Surgical knot tying is one of the most fundamental and important procedures in surgery, and a high-quality knot can significantly benefit the postoperative recovery of the patient. However, a longtime operation may easily cause fatigue to surgeons, especially during the tedious wound closure task. In this paper, we present a vision-based method to automate the suture thread grasping, which is a sub-task in surgical knot tying and an intermediate step between the stitching and looping manipulations. To achieve this goal, the acquisition of a suture's three-dimensional (3D) information is critical. Towards this objective, we adopt a transfer-learning strategy first to fine-tune a pre-trained model by learning the information from large legacy surgical data and images obtained by the onsite equipment. Thus, a robust suture segmentation can be achieved regardless of inherent environment noises. We further leverage a searching strategy with termination policies for a suture's sequence inference based on the analysis of multiple topologies. Exact results of the pixel-level sequence along a suture can be obtained, and they can be further applied for a 3D shape reconstruction using our optimized shortest path approach. The grasping point considering the suturing criterion can be ultimately acquired. Experiments regarding the suture 2D segmentation and ordering sequence inference under environmental noises were extensively evaluated. Results related to the automated grasping operation were demonstrated by simulations in V-REP and by robot experiments using Universal Robot (UR) together with the da Vinci Research Kit (dVRK) adopting our learning-driven framework.