This paper studies the problem of secure state estimation of a linear time-invariant (LTI) system with bounded noise in the presence of sparse attacks on an unknown, time-varying set of sensors. At each time, the attacker has the freedom to choose an arbitrary set of no more than p sensors and manipulate their measurements without restraint. To this end, we propose a secure state estimation scheme and guarantee a bounded estimation error irrespective of the attack signals subject to 2p-sparse observability and a mild, technical assumption that the system matrix has no degenerate eigenvalues. The proposed scheme comprises a design of decentralized observers for each sensor based on the local observable subspace decomposition. At each time step, the local estimates of sensors are fused by a median operator to obtain a secure estimation, which is then followed by a local detection-and-resetting process of the decentralized observers. The estimation error is shown to be upper-bounded by a constant which is determined only by the system parameters and noise magnitudes. Moreover, we design the detector threshold to ensure that the benign sensors never trigger the detector. The efficacy of the proposed algorithm is demonstrated by its application on a benchmark example of IEEE 14-bus system. We show that our proposed scheme can effectively tolerate sparse attacks on an unknown set of sensors, ensuring a bounded estimation error and effectively detecting and resetting the attacked sensors.