Most edge-preserving image restoration algorithms preserve discontinuities that are larger than a prescribed threshold value, therefore noise components whose differences in neighboring pixels are larger than the threshold become amplified unintentionally. We propose a noise-adaptive edge-preserving image restoration algorithm based on a Markov random field image model. The proposed potential function is controlled by the weighting function to adaptively incorporate the discontinuities into the solution. To avoid undesirable amplification of the noise, we introduce a noise-adaptive threshold to each pixel difference. As a result, the potential function varies its shape from a quadratic form to a concave form according to the amount of noise added to each pixel. In doing so, high-frequency components caused by strong noise are relatively more smoothed as with the quadratic potential function used, while edge components that have a small noise intensity are well preserved. The smoothing functional to be minimized is formulated to have a global minimizer in spite of its nonlinearity by enforcing the convergence and convexity requirements. The effectiveness of the proposed algorithm is demonstrated experimentally.