Non-stationary MRI noise occurs in sparse and non-uniform k-space. Weighted least squares regression has been used to handle data with non-stationary noise. A minimum-variance weighting function may reduce the variance (image noise) of the image, and it may also relax the regularization needed for MRI reconstruction. To obtain the optimal weighting in non-uniform MRI reconstruction, this study uses the Monte Carlo method to determine the minimum-variance weighting function in Shepp-Logan phantom and breast MRI. The parameter $$$\alpha=-0.5$$$ provides a weighting function with the minimum-variance in the reconstructed images.