Automating the process of manipulating and delivering sutures during robotic surgery is a prominent problem at the frontier of surgical robotics, as automating this task can significantly reduce surgeons' fatigue during tele-operated surgery and allow them to spend more time addressing higher-level clinical decision making. Accomplishing autonomous suturing and suture manipulation in the real world requires accurate suture thread localization and reconstruction, the process of creating a 3D shape representation of suture thread from 2D stereo camera surgical image pairs. This is a very challenging problem due to how limited pixel information is available for the threads, as well as their sensitivity to lighting and specular reflection. We present a suture thread reconstruction work that uses reliable keypoints and a Minimum Variation Spline (MVS) smoothing optimization to construct a 3D centerline from a segmented surgical image pair. This method is comparable to previous suture thread reconstruction works, with the possible benefit of increased accuracy of grasping point estimation. Our code and datasets will be available at: https://github.com/ucsdarclab/thread-reconstruction.