Background subtraction is needed to extract foreground information from a video sequence for further processing in many applications, such as surveillance tracking. However, due to the presence of a dynamic background and noise, extracting foreground accurately from a video sequence remains challenging. A novel projection method, namely Principal Mode Component Analysis (PMCA), is proposed to capture the most repetitive patterns of a video sequence, which is one of the key characteristics of the video background. The patterns are captured by applying the bootstrapping method together with the statistic mode measure. The bootstrapping method can model the distribution of almost any statistic of the dynamic background and complicated noise. This is different from current methods, which restrict the distribution to a closed-form function. We introduce a mathematical relaxation that can formulate the statistical mode measure for a continuous video data. A fast exhaustive search method is proposed to find the global optimal solution for the PMCA. This fast method adopts a simplification procedure that makes the optimization procedure independent of the video size. The proposed method is computationally much more traceable than existing ones. We compare the proposed method with 10 different methods, including several state-of-the-art techniques, for 19 different real-world video sequences from two popular datasets. Experiment results show that the proposed method performs the best in 16 cases and second best in 2 cases.