This paper analyzes the performance of single-cell massive multiple-input multiple-output (MIMO) systems with non-orthogonal pilots. Specifically, closed-form expressions of the normalized channel estimation error and achievable uplink capacity are derived for both least squares (LS) and minimum mean square error (MMSE) estimation. Then a pilot reconstruction scheme based on orthogonal Procrustes principle (OPP) is provided to reduce the total normalized mean square error (NMSE) of channel estimations. With these reconstructed pilots, a two-step pilot assignment method is formulated by considering the correlation coefficient among pilots to reduce the maximum NMSE. Based on this assignment method, a step-by-step pilot power allocation scheme is further proposed to improve the average uplink signal-to-interference and noise ratio (SINR). At last, simulation results demonstrate the superiority of the proposed approaches.