In this paper robust nonlinear image denoising algorithms are introduced for the distribution which is Gaussian in the center and Laplacian in the tails. The distribution is known as the least favorable ${epsilon}$-contaminated normal distribution that maximizes the asymptotic variance. The proposed filter proves to be the maximum likelihood estimator under the heavy-tailed Gaussian noise environments. It is optimal in the respect of maximizing the efficacy under the above noise environment. Another filter for reducing impulsive noise is proposed by mixing with the myriad filter to propose an amplitude-limited myriad filter. Extensive experiment is conducted with images corrupted with ${alpha}$-stable noise to analyze the behavior and performance of the proposed filters.