Pivot calibration concept for sensor attached mobile c-arms

TL;DR

This work addresses the problem of calibrating a sensor rigidly attached to a mobile C-arm without emitting ionizing radiation. It introduces a radiation-free approach by exploiting the device's mechanical geometry, modeling the C-arm motion as a generalized Torus and extending pivot calibration to a moving pivot locus on the x-circle. The method decomposes the 3D calibration into 2D geometry sub-problems, estimating the torus normal, torus center, pivot locus, and the sensor offset with RANSAC for robustness; it requires only a minimal set of trajectories and shows resilience to pose density and noise. The approach yields a practical, fast, and broadly applicable calibration that enables accurate CAMC AR overlays and could support 2D/3D registration and CBCT-based guidance without radiation exposure, including extensions to isocentric and non-isocentric C-arms.

Abstract

Medical augmented reality has been actively studied for decades and many methods have been proposed torevolutionize clinical procedures. One example is the camera augmented mobile C-arm (CAMC), which providesa real-time video augmentation onto medical images by rigidly mounting and calibrating a camera to the imagingdevice. Since then, several CAMC variations have been suggested by calibrating 2D/3D cameras, trackers, andmore recently a Microsoft HoloLens to the C-arm. Different calibration methods have been applied to establishthe correspondence between the rigidly attached sensor and the imaging device. A crucial step for these methodsis the acquisition of X-Ray images or 3D reconstruction volumes; therefore, requiring the emission of ionizingradiation. In this work, we analyze the mechanical motion of the device and propose an alternatative methodto calibrate sensors to the C-arm without emitting any radiation. Given a sensor is rigidly attached to thedevice, we introduce an extended pivot calibration concept to compute the fixed translation from the sensor tothe C-arm rotation center. The fixed relationship between the sensor and rotation center can be formulated as apivot calibration problem with the pivot point moving on a locus. Our method exploits the rigid C-arm motiondescribing a Torus surface to solve this calibration problem. We explain the geometry of the C-arm motion andits relation to the attached sensor, propose a calibration algorithm and show its robustness against noise, as wellas trajectory and observed pose density by computer simulations. We discuss this geometric-based formulationand its potential extensions to different C-arm applications.

Figures (6)

• Figure 1: a) illustrates the C-arm movement in terms of its translations and rotations, we denote the movement by rotating around c-arm-axis as c-circle (green), that by x-axis as x-circle (red) and the c-circle center as $c_c$. b) demonstrates the c-circle rotates perpendicularly with $c_c$ on the x-circle. This forms a locus of a spindle Torus. For comparison, c) shows a standard Torus generated in the same way as in b) with the major radius $r_{maj}$ larger than the minor radius $r_{min}$. When $r_{maj}$ becomes smaller than $r_{min}$, the standard Torus shrinks into a spindle Torus.
• Figure 2: An example of C-arm Torus is shown in a) and the sensor Torus formed due to the offset $t$ is depicted in b). Since both c-circle and translated c-circle rotate around c-arm-axis, the normal line that passes the center of the translated one tangents the x-circle at $c_{c}$ as illustrated in c).
• Figure 3: An overview of our method: The input is observed separably at step 0 during the $x$-axis and c-arm-axis movements in a defined order for determining the orientation of the C-arm. More specifically, this step observes $j$ sets of $x$-axis movements with $i$ poses, $\{T^{x}_i\}_j$, in the order of moving along the positive $x$-axis, and $m$ sets of c-arm-axis movements with $n$ poses, $\{T^{c}_n\}_m$, with specific sets at $\alpha=-90^o,0^o,90^o$. Step 1 performs circle fittings using $\{T^{x}_i\}_j$ to find the Torus normal $n$ and an center $c^x$, step 2 do the same using $\{T^{c}_n\}_m$ to find another center $c^c$, and the average of $c^x$ and $c^c$ is the estimated Torus center $c$. The locus of the pivot point $c_c$ is defined by a circle centered at $c$ with the Torus major radius $R$, which is solved by taking the average distance between normal line of the fitted circles in step 2 to $c$. In step 4, with $\{T^{c}_n\}_m$ at $\alpha = \ang{-90},\ang{0},\ang{0}$, the Torus orientation is deduced and the known $c_c$ at these axes are used together with the corresponding poses to estimate sensor offset $t$. At last, to increase the robustness to noise, the result is computed iteratively using RANSAC.
• Figure 4: Performance of the calibration method with increasing translation (left) and orientation noise (right)
• Figure 5: Performance of the calibration method with increasing number of observed vertical trajectories (c-circles, left) and horizontal trajectories (x-circles, right).