Ultrasound Calibration

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    Ultrasound Calibration

    Mei-Chuan Chen

    February 10, 2006

    Abstract

    3D ultrasound(US) is a imaging technique that has been recognized as a valuabletool for a variety of clinical applications. This article is devoted to review the methods

    for Ultrasound calibration and aims mainly at the Hopkins Phantom,which introducesa novel method for ultrasound probe calibration based on closed-form formulation andusing minimal US imaging allowing for an immediate result.

    The calibration procedures are mainly search algorithms for the unknown trans-formation parameters to maximize the similarity between the acquired US images inphantom space and phantom model. Four traditional calibration techniques, cross-wirephantom, three-wire phantom,single-wall phantom,cambridge phantom, whose phan-toms are geometrical model based on points and planes, will be introduced and theresults will be compared.In addition, a new methodology based on closed-form solutionwill be described and compared with the traditional to prove that it is a easy and fastto perform.

    1 Introduction

    True 3D imaging modalities, like MRI and CT, are extremely potent with regard to their renderingcapabilities, but are difficult to use for intra-operative procedures, mainly due to obstructivehardware and the latent images. US,however, has been becoming known as a widely popular imageguidance modality,since it is real-time,convenient to use in the operation room and inexpensivecompared to CT and MRT.

    In order to provide the physicians with a 3D real-time visualization of the internal anatomy,individual 2D US images must be assembled into 3D volumes and then the position of surgical toolsare related with respect to the reconstructed US volume.The goal is to preserve true anatomicalshape by means of a consistent reconstruction of 3D volumes

    Using US as a guidance modality for surgical procedures would require tracking the imagingprobe with a magnetic or optical tracking device.A fixed transformation between the US beam andthe tracking device needs to be determined, so that arbitrary image pixels can be referenced in aglobal frame.Obtaining this fixed transformation is referred to as ultrasound calibration.Aftercalibration, a 3D volume is reconstructed by some surface- or voxel-based method ,and the data isvisualized with some appropriate combination of surface extraction, volume rendering, re-slicing,panoramic viewing, or multi-planar techniques.Clearly, the accuracy of calibration is the most sig-nificant factor ,which greatly influences the quality of the reconstructed volume and visualization.

    With regard to the currently known calibration processes, an object of known geometrical prop-erties(phantom) is scanned by the tracked US probe and then various mathematical proceduresare applied to determine the unknown transformation that maximize the similarity between theUS images and the actual phantom.Geometrical model base phantoms like points, plane exist

    and some studies have compared their accuracy and performance.The cross-wire and three-wire

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    phantoms require long time of acquisition and are hard to automate, while the single-wall as inCambridge phantom is automatic repeatable method.

    Hopkins phantom which is based on using a recent closed-form formulation allows for offlineimage processing leading to immediate calibration if the simple docking guide mechanism is in-volved.The basic concept and reconstruction model of ultrasound calibration will be introducedin the section 2.In section 3 and 4,traditional calibration phantoms and Hopkins phantom willbe discussed respectively and the experiment results and the comparison between the differentphantoms will be summarized in section 5.

    2 Reconstruction[3]

    2.1 Processing

    A 3D free-hand examination consists of three stages:scanning, reconstruction and visualization,which is illustrated by figure 1. Before scanning, some sort of position sensor is attached to the

    probe,which is typically the receiver of an electro-magnetic position sensor.Measurements from theposition sensor are used to determine the positions and orientations of the B-scans with respectto the fixed transmitter.In the reconstruction-stage, the set of acquired B-scans and their relativepositions are used to build a regular voxel array.Finally, the voxel array is visualized using ,forexample, volume rendering or surface rendering.

    Figure 1: 3D free-hand ultrasound imaging[3]

    2.2 Mathematical Formulation

    2.2.1 Coordinate systems and Transformation

    In order to modeling the ultrasound calibration and reconstruction, four coordinate systems areneeded, which is showed in figure 2.P is the coordinate system of the B-scan plane.The y-axis isin the beam direction,x-axis in the lateral direction, and z-axis is in the elevational direction, outof the plane of the B-scan.R is the coordinate system of the position sensors receiver,T is the co-ordinate system of transmitter,and C is the coordinate system of reconstruction volume(phantomcoordinates).

    During reconstruction, each pixel in the B-scan has to be located with respect to the recon-struction volume C by means of the transformation between different coordinate systems.First,each

    pixels scan plane location(Px) is transformed to the coordinate system of the receiver R, then

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    to the transmitter Tand finally to the reconstruction volume C.The process is illustrated in thefigure 3.

    Figure 2: Coordinate systems[6]

    Figure 3: Transformation between different coordinate systems[3]

    2.2.2 Mathematical formulation

    The overall transformation can be expressed as the multiplication of homogeneous transformationmatrices:

    Cx =C TT

    T TR R TP Px

    where

    Px =

    Sxu

    Syv

    01

    The standard notation JTI is adopted, which means the transformation from coordinate system

    I to coordinate systemJ.u and v are the column and row indices of the pixel in the cropped image,

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    and Sx and Sy are scale factors with units of mm/pixel.Cx is the pixels location in the coordinatesystem.

    A transformation between two coordinate systems has six degrees of freedom:three rotation(,,)and three translation(x,y,z).The rotation between two coordinate systems is effected by first ro-tating through around the x-axis, then through around the y-axis, and finally through around the z-axis.Using this convention,the homogeneous matrix describing the transformation isillustrated in the figure 4.

    Figure 4: Transformation matrix [3]

    Each of the transformation matrices plays a different role in reconstruction.TTR is deriveddirectly from the position sensor readings.CTT is included largely as a matter of convenience.

    RTPneeds to be determined by calibration. The scale factors Sx and Sy could be derived from theaxis markings on the B-scan. Once Cx has been found for every pixel, the voxels of C can be setaccording to the intensities of the pixels they intersect.

    3 Traditionall calibration phantoms [3]

    In this section, fourtraditionall calibration phantoms are discussed shortly.

    3.1 Cross-wire phantom

    In the cross-wire phantom,two intersecting wires are mounted in a water bath ,with the transmitterplaced at some fixed location with respect to the wires, as shown on the left side of figure 5. Forthe purpose of simplifying the calibration equations ,the origin of C is placed at the intersection ofthe wires.During scanning, the location where the wires cross is scanned repeatedly from differentdirections(see the B-scan in the middle of figure 5), with each B-scan showing a detectable cross.The pixel at the center of the cross should satisfy the calibration equation, as described in thefigure 5.

    Accuracy of calibration with a point object depends on how well the center of the point canbe located, as well as the stationarity of the point with respect to the transmitter.It is commonpractice to locate the point by hand in each B-scan,making the calibration process time-consuming.

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    Figure 5: Cross-wire phantom[3]

    3.2 Three-wire phantom

    In the three-wire phantom, three wires are accurately mounted in orthogonal directions. Thecoordinate system C is placed at the origin of the wires and orient the x, y, z axes along the wires,as shown in the figure 6. By scanning, each wire is scanned, one at a time,along its length from avariety of directions. The wire appears as a detectable dot in the B-scan, as shown in the figure6. For the wire along the x-,y-and z-axis, the pixel at the center of the wire should satisfy thecalibration equation, as shown in the figure 6. For the accuracy of calibration, it depends on theorthogonality, straightness and stationarity of the wires.

    The advantage of the three-wire method compared to the cross-wire method is that it is easierto scan a length of wire than to keep the B-scan centered on a crossing point.

    Figure 6: Three-wire phantom [3]

    3.3 Single-wall phantom

    Insteading of building a phantom with some wires, as described with cross-,and three-wire phan-tom, signal-wall phantom scans the floor of the water bath alone.It is not necessary to constructa special phantom.The accuracy of the calibration depends on the flatness and stationarity of thefloor.The plane should show up as a strong,straight line in the image,and the line can be detectedautomatically.If the coordinate system C is defined to lie in the floor of the water bath, with thez-axis orthogonal to the floor,pixels lying on the line should satisfy the calibration equation, asdescribed in the figure 7.

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    It is proved that the detecting a line in an image is easier than a point in an image and it ispossible to locate a straight line even when portions of the line are corrupted or missing.On thecontrary, the same cannot be said of dots and crosses.In addition, practically speaking, calibration

    with wire-based phantoms seems to be done by means of locating the dots by hand.In contrast,automatic line detection algorithm can be applied to the single-wall calibration so that time canbe saved to collect the images.

    One significant problem of single-wall calibration is caused by the width of the ultrasound beamand the nature of specular reflection.When the beam is not normal to the wall, the first echo toreturn to the probe comes from the edge of the beam closest to the wall, as shown in the figure8(b).In figure 8, point B is encountered by the ultrasound pulse before point A on the centerline.Theecho from B produces a response in the image which does not reflect the true position of the wall.

    Figure 7: Single-wall calibration[3]

    Figure 8: Minimal sequence of motions and beam thickness problem in single-wall calibration[3]

    3.4 Cambridge phantom

    Cambridge phantom is designed to overcome the difficulties experienced with planar calibration,as discussed previously. The phantom consists of two parts:a clamp that fixes around the probeand a thin brass bar mounted between two circular disks,as shown in the figure 9.The idea is thatthe clamp constrains the thin bar to move only in the center of the ultrasound beam.

    The calibration procedure may be summarized as follows:

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    Insert ultrasound probe into clamp and tighten bolts so that the slots through the two sidesof the clamp are aligned with the scan plane of the probe.

    Immerse the phantom in water,slot the clamp over the bar,and scan the bar with the probefrom all possible angles.A clear image of the bar will always be visible in the B-scan

    Because of the clarity of the images and the fact that the phantom produces a line in theB-scan,it is possible to automatically detect the line in each B-scan.

    Once these lines have been located, calibration proceeds is exactly the same as that forsingle-wall phantom.

    Accurate calibration requires a minimal sequence of motions,as shown in the figure 9.The pro-cedure is fully automated and takes less than five minutes to complete,including scanning, linedetection and optimization.It is not possible for the wired-based techniques to do the calibrationso fast, since the dots need to be located by hand in each image. The B-scan images from the

    traditional calibration phantoms are shown in the figure 10 together.

    Figure 9: Cambridge phantom [3]

    Figure 10: Typical B-scans of traditional calibration phantoms[3]

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    4 Hopkins phantom[1][2][4][5]

    With regard to Hopkins phantom,a position sensor is attached to the probe for tagging each im-

    age/volume with its position and orientation in space.The traditional US calibration framework infigure 3 is integrated into the AX= XB framework as in figure 11, using a recent closed form so-lution for the AX= XB problem.It uses minimal US imaging allowing for an immediate result. Inthis section, the original idea will be first discussed, and then the mathematical formulation of thismodel.Finally ,the experiment,calibration setup and protocol, will be described and summarized.

    4.1 Idea

    The closed form solution comes originally from a basic issue, which is to determine the spatialrelationship between a camera mounted into a robot end-effector.This spatial relationship is arigid transformation, that is, a rotation and a translation, known as the hand-eye transformation.

    A classical approach (Tsai and Lenz 1989;Chen 1991;Daniilidis and Bayro-Corrochano 1996;

    Horaud and Dornaika 1995; Shiu and Ahmad 1989; Chou and Kamel 1991; Wang 1992) statesthat when the camera undergoes a motion A = (Ra, ta) and the corresponding end-effector motionis B = (Rb, tb), then they are conjugated by the hand-eye transformation X = (Rx, tx) (Fig. 2).This yields the following homogeneous matrix equation: AX = XB ,where A is estimated, B isassumed to be known, and X is the unknown.This is illustrated in the figure 11( b).

    4.2 Mathematical Formulation

    From the figure 11(a)and (b),the spatial relationship between camera and gripper is applied to thespatial relationship between the coordinate system R and P, and the camera motions is simulatedby the motions of ultrasound probe in the Hopkins phantom.For the camera-robot model, severalmotions might be need to do the calibration, as shown in figure 11 (c).However, the Hopkinsphantom shows that only two motions of US probe is sufficient for getting the good results of

    calibration.

    In the Figure 11 (a),A1, A2 are the transformations of US image coordinate system P withrespect to the reconstruction coordinate system C at poses 1 and 2,respectively.Using A1, A2,thetransformation between poses 1 and 2 is obtained, as A = A2A

    1

    1. B1, B2are the tracking device

    readings for the sensor frame R with respect to tracker reference frame T at poses 1 and 2,whichis given by B = B1

    2B1.Therefore,according the hand-eye calibration,as described previously, it

    yields the homogeneous matrix equation:AX = XB .Where A is estimated from images, B isassumed to be known from the external tracking device,and X is the unknown transformationbetween the US images coordinate system and the sensor frame.

    Figure 11: AX=XB method in the Hopkins Calibration and hand-eye calibration[1][5]

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    The estimated US image frame motion is given by A (),as described in the figure 12,where Rais the rotation of US image frame between pose 1 and 2 and is the unknown scale factor vectorthat relates the translation vector ua in voxel space to US image frame translation vector ta(in

    mm).

    Using AX = XB and the equations from the figure 12,the result that is computed in thefigure 12 is obtained.The figure 13 illustrates only the left side ofAX= XB ,and the right side iscomputed by means of the same calculation.According the result of figrue 13, the rotation equationand translation equation are obtained.

    Figure 12: Mathematical formulation(1)[1]

    Figure 13: Mathematical formulation(2)

    What we will do next step is to reduce the nonlinear model to linear formulation. A new for-mulation,Sylvester equation: UX + XW = T, that is very similar to the homogeneous equationAX= XB is inspired and Hussein et al. proved that Sylvester equation is usually formulated asa linear system by means of the Kronecker product:

    (U I+ IW) vec (V) = vec (T) (1)

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    The computation process how the property of Kronecker product applies to our model is de-scribed in the figure 14.Finally, Andreff et. al[5] proved that two independent motions withnon-parallel axes is sufficient to recover a unique solution for AX= XB.

    Figure 14: Mathematical formulation(3)

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    4.3 Calibration setup and protocol

    4.3.1 Calibration phantom

    Calibration phantom consists of three identical thin(4mm) plastic plates of irregular shape,andthen is submerged in a transparent plastic water tank. For the scanning,SONOLINE Antares USscanner(Siemens Medical Solutions ,USA) with a Siemens VF 10-5 linear array probe Held in arigid attachment mounted on an adjustable arm is used, as shown in the figure 15 (a).

    The plastic plates are machined together to ensure their congruency and then they are positionedand fixed on a flat surface using Lego blocks, as shown in the figure 15 (b).

    Figure 15: Hopkins phantom[1]

    4.3.2 Images acquisition

    For the images acquisition, an optical pointer pivoted to obtain an accurate estimate of the desired3D point is used to collect 3D points of each of the plates for offline processing. Figure 16 presentsthat 3D points are first registered to provide a local coordinate system for each thin plates,and thenthe relative transformations between each pair of plates can be calculate from the local coordinatesystems.What important is that the poses of the three plates should be arranged carefully to givethe optimal results for the two motions required by AX= XB,based on previous experiments.

    Figure 16: Process for acquiring images[1]

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    4.3.3 US probe calibration

    By calibrating, A1, A2 have to be estimated by means of the method discussed in the paragraphImages acquisition.That is,A1 and A2 transformation matrices are the relative transformationsbetween the plates of the calibration phantom with a positional offset based on the pixel coordi-nates of the phantom in the acquired US image.Two protocols are tested for computing A1 andA2 matrices:

    Move the US tracked probe such that the probe is parallel to the thin plate and the imageplane of US images shows the middle of the thin plate.

    Apply the same protocol, but collect multiple tracking data from one end of the thin plateto the other end and using an averaging technique to find the center of the thin plate.

    The result of scanning can be seen in the figure 15 (b).It yields sharp US images that caneven be processed automatically.Three sets of tracking and US image data are sufficient to solve

    the mathematical formulation,and an additional 3 sets of data(48 datasets) will ensure a well-conditioned problem and produce comparable results to previous calibration.

    5 Experiment,results and comparison

    5.1 Methods and results

    Two different sets of experiments have been designed to test the calibration technique.The first setof experiments use synthetic data ,and the second set of experiments use real ultrasound sequences.

    Simulation data is generated to test the numerical stability of the closed form formulation, andartificial noise is added to the data points in order to mimic the error of the tracker and account

    for the effects of ultrasound image properties. The table in the figure 17(a) shows the average errorand standard deviation of the recovered translation vector for different calibration sequences.Thesequences are generated using synthetic data with added noise of different levels.

    Gathered data using a tracked ultrasound probe are acquired to check the repeatability ofthe calibration setup.Real US data was acquired in 2 poses for each of the calibration plates.Thealgorithm was tested on 48 unique combinations of 6 different poses (two poses per plate).The resultwas shown in the figure 17(b).The standard deviation reflects the repeatability of this method.

    Right after the data collection, the calibration algorithm executes almost immediately.The majorsource of expected error stems from the misalignment of the ultrasound probe to the plane of thethin plate. In order to avoid this error, a planned docking station will be developed, as shown inthe figure 18.According to the reference 5, when the probe is held in a docking attachment, theUS images need to be collected and processed only once during the lifetime of the probe.

    5.2 Comparison with the other Techniques

    Figure 19 shows the comparison of reconstruction precision and accuracy between the traditionalcalibration phantoms and the Hopkins phantom. The mean point reconstruction precision of Hop-kins phantom is higher than the traditional calibration phantoms significantly. For thetraditionallcalibration phantoms, Cambridge phantom shows the better calibration result than the others.Forexample, Cambridge phantom got less root mean square error than the others.

    In summary, cross-wire and three-wire phantoms require long time of acquisition,typically fourto five hours, because of the segmentation of points by hand.They suffer from relatively poorrepeatability. In contrast, single-wall phantom and Cambridge phantom is automatic repeatable

    method,since the lines they produce in the B-scan can be detected reliably and automatically using

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    Figure 17: Results of calibration [1]

    Figure 18: Docking station[1]

    the line detection algorithm. It speeds up the calibration process greatly.The performance of theCambridge phantom was significantly better than any of the other methods, while calibration wasperformed in a matter of minutes (less than five minutes).

    Compared to the traditional calibration phantoms, Hopkins phantom indicate significant po-tential in using a simple calibration phantom in conjunction with the AX = XB closed formformulation. It uses optical digitization with a calibrated pointer to replace with a great ex-tent the traditional segmentation of points/planes in US images.The tracked pointer appeared tointroduce significantly less error than the resolution of the US image caused in the traditionalapproaches.It also provided very accurate calibration results using significantly fewer US images

    and requires only minimal image segmentation.

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    Figure 19: Comparison between different calibration phantoms[6]

    References

    [1] Emad Boctor, Anand Viswanathan, Michael Choti, Russell H. Taylor, GaborFichtinger, Gregory Hager, A novel closed form solution for ultrasound calibra-tion, IEEE Int Symp. On Biomedical Imaging, 2004

    [2] Emad Boctor, Anand Viswanathan, Michael Choti, Russell H. Taylor, GaborFichtinger, and Gregory Hager,A Novel Closed Form Solution for Ultrasound

    Calibration, IEEE Int Symp. On Biomedical Imaging, 2004

    [3] R. W. Prager, Rohling R. N., Gee A. H., and Berman L.,Rapid Calibration for3-D Freehand Ultrasound, US in Med. Biol., 24(6):855-869, 1998.

    [4] John W. Brewer, Kronecker Products and Matrix Calculus in System Theory,IEEE Trans. Circuits and systems, 25(9) Sep.1978.

    [5] Nicolas Andreff and Radu Horaud and Bernard Espiau,Robot Hand-Eye Cal-ibration Using Structure from Motion, International J. of Robotics Research,20(3), pp 228-248, 2001.

    [6] Laurence Mercier, Thomas Lang, Frank Linkseth,and D.Louis Collins,A ReviewOf Calibration Techniques For Freehand 3-D Ultrasound Systems, Ultrasoundin Med. & Biol., Vol. 31, No. 4, pp. 449 471, 2005.

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