A Weibull gradient prior for image restoration

In image restoration, image priors with a heavy-tailed gradient distribution found in natural images have been successfully applied to address ill-posed problems such as denoising, deblurring, and super-resolution. However, existing literature has limitations in fitting well with image gradient magnitudes using widely used distributions. This paper introduces the Weibull distribution as a potential model that can better describe image gradient magnitudes compared to existing methods. Specifically, the Weibull distribution provides the best fit for describing small gradient magnitudes. Within the framework of maximum a posteriori estimation, a recovery model based on the Weibull gradient prior is introduced. The shape parameter of the Weibull prior is automatically estimated during the iterative process using the method of moments, allowing the model to better conform to the real images’ gradient distribution. The proposed model is computed using the alternating direction method of multipliers. Moreover, the properties of the solution to the sub-problem related to the Weibull prior are analyzed, and an efficient numerical algorithm for solving the multivariate proximal operator is proposed. Experimental results validate that the proposed model exhibits superior performance and is particularly well-suited for piecewise constant images compared to competing methods.