The research field of visual-inertial odometry has entered a mature stage in recent years. However, unneglectable problems still exist. Tradeoffs have to be made between high accuracy and low computation for users. In addition, notation confusion exists in quaternion descriptions of rotation; although not fatal, this may results in unnecessary difficulties in understanding for researchers. In this paper, we develop a visual-inertial odometry which gives consideration to both precision and computation. The proposed algorithm is a filter-based solution that utilizes the framework of the noted multi-state constraint Kalman filter. To dispel notation confusion, we deduced the error state transition equation from scratch, using the more cognitive Hamilton notation of quaternion. We further come up with a fully linear closed-form formulation that is readily implemented. As the filter-based back-end is vulnerable to feature matching outliers, a descriptor-assisted optical flow tracking front-end was developed to cope with the issue. This modification only requires negligible additional computation. In addition, an initialization procedure is implemented, which automatically selects static data to initialize the filter state. Evaluations of proposed methods were done on a public, real-world dataset, and comparisons were made with state-of-the-art solutions. The experimental results show that the proposed solution is comparable in precision and demonstrates higher computation efficiency compared to the state-of-the-art.