The digital reconstruction of single neurons from 3D confocal microscopic images is an important tool for understanding the neuron morphology and function. However the accurate automatic neuron reconstruction remains a challenging task due to the varying image quality and the complexity in the neuronal arborisation. Targeting the common challenges of neuron tracing, we propose a novel automatic 3D neuron reconstruction algorithm, named Rivulet, which is based on the multi-stencils fast-marching and iterative backtracking. The proposed Rivulet algorithm is capable of tracing discontinuous areas without being interrupted by densely distributed noises. By evaluating the proposed pipeline with the data provided by the Diadem challenge and the recent BigNeuron project, Rivulet is shown to be robust to challenging microscopic imagestacks. We discussed the algorithm design in technical details regarding the relationships between the proposed algorithm and the other state-of-the-art neuron tracing algorithms.