Regularization for denoising in magnetic resonance imaging (MRI) is typically achieved using convex regularization functions. Recently, deep learning techniques have been shown to provide superior denoising performance. However, this comes at the price of lack of explainability, interpretability and stability, which are all crucial to MRI. In this work, we present a constructive approach for designing weakly-convex regularization functions for MR image denoising. We show that our technique performs on par with state-of-the-art denoisers for diffusion-weighted MR image denoising. Our technique can be applied to design weakly-convex convolutional neural networks with prototype activation functions that impart interpretability and are provably convergent. We also show that our technique exhibits fewer denoising artifacts by demonstrating its effect on brain microstructure modelling.