With the advancement of society, increasing attention to individual health has been accompanied by growing investments in healthcare technologies [1,2]. Engineering and technical professionals increasingly offer technical support to physicians based on specific clinical needs. The close cooperation between technology experts and medical professionals has significantly accelerated the development of medical technology, leading to the emergence of numerous innovative techniques. Among these innovations, image-guided surgical techniques [3–6], a cutting-edge minimally invasive treatment, have become widely adopted.

CT scans, fluoroscopy, magnetic resonance imaging (MRI), or ultrasound images are usually used to guide needle insertion in minimally invasive medical procedures. The use of CT has limitations, primarily due to the high doses of ionizing radiation patients are exposed to during the procedure [7]. Instruments made of non-magnetic and dielectric materials are restricted to permeabilization procedures [8]. Ultrasound is considered a safe and readily available imaging modality for observing needles and targets during surgery [9–11].

Calibration of surgical instruments is essential for surgical navigation before surgery [12–14]. The accuracy of instruments calibration determines the overall precision of surgical navigation, playing a critical role in the effectiveness and safety of surgery. Thus, the positioning and calibration of the probe needle in the surgical navigation systems remains a research focus. Traditional hand-eye calibration methods are challenging to implement in the confined spaces of the body. To address this challenge, researchers have proposed an autonomous instrument calibration method using a monocular camera [15]. This method utilizes visual feedback to track the instrument’s motion trajectory, followed by a two-stage parameter estimation process to achieve autonomous calibration. This approach overcomes the limitations of hand-eye observation, enhancing the safety of minimally invasive surgery. Some researchers have shifted their focus to surgical instruments, such as robotic arms, applying image-guided techniques to track their paths [16,17]. An economical, high-precision optical tracking system has been introduced by researchers. Within this system, Bumblebee2 stereo cameras are utilized to capture images of objects marked with distinctive identifiers. The central pixel coordinates of these markers are extracted, and specialized image processing techniques are applied to reconstruct the coordinates of the surgical instrument’s tip, facilitating tracking. This approach significantly enhances the system’s resilience to interference.

In recent years, the precision of surgical navigation systems has increasingly relied on accurate localization of the needle tip and robust calibration of ultrasound probes. However, most existing methods approach these two processes independently, leading to cumulative spatial errors during interventional procedures. For instance, Beigi et al. [18] developed techniques to enhance needle visibility and localization in ultrasound images, yet their method focuses solely on image-based improvements without addressing hardware alignment or spatial registration. Recent advancements have also proposed calibration schemes for ultrasound probes using either phantoms or robotic assistance. Wang et al. [19] introduced a dual-arm surgical navigation platform with independent ultrasound probe calibration via fiducial phantoms, achieving reasonable precision but at the cost of system complexity and multi-step workflows. Meanwhile, Seitel et al. [20] employed miniaturized electromagnetic tracking to guide needle insertions in real time, demonstrating accuracy improvements but still requiring separate probe-needle calibration steps. More recently, Che et al. [21] utilized deep learning to improve needle tip detection in ultrasound-based navigation, but their framework assumes pre-calibrated devices and does not consider joint optimization. These studies collectively reveal a significant research gap: there remains a lack of integrated calibration models that simultaneously account for both needle tips and ultrasound probe alignment within a unified coordinate system. This fragmentation results in accumulated spatial errors and hinders real-time surgical adaptability.

Despite progress in individual domains of needle tracking and ultrasound probe calibration, a major limitation remains: the absence of a unified, model-driven calibration framework that directly relates the physical space of the surgical instrument to the imaging plane of the ultrasound probe in a mathematically consistent and spatially coherent manner. Existing techniques typically decouple the two processes: first localizing the needle tip through geometric constraints, then calibrating the probe separately using phantoms or empirical image mappings. This two-stage pipeline introduces cumulative transformation errors, increases system complexity, and often requires manual intervention or fiducial markers, which are undesirable in real-time, minimally invasive procedures.

In contrast, our work introduces a joint calibration model that integrates both the needle tip localization and the ultrasound probe-to-image plane calibration within a shared coordinate system, supported by explicit coordinate transformation and optimization models. By leveraging the constrained spherical motion of the instrument during rotation, we formulate a gradient-descent-based solution to estimate the true spatial position of the needle tip. This result is then used as a geometric anchor in an N-line ultrasound calibration model, allowing the system to compute the spatial transformation between the ultrasound image plane and the physical coordinate frame without relying on external phantoms. Our approach thus not only reduces system complexity and accumulated error but also demonstrates submillimeter localization accuracy (0.1644 mm) and low pixel deviation in image alignment (1.2373 mm), validated across 253-frame sequences. These contributions position our method as a practical and precise calibration strategy for real-time ultrasound-guided surgical navigation, especially in scenarios requiring portability, reduced radiation exposure, and high spatial fidelity. It fills a critical gap in the literature by offering an end-to-end mathematical framework that bridges the calibration of hardware (needle and probe) with imaging output in a single optimization loop.

Initially, capturing the tip of the surgical instrument is essential. After the initial capturing of the tip, the next step is to establish the optical positioning systems for both the surgical instrument and its tip. Using optical positioning, the three-dimensional spatial coordinates of three non-collinear feature points on the surgical instrument, along with the coordinates of the tip, are captured. Next, the spatial coordinates of the instrument’s tip point are determined relative to its own coordinate system. As the surgical instrument rotates around its tip, the three non-collinear points trace paths on a concentric spherical surface. Leveraging this spherical motion constraint, the spatial coordinates of the tip can be precisely calibrated within the coordinate system defined by the surgical instrument. This method ensures high accuracy in the calibration process, which is vital for the precision and reliability of surgical navigation systems.

Surgical navigation systems have undergone rapid advancements in recent years, particularly in improving intraoperative precision and reducing patient trauma. Nevertheless, traditional navigation methods based on binocular stereovision often suffer from limited visibility in obstructed anatomical regions and lack robustness in dynamic scenarios [28,29]. To address these limitations, our approach integrates optical tracking, robotic manipulation, and real-time ultrasound guidance into a unified calibration framework.

A critical challenge in ultrasound-guided interventions is the accurate spatial registration between the ultrasound probe and the needle tip, which has typically been addressed in isolation. For instance, Kim et al. [30] proposed a calibration procedure for needle-guide systems attached to ultrasound probes, but their method relied on fixed mechanical assumptions and was not applicable to freehand scenarios. Similarly, Paralikar et al. [31] developed a robot-assisted calibration for the probe itself, but without considering the dynamic spatial behavior of the needle tip.

Other studies such as Carbajal et al. [32] utilized N-wire phantoms for ultrasound probe calibration, but treated needle localization as a separate or manually controlled process, introducing potential cumulative error. In contrast, our method achieves joint calibration, in which the accurately optimized needle tip coordinates are directly employed to solve the N-line model of the ultrasound probe. This integration ensures tighter spatial coupling, minimizes registration errors, and eliminates reliance on external fiducial phantoms.

Our experimental results demonstrate that this integrated approach significantly enhances calibration precision. The needle tip localization error remains under 0.25 mm across 253 captured frames, and the ultrasound probe calibration achieves a pixel deviation of only 1.2373 mm. Furthermore, the alignment of the calibrated ultrasound image with a known planar surface shows deviation below 1 mm, reinforcing the spatial accuracy of the proposed method. Compared to previous methods, this represents a substantial improvement in both accuracy and robustness under freehand or robotic setups.

In summary, the proposed joint calibration framework fills a notable gap in the current literature by unifying two previously separate procedures—needle tip and probe calibration—into a single optimization-based workflow. This method has strong potential for application in real-time surgical navigation, especially in 2D ultrasound-based systems that require precise spatial coordination without the aid of ionizing radiation.

Nevertheless, several limitations remain. The current experiments were conducted in a phantom environment using an optical tracking system, industrial camera, and robotic arm, we clarify that the proposed joint calibration method has not yet been deployed or tested on a fully integrated surgical navigation system in clinical or pre-clinical settings. The implementation was performed within a controlled prototype environment designed to simulate real-world constraints, including hardware synchronization and image acquisition workflows that align with those of standard navigation systems. In addition, while gradient descent with a fixed step size proved effective in our study, we acknowledge that more advanced optimization strategies, such as particle swarm optimization or other metaheuristic algorithms, may further enhance robustness in more complex or nonlinear calibration scenarios. Recent reviews and algorithmic studies on robot calibration highlight the effectiveness of data-driven and evolutionary approaches for improving optimization performance, which could inspire future extensions of our framework [33,34].

The entire calibration pipeline was implemented end-to-end, covering both needle tip localization and ultrasound probe-to-image registration. However, real-time integration into a functioning navigation system with user interface, intraoperative imaging, and surgeon feedback loop remains a direction for future development. The current results demonstrate the feasibility and internal consistency of the proposed model, providing a solid computational foundation for future system-level integration. We acknowledge that validation under more complex surgical conditions, such as soft tissue deformation, occlusion, or dynamic anatomical changes, will be necessary to demonstrate robustness in realistic environments. As such, future work will focus on extending the system to animal models or hybrid OR setups, and addressing practical challenges like tracking loss, calibration drift, and probe misalignment through sensor fusion and online error correction methods.

In this research, we introduce an advanced method for precise needle tip localization using a sophisticated rotation algorithm. This technique involves rotating the needle tip by 130 degrees around a fixed pivot, which enables the acquisition of multiple images of four optical spheres within the camera’s coordinate system. These images are then employed to refine the template coordinate system through gradient descent optimization. Subsequently, the coordinates within the template system are transformed into the instrument’s coordinate system. Further optimization of these coordinates is performed within the camera’s coordinate system. Notably, as the number of captured images increases, the calibration error remains consistently low, demonstrating remarkable stability and accuracy.

We then utilize the optimized needle tip coordinates to develop a joint calibration algorithm for the ultrasound probe. This calibration involves identifying the two endpoints of each N-line using the calibrated needle tip and solving the N-line equations based on ultrasound images. Additionally, using data from optical spheres captured by the optical positioning framework, we derive a transformation matrix to facilitate the transition from the probe’s reference system to the camera’s coordinate system. The calibration matrix for the ultrasound probe is formulated using a least-squares methodology, ensuring an optimized solution.

In our experimental setup, we used a Sonostar digital ultrasound imaging system equipped with a CProbe probe. The camera used was the MV-MSU130GM2-T industrial camera from MedViewTech. The mechanical arm used in the experiments was the EC63 intelligent lightweight 6-degree-of-freedom modular collaborative robot developed by Suzhou Eletech Robotics Co. Our experimental results validate the proposed method’s efficacy in achieving high-precision calibration for 2D ultrasound surgical navigation, showing stable calibration errors across different numbers of images.

Overall, the study establishes a practical and accurate calibration strategy that addresses a long-standing gap in ultrasound-guided navigation, offering a streamlined workflow with potential for translation into real-time minimally invasive surgical applications.