This study compares the effectiveness of the traditional minimum circle detection strategy, i.e. Welzl's algorithm, and the state-of-the-art nnU-Net in the localization of lumbar Epidural Steroid Injection (ESI) robot markers across different imaging modalities (MRI and CT). Fiducial frames and markers of identical design were used in both settings. To adjust for human errors in the benchmarking process, experiments were conducted to compare computational and manual marking results. Due to the complexity of the CT dataset, the accuracy and sensitivity of 3D nnU-Net were significantly superior to Welzl's algorithm. However, in the relatively simple MRI datasets, Welzl's algorithm outperformed the learning-based method. Subsequent experiments were conducted to validate that the localization accuracy meets the contingency requirements. This finding informs potential improvement in the current clinical workflow and the registration process of surgical robots in general.