The paper proposes a fast fault detection method for radial DC microgrids established on mathematical morphology (MM) denoising filters and detection principles utilizing only local measurements. The proposed fault detection algorithm utilizes the DC fault current signal from a Voltage Source Converter (VSC) and detects a fault. Problems related to communication delays are thus avoided, while preserving the low cost and computational burden. The proposed fault detection framework includes five different MM-based denoising filters, employed as pre-signal processing methods whose features are assessed and compared. The influence of selecting an appropriate type and length of a structuring element (SE) on the proposed method is analyzed. The developed fault detection method based on MM is used for extracting transient features and detecting pole-to-pole (PP) and pole-to-ground (PG) faults within milliseconds of their occurrence. The proposed method is demonstrated using simulation tests in MATLAB/Simulink. The results indicate that the proposed method enables reliable, accurate and fast detection of both the PP and the PG faults. Computer simulations are carried out to verify that the proposed method distinguishes faults from overload.