# Optimization-Based Calibration for Intravascular Ultrasound Volume Reconstruction

Karl-Philippe Beaudet, Sidaty El Hadramy, Philippe C Cattin, Juan Verde, Stéphane Cotin

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Karl-Philippe Beaudet, Sidaty El Hadramy, Philippe C Cattin, Juan Verde, Stéphane Cotin. Optimization-Based Calibration for Intravascular Ultrasound Volume Reconstruction. ASMUS 2025: The 6th International Workshop on Advances in Simplifying Medical UltraSound, Sep 2025, Deajon, South Korea. 10.1007/978-3-032-06329-8_6. hal-05226307

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# Optimization-Based Calibration for Intravascular Ultrasound Volume Reconstruction

Karl-Philippe Beaudet* †1,2,4, Sidaty El Hadramy* 1,2,3,4, Philippe C. Cattin³, Juan Verde¹,²,⁴, and Stéphane Cotin¹,²

¹ Inria, Strasbourg, France

² University of Strasbourg, CNRS, INSERM, ICube, UMR7357, Strasbourg, France

³ Department of Biomedical Engineering, University of Basel, Switzerland

⁴ Institute of Image-Guided Surgery, IHU Strasbourg, Strasbourg, France

**Abstract.** Intraoperative ultrasound images are inherently challenging to interpret in liver surgery due to the limited field of view and complex anatomical structures. Bridging the gap between preoperative and intraoperative data is crucial for effective surgical guidance. 3D IntraVascular UltraSound (IVUS) offers a potential solution by enabling the reconstruction of the entire organ, which facilitates registration between preoperative computed tomography (CT) scans and intraoperative IVUS images. In this work, we propose an optimization-based calibration method using a 3D-printed phantom for accurate 3D Intravascular Ultrasound volume reconstruction. Our approach ensures precise alignment of tracked IVUS data with preoperative CT images, improving intraoperative navigation. We validated our method using *in vivo* swine liver images, achieving a calibration error from 0.88 to 1.80 mm and a registration error from 3.40 to 5.71 mm between the 3D IVUS data and the corresponding CT scan. Our method provides a reliable and accurate means of calibration and volume reconstruction. It can be used to register intraoperative ultrasound images with preoperative CT images in the context of liver surgery, and enhance intraoperative guidance.

**Keywords:** Intraoperative Ultrasound · Computer-Assisted Surgery · Optimization · Tracked Ultrasound · 3D Ultrasound

## 1 Introduction

IntraVascular UltraSound (IVUS) is a medical imaging technique that involves an ultrasound (US) probe mounted at the tip of a catheter, which can be navigated inside blood vessels. Although IVUS is commonly used in cardiovascular procedures [11], recent work shows that it also has meaningful applications in liver surgery, where its real-time, minimally invasive, and radiation-free imaging capabilities can offer substantial benefits for intraoperative guidance [15]. Reconstructing a 3D volume from 2D IVUS images helps visualize and/or compute

<footnote>* These authors contributed equally to this work</footnote>

<footnote>† Corresponding author: kp.beaudet@gmail.com</footnote>
spatial relationships between internal structures, facilitating its use in computer-assisted interventions [1, 3, 6]. This process includes estimating relative spatial transformations between 2D frames. Although numerous approaches have been proposed to find these transformations without any tracking system [4, 8, 12], these methods often suffer from cumulative drift errors, limiting their effectiveness in applications that require high precision. A tracking system enables accurate drift-free 3D reconstruction of the US volume. However, when manually mounted on an ultrasound probe, the sensor position relative to the image coordinate system is unknown. Therefore, calibration is necessary to determine the transformation that maps the 2D pixel's coordinates in the US image to the 3D coordinates in the sensor reference frame.

Various methods for calibrating ultrasound probes have been proposed [10]. They can generally be categorized into two main approaches. Image-based calibration [16] typically relies on image registration techniques to compare images of the same region from different views, estimating the spatial transformation that aligns them. These methods are advantageous because they do not require additional hardware. However, they are limited by factors such as image quality [9], lack of distinct features, and the small field of view, especially in the case of IVUS [11]. Another class of methods is tool-based calibration, which relies on external objects [14]. A common approach involves using a tracked stylus with a known position, placed at multiple locations so that its tip appears in the ultrasound image from different angles [10]. Other methods rely on dedicated calibration phantoms with well-defined geometric properties, such as spheres, point targets, crossing wires, or more complex 3D-printed structures [13, 17]. Positioning the phantom within the field of view of the ultrasound probe, the system correlates the known geometry with the acquired image data to determine the transformation. However, existing phantoms were designed for standard probes with less constrained in-plane motion, while IVUS operates with predominantly out-of-plane motion and a constrained environment [3, 11]. Additionally, some methods restrict probe orientation, such as wire phantoms with limited space between walls, leading to under-sampling of the six degrees of freedom. This results in varying calibration accuracy between in-plane and out-of-plane motion, further limiting the applicability of the existing methods for IVUS calibration.

In this work, we propose a calibration method adapted to IVUS probes. Our method relies on the 3D design of a phantom that constrains the environment of the IVUS probe and allows for out-of-plane rotational motion, mimicking the conditions of an IVUS inserted into a blood vessel. Landmarks from the phantom are detected in the acquired images and used to find calibration parameters that maximize the accuracy of the 3D reconstructed volume. To the best of our knowledge, this is the first method to propose a calibration approach designed for an IVUS probe. Our contributions are as follows:
1) A phantom designed for IVUS calibration, considering space constraints and rotational motion.
2) A gradient-based optimization method for IVUS probe calibration.
3) Experiments on in-vivo porcine liver demonstrating the method's reliability and efficiency.
The paper is organized as follows: Section 2 states the mathematical formulation of the problem, Section 3 describes the proposed method and its novel aspects, Section 4 presents the experiments conducted, highlighting the accuracy and efficiency of the proposed approach. Section 5 concludes this work and discusses potential directions for future work.

## 2 Problem formulation

We address the spatial calibration problem for a sensor **manually** mounted near the tip of an IVUS probe, as illustrated in Figure 1. The probe captures 2D images within its field of view. Given the transformation $T_{s→w}$ from the sensor to the world reference frame and the acquired images, the aim is to determine the unknown transformation $T_{i→s}$ from the image to the sensor reference frame. This transformation is represented as a 4 × 4 matrix that includes translation, rotation, and scaling.

<figure><img src="image_2.png" /><figcaption>Fig. 1: Illustration of the tracked IVUS setup. A sensor is mounted near the IVUS probe tip, capturing a 2D image. The figure shows three reference frames (world, sensor, and image) and the transformations: the known $T_{s→w}$ and the unknown $T_{i→s}$, which includes translation, rotation, and scale. We consider here the world reference frame as the one where the sensor coordinates are expressed.</figcaption></figure>

## 3 Materials and methods

We propose a two-step approach: a 3D-printed phantom to constrain the IVUS probe and enable rotational motion, followed by an optimization loop to estimate the calibration matrix. While we use electromagnetic (EM) tracking in this work, the proposed method is adaptable to any system providing translation and orientation information. The following sections detail both steps.
## 3.1 Phantom design

We designed an IVUS-specific calibration phantom, illustrated in Figure 2, that constrains the probe within a tube, enabling controlled rotation and translation along one axis. The phantom incorporates needle-shaped markers arranged in three distinct clusters positioned at 60°, 90°, and 120° to define three IVUS poses. Each cluster comprises five asymmetrically arranged needles aligned with the IVUS image plane to facilitate their identification in the image plane. Their lengths range from 1 to 5 cm. The phantom features a half-cylinder cavity with a 6.5 cm internal radius, ensuring that all markers are visible at ultrasound depths ranging from 7 to 11 cm. The model was printed in PLA+ white with 15% infill using tree supports. The process took 3 hours and 29 minutes on a Bambu Lab X1-Carbon 3D printer.

<figure><img src="image_3.png" /><figcaption>Fig. 2: IVUS calibration phantom with a half-cylinder cavity and three marker clusters at 60°, 90°, and 120° defining distinct poses and proper image orientation.</figcaption></figure>

## 3.2 Calibration process

Using the IVUS-specific phantom described in the previous section, we address the calibration problem formulated in Section 2. This is achieved by solving an optimization problem that finds the calibration matrix, ensuring a minimal discrepancy between the ultrasound-reconstructed volume of the phantom and its 3D model when both are registered. We propose a gradient-based optimization process, outlined in Algorithm 1, where the parameters of the calibration matrix are iteratively adjusted to minimize a defined objective function.

**Step 1: Calibration matrix definition:** The calibration matrix, denoted $C$, is defined with 7 parameters: 3 Euler angles (roll, pitch, yaw), 3 translations $(t_x, t_y, t_z)$ and a scaling factor $s$. It reads as:

$$C = \begin{bmatrix} s \cdot \mathbf{R}_{\text{yaw}} \cdot \mathbf{R}_{\text{pitch}} \cdot \mathbf{R}_{\text{roll}} [t_x, t_y, t_z]^T \\ 0 \\ 1 \end{bmatrix} \quad (1)$$

The calibration matrix $C$ is the unknown we seek to determine. We represent the rotation matrices using Euler angles and optimize over these angles and the
**Algorithm 1 Calibration matrix optimization process**

**Input:** The phantom needle points coordinates ($P_{Phantom}$), the needle points coordinates in the ultrasound images ($P_{US}$), the pose transforms of EM tracking sensor in which the needle points are visible ($T_{Sensor}$), the error threshold ($\epsilon$)

**Output:** An estimation of the calibration matrix $C$

1: params ← (0,0,0,0,0,1) ▷ ($R_x, R_y, R_z, t_x, t_y, t_z, s$)

2: optimizer ← Adam(param, lr)

3: error ← $\infty$

4: **while** error > $\epsilon$ **do**

5: $C \leftarrow$ calibrationMatrix(param) ▷ **Step 1:** Define Calibration matrix

6: $P'_{US} \leftarrow T_{Sensor} \cdot C \cdot P_{US}$ ▷ **Step 2:** Apply calibration matrix

7: $T_{Registration} \leftarrow \text{ICP}(P'_{US}, P_{Phantom})$ ▷ **Step 3:** Compute ICP registration

8: $P''_{US} \leftarrow P'_{US} \cdot T_{Registration}$ ▷ **Step 3:** Apply rigid registration

9: error $\leftarrow \frac{1}{n} \sum_{i=1}^{n} \|P''_{US_i} - P_{Phantom_i}\|_2^2$ ▷ **Step 4:** Evaluate the cost function

10: params ← optimizer error ▷ **Step 5:** Optimization

11: **end while**

12: $C \leftarrow$ calibrationMatrix(param) ▷ Optimized calibration matrix definition

translations. Additionally, using this approach, we guarantee that $C$ remains a calibration matrix at each iteration of the optimization process.

**Step 2: Projection of ultrasound image landmarks:** At each iteration of the optimization process, the updated parameters are used to estimate the calibration matrix $C$ as described in the previous step. This matrix is then applied to project manually detected landmarks from the image reference frame to the world reference frame using the equation: $\mathbf{p}_{world} = \mathbf{T}_{sensor} \cdot \mathbf{C} \cdot \mathbf{p}_{image}$. Here, $\mathbf{T}_{sensor}$ is the transformation matrix provided by the EM tracker, and $\mathbf{p}_{image} = [x, y, 0, 1]^T$ represents the landmark positions in the image reference frame. These landmarks correspond to the needle tips, which can be easily identified and manually selected in the ultrasound images thanks to the phantom's design choice. Thus, the landmarks are transferred to the world reference frame.

**Step 3: ICP registration:** At each optimization iteration, the given calibration matrix is used to transform the landmark positions into the world reference frame ($\mathbf{p}_{world}$). These transformed landmarks are then rigidly registered to their corresponding points in the phantom's 3D model using point-to-point Iterative Closest Point (ICP) [2]. The ICP algorithm iteratively minimizes the alignment error, producing a registration matrix $\mathbf{R}_{ICP}$ that ensures the best possible fit between the two landmark sets. We assume that any remaining misalignment after ICP reflects the error introduced by the calibration matrix at that iteration, providing a direct measure of its accuracy.

**Step 4: Objective function evaluation:** After the ICP registration, we evaluate the objective function defined in Equation 2 to measure the error introduced by the calibration matrix. This function quantifies the residual misalignment between the transformed ultrasound landmarks and their homologous in the phantom's 3D model. The resulting error provides a direct assessment of the calibration matrix's accuracy at the given optimization iteration. In
Equation 2, N represents the total number of landmarks, and $p_{Phantom,i}$ and $p_{transformed,i} = R_{ICP} \cdot p_{world,i}$ are respectively the coordinates of the i-th ground truth and transformed ultrasound landmarks.

$$E_{MSE} = \frac{1}{N} \sum_{i=1}^{N} \| p_{transformed,i} - p_{Phantom,i} \|^2 \quad (2)$$

**Step 5: Optimization process:** The optimization process is performed using gradient descent with the Adam optimizer, where the objective function defined in Equation 2 is minimized over the seven parameters (three translations, three rotations and scaling) to determine the calibration matrix C. We use a learning rate $\alpha$ of 0.05 to balance convergence speed and optimization accuracy.

# 4 Experiments and Results

## 4.1 Data acquisition and implementation details

The calibration algorithm was implemented in Python using PyTorch 2.0.1 and optimized with the Adam optimizer. All computations were performed on an NVIDIA RTX 4070 Ti GPU. US images were acquired using an ACUSON S3000 HELX Touch system with an IVUS (ACUSON AcuNav™ Ultrasound Catheter - 8 Fr / 90 cm) probe, covering depths from 7 to 11 cm. An EM tracking system (trakSTAR, NDI) with a 6-DoF sensor was attached to the IVUS near the transducer. For each depth, a separate acquisition was performed, where the probe was rotated clockwise for 10 seconds while the phantom remained fixed and fully immersed in water. The images (680×480 pixels, 15 fps) were captured using a frame grabber (AV.io HD+, Epiphan Video), yielding approximately 150 images per acquisition. The needle tips of the phantom (landmarks) were manually annotated and paired with their corresponding transformation matrices, with 15 landmarks recorded per acquisition.

## 4.2 Calibration results

For each acquired IVUS depth sequence, the calibration matrix is computed using the proposed approach. The optimization process converges in less than a minute for each sequence. The accuracy of the estimated calibration matrices is assessed by computing the RMSE between the landmark positions in the reconstructed US sequence and their corresponding positions in the 3D phantom. The results of our method are reported in Table 1, together with the results of the baseline PLUS method, which we also tested using the Perklab fCal-2.1 N-wire phantom [7]. However, manipulating the IVUS transducer proved challenging for the baseline method, even for a surgeon experienced in handheld ultrasound calibration. Maintaining the phantom within the field of view while ensuring stable images for capturing target points/needles was not possible. This led to the baseline method failing to complete calibration at depths of 7, 9, and 11 cm, with some attempts taking over an hour. In contrast, as shown in Table 1, our
method achieves results comparable to the baseline while maintaining a typical calibration error of 0.8–1.5 mm [7], which is on the same order of magnitude as the EM tracking system (1 mm), and being significantly faster. The overall time for our approach, including sequence acquisition, landmark annotation, and algorithm convergence, is under 10 mins. Additionally, our method is better suited to the IVUS probe due to the phantom’s design and is more adaptable to different US depths.

<table><caption>Table 1: Results of our method compared with the baseline PLUS method [7] regarding accuracy, speed, and adaptability to varying depths.</caption><thead><tr><th rowspan="2">US Depth (cm)</th><th colspan="2">RMSE (mm)</th><th colspan="2">Time (min)</th></tr><tr><th>Ours</th><th>Baseline [7]</th><th>Ours</th><th>Baseline [7]</th></tr></thead><tbody><tr><td>7</td><td>1.16</td><td>-</td><td>&lt; 10</td><td>-</td></tr><tr><td>8</td><td>0.88</td><td>0.83</td><td>&lt; 10</td><td>≈ 60</td></tr><tr><td>9</td><td>1.80</td><td>-</td><td>&lt; 10</td><td>-</td></tr><tr><td>10</td><td>1.53</td><td>1.12</td><td>&lt; 10</td><td>≈ 60</td></tr><tr><td>11</td><td>1.79</td><td>-</td><td>&lt; 10</td><td>-</td></tr></tbody></table>

## 4.3 Preclinical *in vivo* validation

A porcine model with prior liver ablations and four tumors was used to simulate a clinically realistic liver ablation scenario. This setup presented challenges such as ultrasound shadows from air pockets and ablation zones, obscuring liver parts, representing a worst-case scenario. IVUS acquisitions (single 360° rotation scans) were performed at calibrated depths of 7, 8, 9, 10, and 11 cm. However, only the 8 and 9 cm volumes provided sufficient anatomical detail for analysis. At 7 cm, the field of view was limited, and at 10 and 11 cm, the resolution was inadequate for visualizing smaller vessels, as the key vascular structures were positioned outside of the optimal focal range. The reconstructed IVUS volumes at 8 and 9 cm were computed using the calibration matrix from the proposed approach and the 3D Slicer IGT [5, 7], with an output spacing of 0.25 mm/pixel. Additionally, a contrast-enhanced CT scan was obtained at the end of the experiment to serve as a high-resolution anatomical reference.

To evaluate our approach, the reconstructed IVUS volumes were segmented to extract the portal tree and then registered to the CT scan using liver surface landmarks, ensuring unbiased alignment and improving confidence in the spatial correspondence. The internal liver landmarks from both IVUS and CT volumes were compared, focusing on the Right Lateral Portal Vein (RLPV), Right Medial Portal Vein (RMPV), and Left Medial Portal Vein (LMPV). Ultrasound shadows from air pockets and ablation zones obstructed certain liver regions, preventing the identification of the Left Lateral Portal Vein. An expert surgeon independently identified bifurcations from both the IVUS volumes and the reference CT
scan. Quantitative evaluation was performed by calculating the RMSE between identified vessel bifurcations in the IVUS and CT volumes, as shown in Table 2. While our method yields an accurate reconstruction compared to the reference CT (see Fig 3), some errors remain, which can be attributed, in part, to motion and deformation of the liver occurring between the IVUS and CT acquisitions. It is important to emphasize that our calibration performs optimally at specific depths and focal distances. However, the use of fixed-depth IVUS scanning limits validation, as key structures at varying depths within the liver fall outside the ideal focal ranges —particularly at US depths of 7, 10, and 11 cm—, leading to errors caused by imaging limitations.

<figure><img src="image_4.png" /><figcaption>Fig. 3: **Left:** IVUS volume in axial view; **Center:** IVUS overlaid with CT in axial view; **Right:** 3D reconstruction of the portal tree, based on the CT volume (purple), IVUS volume at 8 cm (green), and IVUS at 9 cm (blue)</figcaption></figure>

<table><caption>Table 2: RMSE between IVUS reconstruction and reference CT for different anatomical landmarks.</caption><thead><tr><td>IVUS Depth (cm)</td><td>RLPV (mm)</td><td>RMPV (mm)</td><td>LMPV (mm)</td><td>RMSE (mm)</td></tr></thead><tbody><tr><td>8</td><td>1.35</td><td>7.14</td><td>6.70</td><td>5.71 ± 2.63</td></tr><tr><td>9</td><td>1.82</td><td>2.88</td><td>4.78</td><td>3.40 ± 1.23</td></tr></tbody></table>

# 5 Conclusion and Discussion

In this work, we present a fast and efficient calibration method for IVUS probes that enables accurate volume reconstruction. Using a custom-designed phantom and a gradient-based optimization algorithm, we estimate the calibration matrix between the image and tracking sensors. Compared to the PLUS method [7], our approach offers similar accuracy but is faster, easier to implement, and adaptable
to different ultrasound depths. Preclinical validation on a porcine liver demon-
strated that our reconstructed IVUS volumes closely match the geometry of
reference CT scans. We also achieved accurate alignment of internal anatomical
landmarks. Optimal visualization of hepatic structures was observed at image
depths of 8 and 9 cm, suggesting the potential of a multi-depth IVUS fusion ap-
proach to account for portal tree variability. Additionally, our findings emphasize
the need for a depth-dependent calibration phantom with markers at the ultra-
sound image's focal distance to improve accuracy. Future improvements include
automating needle tip identification for fully automatic calibration, addressing
sound speed variations, and developing depth-specific phantoms for enhanced
accuracy. Our method also supports accurate 3D imaging, which could be useful
for computer-aided interventions, including autonomous vessel tree identifica-
tion [1].

**Acknowledgments.** This work was partially supported by French state funds man-
aged by the ANR under reference ANR-10-IAHU-02 (IHU Strasbourg). The authors
would like to express their gratitude to the Preclinical Research Unit of the IHU Stras-
bourg for their invaluable support and assistance throughout this study.

**Disclosure of Interests.** The authors have no competing interests to declare that are relevant to the content of this article.

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