Controladores lógicos Programables

We have created a model to describe the hand-eye transformation, between the coordinate system of the optical reference sensor and the camera coordinate system, as a function of the scope’s rotation angle. The reprojection error of the combined intrinsic camera and hand-eye model ranged from 20-100 pixels RMS for two different laparoscopes. The intrinsic camera model, and the hand-eye model are re- evaluated to find the source of the large reprojection error.

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In the intrinsic camera model it was assumed that focal lengths are fixed. Here we have shown that this is not the case. The size and unpredictability of the change in focal length during rotation suggest that the distance between the two compound lens-systems in the laparoscope can change. As a consequence, radial distortion can also change during rotation as it is a function of the focal length. Modeling of the principal point and decentering distortion by an ellipsoid provides a good approximation of the calibrated values. If the fitted ellipsoid is close to a circle, the approximation is more accurate compared to a strong ellipsoidal shape. In the fitted functions, the rotation angle measured by navigation was used for visualization of the modeled values. As the rotation of the image plane is less than the navigation based rotation angle, the model fits better than shown here. The reprojection error for the calibration and validation images is close to one if the extrinsic parameters are estimated from the image.

Fixing the focal length to a single value, and modeling of the principal point by an ellipse, creates hand- eye translations that resemble a circle in a plane. The translations obtained from the original calibrated values produce a distribution that cannot be modeled. The intrinsic camera model is therefore also necessary for hand-eye calibration. The spread in hand-eye positions is in the order of 1 mm in all directions per angle. Ideally, all of the points per angle would coincide in a single point. However, due to measurement errors, there will always be some spread in the positions. The translation component is modeled as a rotating point around a fixed radius to the rotation axis in the center of the circle. The amount of rotation around the rotation axis is not linear to the navigation based rotation and is therefore approximated by a Fourier sum. The mean deviation between the average position per angle and its modeled position is 0.5 mm.

Re-evaluation of the intrinsic and hand-eye model show that the modeled values are a good approximation of the measured values. Based on these results, the overall reprojection errors are expected to be lower and have a smaller range than obtained with the validation images. The difference between the extrinsic parameters obtained from the images and from the hand-eye model show that there is a significant rotation and position error in the hand-eye model. Rotation around the optical axis shows hysteresis of the rotation of the image plane. On a rotation of 200°, the rotation of the image plane is constantly 4% less than the rotation measured using navigation. Since the rotation offset is very consistent, it might be possible to model the hysteresis by including the history of rotation. However, this would require the optical tracking system to record every single rotation of the scope to be accurate. This might be possible in an experimental setup, but in a clinical setting this is not realistic. Changing the rotation direction between acquisition of calibration and validation images creates a movement of the camera coordinate system in the laparoscope. As a result, there is position error of up to 10 mm in the extrinsic parameters obtained using navigation. Rotation offset (hysteresis), displacement of the camera coordinate system, and changing focal length can all be explained by a movement of the inner compound lens-system including image sensor relative to the handle of the laparoscope. The unexpected hand-eye translation behavior that could not be explained in chapter 3 can also be caused by this. In order to create model for the laparoscope it is necessary to know the pose of both compound lens- systems in the tip of the laparoscope. The outer lens-system is still assumed fixed in relation to the scope’s cylinder and can be tracked by an optical sensor attached to this cylinder. As the inner lens- system is not fixed in relation to the handle, as initially assumed, there is no option to track its pose based on anything accessible on the outside of the laparoscope. This is the result of the configuration in chip-on-the-tip laparoscopes. In conventional systems, the laparoscope consists of an interchangeable lens-system that is attached to a camera-head. This conventional setup allows external tracking of the individual components of the laparoscope, thereby eliminating the source of our problem.

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Three different laparoscopes were used in the experiments performed during this research. All laparoscopes showed that the image sensor is not fixed in relation to the handle, thereby making it impossible to track the two components of the camera system. However, none of the laparoscopes was calibrated twice. It is possible that the movement between the two components is predictable and can be correlated to the rotation of the laparoscope. Further research will need to be conducted to verify if this correlation exists. If it exists, it would still be difficult to model the behavior based on optical tracking as the tracking system would need to record every single rotation of the scope. Including rotation history in the model would also make the model very complex and prone to errors.