We present a solution method for adaptively smoothing magnetic resonance (MR) images while preserving discontinuities. We assume that the spatial behavior of MR data can be captured by a first order polynomial defined at every pixel. The formulation itself is similar to Leclerc's work on piecewise-smooth image segmentation, but we use the graduated non- convexity (GNC) algorithm as an optimizing tool for obtaining the solution. This requires initial values for polynomial coefficients of order greater than zero. These values are obtained by using ideas similar to that found in robust statistics. This initial step is also useful in determining the variance of the noise present in the input image. The variance is related to an important parameter α required by the GNC algorithm. Firstly, this replaces the heuristic nature of α with a quantity that can be estimated. Secondly, it is useful especially in situations where the variance of the noise is not uniform across the image. We present results on synthetic and MR images. Though the results of this paper are given using first order polynomials, the formulation can handle higher order polynomials.