recover the missing information of images. The main focus of this paper is on combining hyper-Laplacian priors with a pupil and its rotated pupils to compensate PSF information and improve the image quality in OSA systems. A scheme of rotating the pupil that has double apertures is analyzed. The cost function relative to multiple degraded images and PSFs obtained by rotating the pupil is established. The alternating minimization algorithm consisting of two phases is implemented to acquire restored images. In one phase, the non-convex part of the problem is solved. In the other phase, the fast Fourier transforms (FFTs) are used to solve a quadratic equation in the frequency domain. Using the peak signal-to noise ratio (PSNR), a quantitative analysis is provided. Simulation results show that hyper-Laplacian priors combined with rotating pupils can restore images better than a hyper-Laplacian prior of single exposure in an OSA system. Taking spoke-square image as the test image, the PSNR is 28.34 dB with two rotations and 23.52 dB without rotation. Moreover, the numbers of rotating the pupil that lead to different changes of the image quality are demonstrated.