This technical note is concerned with the secure state estimation problem of a linear discrete-time Gaussian system in the presence of sparse integrity attacks. \(m\) sensors are deployed to monitor the state and \(p\) of them can potentially be compromised by an adversary, whose data can be arbitrarily manipulated by the attacker. We show that the optimal Kalman estimate can be decomposed as a weighted sum of local state estimates. Based on these local estimates, we propose a convex optimization based approach to generate a more secure state estimate. It is proved that our proposed estimator coincides with the Kalman estimator with a certain probability when all sensors are benign. Besides, we establish a sufficient condition under which the proposed estimator is stable against the $(p,m)$-sparse attack. A numerical example is provided to validate the secure state estimation scheme.