3D trajectory reconstruction is a crucial task in Autonomous Driving, Augmented Reality and Scientific Training. However, in critical configurations where the camera is in smooth motion, this task remains challenging due to the lack of depth information. In this paper, we study the 3D reconstruction of human motion from 2D projections, aiming to reconstruct 3D tra-jectories more accurately in the critical configuration(i.e. Scenes with smooth camera motions), which remedy the limitation of the existing methods. We proposed a new function to search the sparse solution in the discrete cosine transformation domain with $\ell_{1}$ optimization, in order to build a good reconstruct space for critical configurations, which transforms the underconstrained system into a convex optimization problem and solve it with a new algorithm for the first time as we known. By improving the Cross-Validation, we propose a reconstruction criterion to adaptively select coefficient weights $\beta$ and the number of basis vectors $K$ for each point, which makes the reconstruction satisfy a good situation. Experiments on synthetic datasets made on Human3.6m and real datasets show that our method is superior to existing methods in 3D reconstruction under smooth camera motion.