Memristive devices, with a fusion of memory and logic functions, provide good opportunities for configuring new concepts computing. However, progress towards paradigm evolution has been delayed due to the limited understanding of the underlying operating mechanism. The stochastic nature and fast growth of localized conductive filament bring difficulties to capture the detailed information on its growth kinetics. In this work, refined programming scheme with real-time current regulation was proposed to study the detailed information on the filament growth. By such, discrete tunneling and quantized conduction were observed. The filament was found to grow with a unit length, matching with the hopping conduction of Cu ions between interstitial sites of HfO2 lattice. The physical nature of the formed filament was characterized by high resolution transmission electron microscopy. Copper rich conical filament with decreasing concentration from center to edge was identified. Based on these results, a clear picture of filament growth from atomic view could be drawn to account for the resistance modulation of oxide electrolyte based electrochemical memristive elements.