Automating suturing during robotically-assisted surgery reduces the burden on the operating surgeon, enabling them to focus on making higher-level decisions rather than fatiguing themselves in the numerous intricacies of a surgical procedure. Accurate suture thread reconstruction and grasping are vital prerequisites for suturing, particularly for avoiding entanglement with surgical tools and performing complex thread manipulation. However, such methods must be robust to heavy perceptual degradation resulting from heavy noise and thread feature sparsity from endoscopic images. We develop a reconstruction algorithm that utilizes quadratic programming optimization to fit smooth splines to thread observations, satisfying reliability bounds estimated from measured observation noise. Additionally, we craft a grasping policy that generates gripper trajectories that maximize the probability of a successful grasp. Our full image-to-grasp pipeline is rigorously evaluated with over 400 grasping trials, exhibiting state-of-the-art accuracy. We show that this strategy can be applied to the various techniques in autonomous suture needle manipulation to achieve autonomous surgery in a generalizable way.