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Comparison of Freehand 3-D Ultrasound Calibration Techniques Using a Stylus
Ultrasound in Med. & Biol., Vol. 34, No. 10, pp. 1610 –1621, 2008 Copyright © 2008 World Federation for Ultrasound in Medicine & Biology Printed in the USA. All rights reserved 0301-5629/08/$–see front matter
doi:10.1016/j.ultrasmedbio.2008.02.015
● Original Contribution COMPARISON OF FREEHAND 3-D ULTRASOUND CALIBRATION TECHNIQUES USING A STYLUS PO-WEI HSU, GRAHAM M. TREECE, RICHARD W. PRAGER, NEIL E. HOUGHTON, and ANDREW H. GEE Department of Engineering, University of Cambridge, Cambridge, United Kingdom (Received 30 June 2007; revised 9 February 2008; in final form 21 February 2008)
Abstract—In a freehand 3-D ultrasound system, a probe calibration is required to find the rigid body transformation from the corner of the B-scan to the electrical center of the position sensor. The most intuitive way to perform such a calibration is by locating fiducial points in the scan plane directly with a stylus. The main problem of this approach is the difficulty in aligning the tip of the stylus with the scan plane. The thick beamwidth makes the tip of the stylus visible in the B-scan, even if the tip is not exactly at the elevational center of the scan plane. We present a novel stylus and phantom that simplify the alignment process for more accurate probe calibration. We also compare our calibration techniques with a range of styli. We show that our stylus and cone phantom are both simple in design and can achieve a point reconstruction accuracy of 2.2 mm and 1.8 mm, respectively, an improvement from 3.2 mm and 3.6 mm with the sharp and spherical stylus. The performance of our cone stylus and phantom lie between the state-of-the-art Z-phantom and Cambridge phantom, where accuracies of 2.5 mm and 1.7 mm are achieved. (E-mail: [email protected]) © 2008 World Federation for Ultrasound in Medicine & Biology. Key Words: Freehand three-dimensional ultrasound, Calibration.
space. For brevity, we will use the notation TB¢A to mean a rotational transformation followed by a translation from the coordinate system A to coordinate system B. Three-dimensional US calibration has been an active research topic for several years (Mercier et al. 2005). The usual approach is to scan an object with known dimensions (a phantom). These scans place constraints on the eight calibration parameters: 2 image scales, 3 translations in the direction of the x, y and z axes and the three rotations—azimuth, elevation and roll—about these axes. The simplest phantom is probably a point target (Detmer et al. 1994; State et al. 1994; Amin et al. 2001; Gooding et al. 2005; Barratt et al. 2006). The point is scanned from different positions and orientations and its location marked in the B-scans. The segmented points can be mapped to the sensor’s coordinate system by using an assumed calibration and then to the world coordinate system from the position sensor readings. If the assumed calibration is correct, the points from the different B-scans should have the same coordinates in 3-D space. This places constraints on the calibration parameters. The calibration is solved by an iterative optimization technique.
INTRODUCTION Freehand 3-D ultrasound (US) (Fenster et al. 2001) is a 3-D medical imaging system with many clinical applications in anatomical visualization, volume measurements, surgery planning and radiotherapy planning (Gee et al. 2003). As a conventional 2-D US probe is swept over the anatomy of interest, the trajectory of the probe is recorded by the attached position sensor. The volume of the anatomy can be constructed by matching the US data with its corresponding position in space. However, the position sensor measures the 3-D location of the sensor, S, rather than the scan plane, P, relative to an external world coordinate system, W, as shown in Fig. 1. It is therefore necessary to find the position and orientation of the scan plane with respect to the electrical center of the position sensor. This rigid-body transformation, TS¢P, is determined through a process frequently referred to as “probe calibration.” In general, a transformation involves both a rotation and a translation in 3-D Address correspondence to: Po-Wei Hsu, Department of Engineering, University of Cambridge, Trumpington Street, Cambridge, Cambridgeshire CB2 1PZ, United Kingdom. E-mail: [email protected] 1610
Freehand 3-D US calibration ● P.-W. HSU et al.
Mobile part of position sensor TW
S S
TS
P
P
z
y
World coordinate system
Probe x
Scan plane
W Fig. 1. The coordinates associated with the scan plane and the mobile part of the position sensor.
Calibrating with the point phantom has three major disadvantages. First, it is very difficult to align the point phantom precisely with the scan plane. The finite thickness of the US beam makes the target visible in the B-scans, even if the target is not exactly at the elevational center of the scan plane. This error can be several millimeters depending on the US probe and the skill of the user. Second, automatic segmentation of isolated points in US images is seldom reliable. As a result, the point phantom is often manually or semi-automatically segmented in the US images. This makes the calibration process long and tiresome. Finally, the phantom needs to be scanned from a sufficiently diverse range of positions, so that the resulting system of constraints is not underdetermined with multiple solutions (Prager et al. 1998). Over the last decade, much research has been undertaken to make probe calibration more reliable and, at the same time, easier and quicker to perform. These phantoms include a plane (Prager et al. 1998; Rousseau et al. 2005), a 2-D alignment phantom (Sato et al. 1998; Gee et al. 2005), Z-fiducial phantom (Comean et al. 1998; Pagoulatos et al. 2001; Bouchet et al. 2001; Lindseth et al. 2003; Chen et al. 2006; Hsu et al. 2008) and other wire phantoms (Boctor et al. 2003; Dandekar et al. 2005). As in the case with a point phantom, when these phantoms are scanned, constraints are placed on the calibration parameters, which are then solved either iteratively or algebraically depending on the phantom used. Mercier et al. (2005) and Rousseau et al. (2006) compared some of these techniques. A 3-D localizer, often called a pointer or a stylus, can be used to aid probe calibration. It is essentially a point target connected to another position sensor at the end of the stylus, whose tip can be spherical or sharpened to a point. The rigid-body transform between the tip of the stylus and the position sensor is usually supplied by the manufacturer (Muratore and Galloway 2001). In the case where the transformation is not available, it can be
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determined by a simple pointer calibration (Leotta et al. 1997). During a pointer calibration, the stylus is rotated about its tip while the position sensor’s readings are recorded. Because the tip of the stylus must be mapped to the same location in 3-D space, this places constraints on the possible locations of the stylus’ tip, whose location can be determined by an iterative optimization algorithm. In any case, the location of the stylus’ tip is known in 3-D space. The location of the point phantom can therefore be determined by pointing the stylus at the phantom. If the scales of the B-scan are known (Hsu et al. 2006), the calibration parameters can be solved in a closed-form by least-squares minimization (Arun et al. 1987). Because a stylus can be used to locate points in 3-D space, Muratore and Galloway (2001) calibrated their probe by locating points directly in the scan plane. Assuming at least three noncollinear points have been located, calibration can be solved by least-squares optimization as described previously. This technique, nevertheless, requires two targets to be tracked simultaneously. Furthermore, it is difficult to align the tip of the stylus with the scan plane. Khamene and Sauer (2005) improved on this technique by imaging a rod transversely. Both ends of the rod are pointer calibrated to define the location and orientation of the rod in space. Each image of the rod sets up a constraint on the calibration parameters. Probe calibration can then be found using optimization techniques. In this paper, we study the relative merits of a particular class of calibration algorithms that locate points in the B-scan with a stylus. We begin in the next section by outlining different calibration techniques, including our novel cone phantom and cone stylus. We then performed a series of experiments to evaluate the precision and accuracy of each technique. The various calibration algorithms are compared and discussed in the final section. MATERIALS AND METHODS The calibration phantoms Figure 2 shows the four calibration styli and the cone phantom. Figure 2, a and b are standard Polaris (Northern Digital Inc., Waterloo, ON, Canada) styli with a sharp and a spherical tip. Figure 2c shows a rod stylus similar to the one used by Khamene and Sauer (2005). Figure 2, d and e show the cone phantom and the cone stylus that we have designed. The cone phantom is an improvement on one of our previous phantoms (Hsu et al. 2007). Although this phantom is not a stylus physically, it works on the same principle as a stylus by locating points with known 3-D locations in the scan plane. These are the four styli and the cone phantom that we will use for probe calibration comparison. During
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Fig. 2. The calibration styli and phantom.
Freehand 3-D US calibration ● P.-W. HSU et al.
probe calibration, we will scan a point, x, that is on the styli/phantom. This point xP⬘ ⫽ (u, v, 0)t can be segmented in the US image, where u and v are the column and row indices of the cropped image in pixels. This point is changed to metric units by an appropriate scaling factor
冢 冣 su 0 0
r2
L
xW ⫽ TW¢STS¢PxP,
(1)
where TW¢S is given by the Polaris (Northern Digital Inc.) readings. Polaris styli Figure 3 shows the coordinate systems involved when calibrating with either of the Polaris (Northern Digital Inc.) styli. During pointer calibration, the location of the stylus’ tip, xL, is found in the stylus’ coordinate system, L. This point can be mapped to 3-D space by: xW ⫽ TW¢LxL,
(2)
where the transformation TW¢L can be read off the Polaris. If we place the stylus’ tip in the scan plane, its location in space is also given by eqn (1). This means that only the calibration TS¢P and possibly the scale factors Ts are unknown in the two expressions for xW. Calibration can therefore be found by equating eqns (1) and (2), i.e., minimizing
f1 ⫽ f2 ⫽
兺 ⱍT
ⱍ
TS¢PTsxiP⬘ ⫺ TW¢LixL ,
W¢si
where | · | denotes the usual Euclidean norm on ⺢3. The above equation is minimized by using the iterative Levenberg-Marquardt algorithm (More 1977). Rod stylus Figure 2c shows the rod stylus that is used to follow the approach by Khamene and Sauer (2005). The corresponding coordinates are shown in Fig. 4. We have attached a 15-cm-long, 1.5-mm-thick rod to a Polaris (Northern Digital Inc.) active marker. Both ends of the rod are sharpened for accurate pointer calibration. Again, the end points of the rod rL1 and rL2 are found by two separate pointer calibrations, and their coordinates in 3-D space are TW¢LrL1 and TW¢LrL2 , respectively. Now, the point of intersection of the rod and the scan plane is given by eqn (1). Furthermore, this point lies on the line formed by the two end points of the stylus. This means that the distance from the point xW to the line r1Wr2W is zero, i.e.,
ⱍ共r
W 2
ⱍ
⫺ r1W兲 ⫻ 共r1W ⫺ xW兲 ⫽0 r2W ⫺ r1W
ⱍ
ⱍ共T
ⱍ
ⱍ
r ⫺ TW¢Lr1L兲 ⫻ 共TW¢Lr1L ⫺ TW¢STS¢PxP兲 ⫽ 0. TW¢Lr2L ⫺ TW¢Lr1L
L W¢L 2
ⱍ
ⱍ
The ⫻ in the above equations denotes the cross product of two vectors in ⺢3. Because |TW¢Lr2L ⫺ TW¢Lr1L| is a constant, calibration can be found by minimizing
S
P
L
i
)
TS
TW
r1
Fig. 4. The coordinate systems involved when calibrating with a rod stylus.
where su and sv are the scales in millimeters per pixel to give xP ⫽ TsxP⬘. If the probe calibration TS¢P is known, then the segmented point can be mapped to world space by:
S
x
World coordinate system W
Ts ⫽ 0 sv 0 , 0 0 0
TW
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L
P
f3 ⫽
兺 ⱍ共T
r ⫺ TW¢Lir1L兲
L W¢Li 2
i
World coordinate system
x
TW
L
W
Fig. 3. The coordinate systems involved when calibrating with a Polaris stylus.
ⱍ
⫻ 共TW¢Lir1L ⫺ TW¢SiTS¢PxiP兲 . However, it is practically impossible to minimize f3 without knowing the scale factors. Therefore, the scale factors need to be determined explicitly and fixed before minimizing f3.
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Cone phantom The cone phantom (Fig. 2d) is primarily comprised of a polypropylene block machined into the shape of two hollow cones with 1-mm wall thickness. The center of the circle where the two cones join is the fiducial point, x, that we will use for probe calibration. The cones have outer diameters varying between 12-mm and 52-mm. They are supported by an aluminium frame structure consisting of two 135 ⫻ 135-cm square plates held together by four bolts. There are three cone-shaped divots, di, in the two blocks to fit a 3-mm ball-pointed Polaris (Northern Digital Inc.) stylus. The position of the divots, dW i , are therefore known in 3-D space. These divots serve as the principal axes of the phantom and are used to determine the phantom position in 3-D space. The location of x is measured with a Mitutoyo (Mitutoyo Corporation, Kawasaki, Japan) coordinate measuring machine (CMM) relative to the divots, i.e., xA ⫽ fCMM(dAi ), where the function fCMM is determined by the CMM and is valid for arbitrary coordinate system A. In particular, xW ⫽ fCMM(dW i ) in world space. All dimensions are precision manufactured by our workshop, with a tolerance of ⫾0.1 mm. If we align the scan plane with the circle where the two cones join, we get a circle with a 12-mm diameter. This circle can be segmented with its center xP⬘ automatically. As we shall see later, the image scales are necessary for reliable segmentation. Let us assume for the moment that the scales, Ts, are known; hence xP ⫽ TsxP⬘ can be computed. Now, recall that we have already found the coordinates of this centre in 3-D space. We can transform this point to the sensor’s coordinate system by using the inverse of the position sensor’s readings: ⫺1 xS ⫽ TW¢S xW ⫺1 ⫽ TW¢S f CMM(dW i ).
For probe calibration, we need to find the single transformation TS¢P that best transforms {xPi} to {xSi} ⫽ ⫺1 兵TW¢S f 共dW i 兲其. This can be found (Arun et al. 1987) by i CMM minimizing f4 ⫽
兺 ⱍT i
共dWi 兲 ⫺ TS¢PxP ⱍ.
⫺1 W¢S CMM
f
i
Cone stylus Figure 2e shows a photograph of our new cone stylus. It consists of a stainless steel shaft, 120 mm in length and 13 mm in diameter. On one end of the shaft, a platform is ground to fit a PassTrax (Traxtal Technologies, Toronto, ON, Canada) position sensor in such a way that the z axis of the sensor is parallel to the shaft. The other end of the shaft is sharpened for accurate
Volume 34, Number 10, 2008
pointer calibration. As previously, this means that the point rL is found by a pointer calibration. The main feature of this stylus is that part is thinned in the shape of two cones, to meet at the critical point x. This point is precisely 20 mm from the stylus’ tip. Here, the diameter is 1.5 mm. The stylus was precision manufactured by our workshop with a tolerance of ⫾0.1 mm. From the known geometric model of the stylus, xL L ⫽ r ⫺ (0, 0, 20)t mm. If the position sensor cannot be placed with its z axis parallel to the shaft, the same technique can still be used provided that both ends of the shaft are pointer calibrated so that the orientation of the shaft is known. We now place fiducial marks by aligning this critical point with the scan plane. From here, the calibration process is identical to that for a Polaris (Northern Digital Inc.) stylus, the only difference being that we are aligning the critical point, and not the stylus’ tip. The calibration parameters are found by minimizing f5 ⫽
兺 ⱍT i
ⱍ
TS¢PTsxiP ⫺ TW¢Li共rL ⫺ 共0, 0, 20兲t兲 . ⬘
W¢Si
Segmentation Figure 5 shows typical US images of the various styli/phantom. The rod stylus does not need to be aligned with the scan plane, and the calibration is performed by poking the rod into the scan plane in different places. The sharp stylus is aligned visually so that the stylus’ tip just appears in the US image. The cone phantom is also aligned visually by moving the probe up and down its length until we get a circle with the smallest radius. The spherical and cone stylus are aligned until we see a strong reflection, as shown in Fig. 5, b and c. This happens when the stylus is perpendicular to the scan plane and a strong reflection is obtained. Our cone stylus has a small flat region between the two inverted cones to ensure a strong reflection once the stylus is aligned. A similar point spread function is seen in Fig. 5c; this occurs if the rod is perpendicular to the scan plane. Because we need to scan the rod at different angles, an image similar to Fig. 5a is seen when the rod is not perpendicular to the scan plane. Because the styli are made from stainless steel, the US images are pointspread functions and not a circle. The radius of the styli cannot be determined from the US images. However, as the cone stylus is moved through the scan plane, we can see the varying radius of the stylus and this assists the user to align the stylus with the scan plane. The cone phantom is designed so that we can use the radius of the phantom to assist segmentation. When segmenting the image of the four styli, we first search through the entire image to find the pixel with maximum intensity. This pixel serves as a starting point for the segmentation algorithm. We are anticipating a
Freehand 3-D US calibration ● P.-W. HSU et al.
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Fig. 5. Typical images of the different styli and phantom with the segmentation result imposed.
strong reflection and an image of the point-spread function, which will be symmetrical about the direction of propagation of the US waves. This means that for a linear probe, there should be a vertical line of symmetry. To
allow for curvilinear probes, we iterate this line through all possible angles and across a horizontal distance about 10% of the image width, near the pixel with maximum intensity. The angles are incremented in steps of 1°.
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RESULTS
Group 1 Probe Calibration 1-1
Pointer Calibration 2
Probe Calibration 1-2
Probe Calibration 1-3
Probe Calibration 1-4
Probe Calibration 1-5
Probe Calibration 5-4
Probe Calibration 5-5
Group 2
...
... Pointer Calibration 5
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Group 5 Probe Calibration 5-1
Probe Calibration 5-2
Probe Calibration 5-3
Fig. 6. The calibration protocol. Five pointer calibrations were performed. For each pointer calibration, five probe calibrations were performed.
During each iteration, the correlation of the image on either side of the line is computed. Once we have found the line of symmetry corresponding to the maximum symmetric autocorrelation, the required point is assumed to be on this line. We sum the intensity across each depth on this line and find the depth that has the highest intensity. The point on the line of symmetry corresponding to the brightest depth is segmented. This algorithm is used to segment the image of each stylus, and the results are imposed on Fig. 5a– c and e. The line of symmetry is shown as well. The near-vertical line of symmetry verifies that the point has been segmented correctly. Because the top, rather than the center of the surface is segmented, a correction is performed in the optimization process to shift the segmented point downwards by half the stylus thickness. We have measured the spherical tip of the Polaris (Northern Digital Inc.) stylus with a digital micrometer to be 3.0 mm in diameter. The thickness of the rod and cone stylus is 1.5 mm. To calibrate our linear probes in room temperature water so that the temperature is more stable, we now post-process the segmented points by shifting them upwards towards the probe face. Assuming that the user will measure the water temperature, the speed of sound in water at this temperature can be computed (Bilaniuk and Wong 1993). We then shift the pixels upwards by the sound speed in water temperature correction factor . 1540 For the cone phantom, we first place a user-defined threshold on the B-scan to remove any unwanted scatters, resulting in a binary image consisting mainly of the circle that is to be segmented. We now shift each pixel upwards by the temperature correction factor. Edge detection is performed by applying a Sobel filter to overlapping 3 ⫻ 3 blocks of the image. If the image scales are known, we can apply the Hough transform (Hough 1959) on the resulting edges to detect a circle with a 12-mm diameter.
To measure the calibration quality of the different styli and the phantom, we calibrated a Diasus (Dynamic Imaging Ltd., Birmingham, UK) 5–10 MHz linear array probe. The analog radiofrequency (RF) US data, after receive focusing and time– gain compensation, but before log compression and envelope detection, was digitized using a Gage CompuScope (GaGe Applied Technologies Inc., Lockport, IL, USA) 14200 PCI 14-bit analog to digital converter, and transferred at 25 frames per second to a Pentium® 4 2.80-GHz personal computer running Microsoft Windows XP (Microsoft Corp., Redmond, WA, USA). Because we have access to the RF data, the image scales could be set at our discretion. The B-scans were displayed at 0.1 mm/pixel, with a cropped size of 30.0 ⫻ 38.1 mm. The probe was tracked using a PassTrax (Traxtal Technologies) target for the Polaris (Northern Digital Inc.) optical tracking system. A pointer calibration was initially performed on each stylus. Five probe calibrations were followed. The five probe calibrations are used to evaluate the repeatability of the calibration. During each probe calibration, 20 points were located throughout the image with the stylus. To evaluate the impact of the pointer calibration on probe calibration, this whole routine was again repeated five times, each time with a different pointer calibration, as shown in Fig. 6. For probe calibrations with the cone phantom, a pointer-calibrated stylus (Fig. 2b) was used to locate the phantom in space. Ten readings were taken at each divot, and the mean was used for probe calibration. Five calibrations, each with 20 images of the cone spread throughout the B-scan, were performed. Again, this process was repeated five times, each time recalibrating the stylus and relocating the phantom by its divots. Thus, a total of 25 calibrations were computed for each stylus/ phantom. Precision One way to assess the calibration quality is by computing its precision. A point, pP, in the scan plane
1.0
Exclude pointer calibration variation
2.1 5.0
Include pointer calibration variation
0.8
Precision (mm)
Pointer Calibration 1
Ultrasound in Medicine and Biology
0.6
0.4
0.2
Sharp Stylus
Spherical Stylus
Rod Stylus
Cone Phantom
Cone Stylus
Fig. 7. Precision of the probe calibrations.
Freehand 3-D US calibration ● P.-W. HSU et al. Lateral 1.4
Axial
5.0
Elevational
1.2
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ponents in the lateral, axial and elevational directions. Figure 8 shows the mean variation of the four corners and the center of the B-scan in these three principle axes.
1.0
0.6
0.4
0.2
Sharp Stylus
Spherical Stylus
Cone Phantom
Rod Stylus
Cone Stylus
Fig. 8. Mean variation of the four corners and the center of the B-scan in the three principle axes of the scan plane.
can be mapped to the sensor’s coordinate system after probe calibration, because pS ⫽ TS¢PpP. If all the calibrations are identical, the point, pS, should remain stationary. Precision is therefore measured by calculating 1 ⫽ N
N
兺 共p
Si
⫺ pSi兲,
|TW¢LiwL ⫺ TW¢STS¢PTswP⬘|.
i⫽1
where pSi is the point in the sensor’s coordinate system mapped by the ith calibration, pSi is used to denote the arithmetic mean of (pSi) and N is the number of calibrations. It is obvious that the above measure is dependent on the point chosen in the B-scan. We will therefore measure the variation of the four corners and the center of the B-scan. To investigate whether the pointer calibration is sufficiently accurate and measure its impact on the calibration result, we compute two precision measures. First, the precision for each group of the five calibrations specific to the same pointer calibration is calculated. These five precisions for the five groups of calibrations are then averaged. Specifically, 1 1 ⫽ 25
Accuracy Although precision measures calibration reproducibility, this does not reflect the accuracy of the calibration, because there may be a systematic error. We assess the accuracy of our calibrations by measuring the point reconstruction accuracy of a point target. We scan the tip of a wire that is 1.5 mm thick in water at room temperature. The image of the wire tip, w, is reconstructed in space by using the obtained calibrations. The 3-D location of the wire tip is found by using an independent stylus. The root-mean-square error of the pointer calibration is 0.8 mm. Ten readings of the wire tip were obtained to eliminate errors from the stylus’ readings. The true location of the wire tip is taken to be the mean of the stylus’ readings. Accuracy is measured by the amount of mismatch between the reconstructed image and the true location, i.e.,
5
Point reconstruction accuracy may depend on the position of the probe and the position of the wire tip in the B-scan. We therefore capture images of the wire tip at five different locations in the B-scans—near the four corners and the center of the B-scan. The wire tip is aligned with the scan plane by hand, and segmented manually from the B-scans. The probe is held perpendicular to the wire and rotated through a full revolution about the lateral axis at six different locations. Furthermore, five images of the wire are taken at each probe position and at each location in the B-scans. A total of 150 images of the wire tip are captured. The point reconstruction accuracy is shown in Fig. 9, where the graph shows the mean of the 30 repetitions at each position in the B-scan.
5
兺 兺 共p
Si j
⫺ pSj i兲,
j⫽1 i⫽1
th
Si j
where p denotes the point mapped by the i probe calibration associated with the jth pointer calibration, and pSj i is the mean of 共pSj 1, pSj 2, ···, pSj 5兲, for a given, j. The results are shown in Table A1. Table A2 shows the precision without differentiating the 25 calibrations, i.e.,
8.0
Point Reconstruction Accuracy
7.0
Bias Spread
6.0
Accuracy (mm)
Precision (mm)
0.8
5.0
4.0
3.0
2 ⫽
1 25
25
兺 共p
Si
⫺ pSi兲.
2.0
1.0
i⫽1
The mean variations of the four corners and the center of the B-scan are also given in Fig. 7. Each of these variations can be broken down into their corresponding com-
Sharp Stylus
Spherical Stylus
Rod Stylus
Cone Phantom
Cone Stylus
Fig. 9. Point reconstruction accuracy, bias and spread of the point reconstructions.
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s5
1
true location bia s2
spread
centroid
ad
bias 3
s4
re sp
bia
s bia
spread
bia
bias 6
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Fig. 10. Calculation of bias and spread of the point reconstruction accuracy.
Point reconstruction accuracy consists of a bias and a spread. The bias consists of calibration error and other systematic errors, such as systematic misalignment, systematic segmentation, sound speed errors and an incorrect position of the true location. Because we have placed our probe at six different positions, any systematic error will cause the reconstructed points to be clustered into six groups, as shown in Fig. 10. For each probe position, we compute the centroid of the points belonging to that cluster. The bias is measured to be the distance from the centroid to the true location. Figure 9 also shows the mean of the biases and their spread. The spread is measured by the root-mean-square distance from each reconstructed point to the corresponding centroid, i.e., the standard deviation of the points within each cluster. Table A3 shows the biases and spread in the form bias ⫾ spread. DISCUSSION Figure 7 shows the precision of the probe calibrations with a fixed pointer calibration. These precisions can be achieved if the pointer calibration is supplied by the manufacturer of the stylus. Probe calibration precision that includes variations as a result of pointer calibration imprecision is also shown in the same figure. The slight increase in error of the sharp and cone stylus is caused by pointer calibration imprecision. The precision of the spherical stylus improved slightly. This means that the pointer calibration of the spherical stylus is very precise and the main source of error is caused by probe calibration imprecision. The precision of the cone phantom remains unchanged; thus phantom positioning in 3-D space with the spherical stylus is repeatable. From the two precision measures, we see that the rod stylus performs considerably worse than the other styli/phantom. There is also a discrepancy between the two precision measures. This means that, although for a particular pointer calibration a probe calibration precision of about 2 mm can be achieved, a significantly
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worse precision of 5 mm should be expected when given an arbitrary probe calibration with an uncertain pointer calibration. The variations in the three principle axes as a result of calibration imprecision are given in Fig. 8. If the calibration is correct, then variations in the lateral direction will be primarily segmentation errors. The variation in the axial direction will be caused by a combination of segmentation and sound speed errors, and the elevational direction variations caused by alignment errors. In all calibrations, the variation is greatest in the elevational direction. This agrees with our knowledge of such techniques, where aligning the scan plane with the stylus/ phantom is difficult. The cone phantom gives the best precision in the lateral direction. This is because a circle inherently contains much more information than an isolated point. This makes segmentation easier and more precise. The relatively high elevational error signifies that aligning the cones phantom is harder than the spherical and the cone stylus. This is because aligning such styli is aided by the strong reflection when the styli are aligned perfectly. The rod stylus performs poorly, as previously discussed. The biases are only slightly less than the point reconstruction accuracy shown in Fig. 9. The spreads of the point reconstruction error are much less than the bias. This means that the majority of the point reconstruction errors lies in systematic errors, such as calibration, systematic misalignment, systematic segmentation and sound speed errors. The spreads given in the table are slightly larger, but follow closely to the precision, including pointer calibration variation given in Fig. 7. This spread is a combination of variations as a result of calibration, position sensor, segmentation and stylus/ phantom misalignment. The relatively high error in the point reconstruction accuracy using the calibrations from the rod stylus further suggests that such a stylus is unreliable for probe calibration. Both the sharp and spherical stylus have similar point reconstruction accuracies, being worse than the cone phantom and cone stylus. This difference is mainly attributed to the fact that it is difficult to align the scan plane with the tip of the sharp and spherical styli. Although the spherical stylus produces a better reflection, which can suggest whether we have aligned its tip correctly, this information is not found to be helpful because the calibration accuracies have not been improved. In fact, the accuracy of a spherical stylus is worse than the sharp stylus. This is because of the 3-mm-thick spherical tip of the stylus. Even with a good visual feedback, it is still difficult to align the tip of the spherical stylus with the scan plane. The cone stylus produces better accuracies than other styli, because any misalignment of the stylus with the scan plane is readily
Freehand 3-D US calibration ● P.-W. HSU et al.
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Table 1. Probe calibration factors for the different styli and the cone phantom Factor
Sharp
Spherical
Rod
Cone phantom
Cone stylus
Precision Accuracy Image scales Easy to use Rapid calibration Easy to make
0.61 3.21 Not required Easy Rapid Very easy
0.44 3.63 Not required Easy Rapid Very easy
5.04 7.75 Required Very easy Very rapid Very easy
0.59 1.77 Required Moderate Moderate Moderate
0.61 2.18 Not required Easy Rapid Easy
visible in the B-scans. The cone phantom produces the best accuracy. It is easier to segment a circle accurately, and we are thus able to locate accurate fiducial points in the scan plane. The poor performance of the rod stylus is a result of its design and calibration routine, which requires us to poke the scan plane in different orientations with the rod. If the scan plane is poked with the rod parallel to the normal of the scan plane, then the scan plane can be anywhere along the length of the rod. Thus, ideally the scan plane should be poked from different directions, each as oblique as possible to the normal of the scan plane. However, the image of the rod degrades as its inclination with the normal increases, resulting in segmentation errors. Therefore, for a good calibration, we need both a good image of the rod and a high inclination of the rod relative to the normal of the scan plane. These are two incompatible conditions. As a result, an imprecise and inaccurate calibration is obtained. Although precision and accuracy may be the most important measures to judge a calibration technique, there are other factors that should be noted. First, the image scales need to be known to segment the circle for the cone phantom. We have this information because we have access to the RF data. This may not be the case for other research groups, in which case the scales may need to be determined by another technique, such as using the US machine’s distance measurement tool (Hsu et al. 2006). The US settings, such as transmitter and receiver gain and time gain compensator, need to be set manually by the user for optimal segmentation. The user also needs to ensure that the correct circle has been segmented by the algorithm and replace any obvious incorrect segmentations. The stylus has the advantage that segmentation is not highly dependent on the US machine settings. The image scales can also form part of the optimization process, when functions f1 ⫽ f2, f3, f5 are minimized. However, it appears that function f3, for the rod, has multiple local minima. In our calibrations, it is necessary to fix the image scales and choose a starting point sufficiently near the solution for the optimization to converge to the correct minimum. Nevertheless, be-
cause the stylus does not need to be aligned with the scan plane, calibration is easy and rapid to perform. A summary of the various factors that should be taken into account when choosing a calibration technique is given in Table 1. This table is drawn up based on our experiences working with these styli/phantoms. Precision and accuracy are drawn from our experiments. All other factors are graded into five classes: very poor, poor, moderate, good and very good, except the need for the image scales, which is simply classed as required or not required. From the table it can be deduced that the rod stylus lies at one end of the scale, offering speed and simplicity as its main advantages, but failing to ensure a reliable, precise and accurate calibration. The cone stylus clearly surpasses the other styli, offering better accuracy and retaining calibration simplicity. The cone phantom produces the most accurate calibration, but the phantom is considerably larger and requires more skill to use. Table 2 shows the same factors for other calibration phantoms that our group has used in the past. This allows us to compare our cone stylus and the cone phantom with other phantoms we have used. The precision and accuracy achievable with the cone stylus is similar to those achievable with the plane and the Cambridge phantom, whereas the cone stylus is much more simple to manufacture and use. The mechanical instrument (Gee et al. 2005) remains the most accurate calibration technique, but it is difficult and expensive to manufacture. Calibration is most rapid when the Z-phantom is used, because the phantom does not need to be aligned with the scan plane. However, the Zphantom can only produce calibrations with a reasonable accuracy. CONCLUSION We have compared different techniques to calibrate freehand 3-D US probes using a stylus. The rod stylus is simple and quick to use, but produces poor precision and accuracy. This stylus may be useful to obtain a quick estimate of the calibration, although this may be needed in the first place for the optimiza-
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Table 2. Probe calibration factors for the other phantoms Factor
Plane
Cambridge
Instrument Gee et al. (2005)
Z-phantom Hsu et al. (2008)
Precision Accuracy Image scales Easy to use Rapid calibration Easy to make
0.57 2.28 Not required Very difficult Moderate Very easy
0.88 1.67 Not required Very difficult Moderate Moderate
0.19 0.35* Not Required Moderate Moderate Very difficult
0.78 2.5 Required Easy Very rapid Difficult
* This is not point reconstruction accuracy; hence it is not directly comparable with the other accuracies. tion when using such a stylus. The cone stylus is clearly an improvement on both the sharp and the spherical stylus, with a small modification to the design. Better accuracies are achieved, while maintaining its ease of use and calibration simplicity. The cone phantom produces the best calibration accuracy at the expense of a more sophisticated phantom, segmentation and calibration protocol. We have shown that the point reconstruction accuracy achievable with the cone stylus and cone phantom are 2.2 mm and 1.8 mm, respectively, surpassing the 3.1 mm and 3.6 mm achievable with the sharp and spherical stylus. The performance of our cone stylus and phantom lie between the state-of-the-art Z-phantom and Cambridge phantom, with which accuracies of 2.5 mm and 1.7 mm are achieved. Another advantage of the cone phantom over the stylus approaches is that calibration does not require two targets to be tracked simultaneously. REFERENCES Amin DV, Kanade T, Jaramaz B, et al. Calibration method for determining the physical location of the ultrasound image plane. In: Med Image Comput Comput Assist Interv Int Conf Med Image Comput Comput Assist Interv, Lecture Notes in Computer Science. Springer. 2001. Vol. 2208, pp. 940 –947. Arun KS, Huang TS, Blostein SD. Least-squares fitting of two 3-D point sets. IEEE Trans Pattern Analys Mach Intel 1987;9:698 –700. Barratt DC, Penney GP, Chan CSK, et al. Self-calibrating 3D-ultrasound-based bone registration for minimally invasive orthopedic surgery. IEEE Trans Med Imaging 2006;25:312–323. Bilaniuk N, Wong GSK. Speed of sound in pure water as a function of temperature. J Acoust Soc Am 1993;93:1609 –1612. Boctor M, Jain A, Choti MA, Taylor RH, Fichtinger G. A rapid calibration method for registration and 3D tracking of ultrasound images using spatial localizer. In: Proceedings of SPIE. 2003. Vol. 5035, pp. 521–532. Bouchet LG, Meeks SL, Goodchild G, Bova FJ, Buatti JM, Friedman WA. Calibration of three-dimensional ultrasound images for image-guided radiation therapy. Phys Med Biol 2001;46:559 –577. Chen TK, Abolmaesumi P, Thurston AD, Ellis RE. Automated 3D freehand ultrasound calibration with real-time accuracy control. In: Med Image Comput Comput Assist Interv Int Conf Med Image Comput Comput Assist Interv. Lecture Notes in Computer Science. Springer. 2006. Vol. 4190, pp. 899 –906. Comean RM, Fenster A, Peters TM, Integrated MR and ultrasound imaging for improved image guidance in neurosurgery. In: Proceedings of SPIE. 1998. Vol. 3338, pp. 747–754.
Dandekar S, Li Y, Molloy J, Hossack J. A phantom with reduced complexity for spatial 3-D ultrasound calibration. Ultrasound Med Biol 2005;31:1083–1093. Detmer PR, Bashein G, Hodges T, Detmer PR, Bashein G, Hodges T, Beach KW, Filer EP, Burns DH, Stradness DE. 3D ultrasonic image feature localization based on magnetic scanhead tracking: In vitro calibration and validation. Ultrasound Med Biol 1994;20:923– 936. Fenster A, Downey DB, Cardinal HN. Three-dimensional ultrasound imaging. Phys Med Biol 2001;46:R67–R99. Gee AH, Houghton NE, Treece GM, Prager RW. A mechanical instrument for 3D ultrasound probe calibration. Ultrasound Med Biol 2005;31:505–518. Gee AH, Prager RW, Treece GH, Berman LH. Engineering a freehand 3D ultrasound system. Pattern Recogn Lett 2003;24:757–777. Gooding MJ, Kennedy SH, Noble JA. Temporal calibration of freehand three-dimensional ultrasound using image alignment. Ultrasound Med Biol 2005;31:919 –927. Hough PVC. Machine analysis bubble chamber pictures. In: International Conference on High Energy Accelerators and Instrumentation. CERN. 1959, pp. 554 –556. Hsu PW, Prager RW, Gee AH, Treece GM. Rapid, easy and reliable calibration for freehand 3D ultrasound. Ultrasound Med Biol 2006; 32:823– 835. Hsu PW, Prager RW, Gee AH, Treece GM. Real-time freehand 3D ultrasound. Ultrasound Med Biol 2008;34:239 –251. Hsu PW, Prager RW, Houghton NE, Gee AH, Treece GM. Accurate fiducial location for freehand 3D ultrasound calibration. In: Proceedings of SPIE. 2007. Vol. 6513. Khamene A, Sauer F. A novel phantom-less spatial and temporal ultrasound calibration method. In: Med Image Comput Comput Assist Interv Int Conf Med Image Comput Comput Assist Interv, Lecture Notes in Computer Science. Springer-Verlag, 2005, Vol. 3750, pp. 65–72. Leotta DF, Detmer PR, Martin RW. Performance of a miniature magnetic position sensor for three-dimensional ultrasound imaging. Ultrasound Med Biol 1997;23:597– 609. Lindseth F, Tangen GA, Lango T, Bang J. Probe calibration for freehand 3-D ultrasound. Ultrasound Med Biol 2003;29:1607– 1623. Mercier L, Lango T, Lindsesth F, Collins DL. A review of calibration techniques for freehand 3-D ultrasound systems. Ultrasound Med Biol 2005;31:449 – 471. More JJ. The Levenberg-Marquardt algorithm: Implementation and theory. In: Numerical Analysis, Lecture Notes in Mathematics. Springer-Verlag, 1977, Vol. 630, pp. 105–116. Muratore DM, Galloway RL Jr. Beam calibration without a phantom for creating a 3-D freehand ultrasound system. Ultrasound Med Biol 2001;27:1557–1566. Pagoulatos N, Haynor DR, Kim Y. A fast calibration method for 3-D tracking of ultrasound images using a spatial localizer. Ultrasound Med Biol 2001;27:1219 –1229. Prager RW, Rohling RN, Gee AH, Berman L. Rapid calibration for 3-D freehand ultrasound. Ultrasound Med Biol 1998;24:855– 869.
Freehand 3-D US calibration ● P.-W. HSU et al. Rousseau F, Hellier P, Barillot C. Confhusius: A robust and fully automatic calibration method for 3D freehand ultrasound. Med Image Analyst 2005;9:25–38. Rousseau F, Hellier P, Letteboer MMJ, Niessen WJ, Barillot C. Quantitative evaluation of three calibration methods for 3-D freehand ultrasound. IEEE Trans Med Imaging 2006;25: 1492–1501.
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Sato Y, Nakamoto M, Tamaki Y, Sasama T, Sakita I, Nakajima Y, Monden M, Tamura S. Image guidance of breast cancer surgery using 3-D ultrasound images and augmented reality visualization. IEEE Trans Med Imaging 1998;17:681– 693. State A, Chen DT, Tector C, Brandt A, Chen H, Ohbuchi R, Bajura M, Fuchs H. Case study: Observing a volume rendered fetus within a pregnant patient. In: Proceedings of the Conference on Visualization ’94, IEEE Visualization. IEEE Computer Society Press California, 1994, pp. 364 –368.
APPENDIX Table A1. Precision of the probe calibrations specific to each stylus pointer calibration in millimeters. The value assumes that the pointer calibration is known and excludes any variations due to pointer calibration imprecision Point
Sharp
Spherical
Rod
Cone phantom
Cone stylus
Top left Top right Bottom left Bottom right Centre Mean
0.52 0.53 0.53 0.47 0.51 0.51
0.53 0.51 0.54 0.47 0.51 0.51
4.35 1.31 1.61 1.55 1.46 2.05
0.33 0.54 1.08 0.71 0.30 0.59
0.50 0.54 0.55 0.47 0.51 0.51
Table A2. Precision of undifferentiated probe calibrations in millimeters. The precision includes variations due to pointer calibration imprecision and probe calibration imprecision Point
Sharp
Spherical
Rod
Cone phantom
Cone stylus
Top left Top right Bottom left Bottom right Centre Mean
0.64 0.64 0.55 0.72 0.50 0.61
0.57 0.45 0.40 0.45 0.31 0.44
4.36 4.38 5.69 5.77 5.02 5.04
0.39 1.14 0.82 0.33 0.27 0.59
0.63 0.71 0.71 0.58 0.45 0.61
Table A3. Bias and spread of the point reconstructions. The figures are quoted as bias ⫾ spread in millimeters Point
Sharp
Spherical
Rod
Cone phantom
Cone stylus
Top left Top right Bottom left Bottom right Centre Mean
4.03 ⫾ 0.96 3.57 ⫾ 1.05 2.71 ⫾ 0.70 2.78 ⫾ 0.80 2.58 ⫾ 0.80 3.14 ⫾ 0.86
4.20 ⫾ 0.89 3.62 ⫾ 0.94 3.40 ⫾ 0.58 3.65 ⫾ 0.56 3.02 ⫾ 0.68 3.58 ⫾ 0.73
7.75 ⫾ 4.88 6.93 ⫾ 4.88 7.32 ⫾ 5.67 7.27 ⫾ 5.67 6.85 ⫾ 5.17 7.22 ⫾ 5.26
1.48 ⫾ 0.58 1.69 ⫾ 0.96 1.70 ⫾ 0.63 1.80 ⫾ 0.45 1.83 ⫾ 0.50 1.70 ⫾ 0.62
2.68 ⫾ 0.94 2.28 ⫾ 1.02 1.81 ⫾ 0.80 2.12 ⫾ 0.69 1.37 ⫾ 0.77 2.05 ⫾ 0.84
doi:10.1016/j.ultrasmedbio.2008.02.015
● Original Contribution COMPARISON OF FREEHAND 3-D ULTRASOUND CALIBRATION TECHNIQUES USING A STYLUS PO-WEI HSU, GRAHAM M. TREECE, RICHARD W. PRAGER, NEIL E. HOUGHTON, and ANDREW H. GEE Department of Engineering, University of Cambridge, Cambridge, United Kingdom (Received 30 June 2007; revised 9 February 2008; in final form 21 February 2008)
Abstract—In a freehand 3-D ultrasound system, a probe calibration is required to find the rigid body transformation from the corner of the B-scan to the electrical center of the position sensor. The most intuitive way to perform such a calibration is by locating fiducial points in the scan plane directly with a stylus. The main problem of this approach is the difficulty in aligning the tip of the stylus with the scan plane. The thick beamwidth makes the tip of the stylus visible in the B-scan, even if the tip is not exactly at the elevational center of the scan plane. We present a novel stylus and phantom that simplify the alignment process for more accurate probe calibration. We also compare our calibration techniques with a range of styli. We show that our stylus and cone phantom are both simple in design and can achieve a point reconstruction accuracy of 2.2 mm and 1.8 mm, respectively, an improvement from 3.2 mm and 3.6 mm with the sharp and spherical stylus. The performance of our cone stylus and phantom lie between the state-of-the-art Z-phantom and Cambridge phantom, where accuracies of 2.5 mm and 1.7 mm are achieved. (E-mail: [email protected]) © 2008 World Federation for Ultrasound in Medicine & Biology. Key Words: Freehand three-dimensional ultrasound, Calibration.
space. For brevity, we will use the notation TB¢A to mean a rotational transformation followed by a translation from the coordinate system A to coordinate system B. Three-dimensional US calibration has been an active research topic for several years (Mercier et al. 2005). The usual approach is to scan an object with known dimensions (a phantom). These scans place constraints on the eight calibration parameters: 2 image scales, 3 translations in the direction of the x, y and z axes and the three rotations—azimuth, elevation and roll—about these axes. The simplest phantom is probably a point target (Detmer et al. 1994; State et al. 1994; Amin et al. 2001; Gooding et al. 2005; Barratt et al. 2006). The point is scanned from different positions and orientations and its location marked in the B-scans. The segmented points can be mapped to the sensor’s coordinate system by using an assumed calibration and then to the world coordinate system from the position sensor readings. If the assumed calibration is correct, the points from the different B-scans should have the same coordinates in 3-D space. This places constraints on the calibration parameters. The calibration is solved by an iterative optimization technique.
INTRODUCTION Freehand 3-D ultrasound (US) (Fenster et al. 2001) is a 3-D medical imaging system with many clinical applications in anatomical visualization, volume measurements, surgery planning and radiotherapy planning (Gee et al. 2003). As a conventional 2-D US probe is swept over the anatomy of interest, the trajectory of the probe is recorded by the attached position sensor. The volume of the anatomy can be constructed by matching the US data with its corresponding position in space. However, the position sensor measures the 3-D location of the sensor, S, rather than the scan plane, P, relative to an external world coordinate system, W, as shown in Fig. 1. It is therefore necessary to find the position and orientation of the scan plane with respect to the electrical center of the position sensor. This rigid-body transformation, TS¢P, is determined through a process frequently referred to as “probe calibration.” In general, a transformation involves both a rotation and a translation in 3-D Address correspondence to: Po-Wei Hsu, Department of Engineering, University of Cambridge, Trumpington Street, Cambridge, Cambridgeshire CB2 1PZ, United Kingdom. E-mail: [email protected] 1610
Freehand 3-D US calibration ● P.-W. HSU et al.
Mobile part of position sensor TW
S S
TS
P
P
z
y
World coordinate system
Probe x
Scan plane
W Fig. 1. The coordinates associated with the scan plane and the mobile part of the position sensor.
Calibrating with the point phantom has three major disadvantages. First, it is very difficult to align the point phantom precisely with the scan plane. The finite thickness of the US beam makes the target visible in the B-scans, even if the target is not exactly at the elevational center of the scan plane. This error can be several millimeters depending on the US probe and the skill of the user. Second, automatic segmentation of isolated points in US images is seldom reliable. As a result, the point phantom is often manually or semi-automatically segmented in the US images. This makes the calibration process long and tiresome. Finally, the phantom needs to be scanned from a sufficiently diverse range of positions, so that the resulting system of constraints is not underdetermined with multiple solutions (Prager et al. 1998). Over the last decade, much research has been undertaken to make probe calibration more reliable and, at the same time, easier and quicker to perform. These phantoms include a plane (Prager et al. 1998; Rousseau et al. 2005), a 2-D alignment phantom (Sato et al. 1998; Gee et al. 2005), Z-fiducial phantom (Comean et al. 1998; Pagoulatos et al. 2001; Bouchet et al. 2001; Lindseth et al. 2003; Chen et al. 2006; Hsu et al. 2008) and other wire phantoms (Boctor et al. 2003; Dandekar et al. 2005). As in the case with a point phantom, when these phantoms are scanned, constraints are placed on the calibration parameters, which are then solved either iteratively or algebraically depending on the phantom used. Mercier et al. (2005) and Rousseau et al. (2006) compared some of these techniques. A 3-D localizer, often called a pointer or a stylus, can be used to aid probe calibration. It is essentially a point target connected to another position sensor at the end of the stylus, whose tip can be spherical or sharpened to a point. The rigid-body transform between the tip of the stylus and the position sensor is usually supplied by the manufacturer (Muratore and Galloway 2001). In the case where the transformation is not available, it can be
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determined by a simple pointer calibration (Leotta et al. 1997). During a pointer calibration, the stylus is rotated about its tip while the position sensor’s readings are recorded. Because the tip of the stylus must be mapped to the same location in 3-D space, this places constraints on the possible locations of the stylus’ tip, whose location can be determined by an iterative optimization algorithm. In any case, the location of the stylus’ tip is known in 3-D space. The location of the point phantom can therefore be determined by pointing the stylus at the phantom. If the scales of the B-scan are known (Hsu et al. 2006), the calibration parameters can be solved in a closed-form by least-squares minimization (Arun et al. 1987). Because a stylus can be used to locate points in 3-D space, Muratore and Galloway (2001) calibrated their probe by locating points directly in the scan plane. Assuming at least three noncollinear points have been located, calibration can be solved by least-squares optimization as described previously. This technique, nevertheless, requires two targets to be tracked simultaneously. Furthermore, it is difficult to align the tip of the stylus with the scan plane. Khamene and Sauer (2005) improved on this technique by imaging a rod transversely. Both ends of the rod are pointer calibrated to define the location and orientation of the rod in space. Each image of the rod sets up a constraint on the calibration parameters. Probe calibration can then be found using optimization techniques. In this paper, we study the relative merits of a particular class of calibration algorithms that locate points in the B-scan with a stylus. We begin in the next section by outlining different calibration techniques, including our novel cone phantom and cone stylus. We then performed a series of experiments to evaluate the precision and accuracy of each technique. The various calibration algorithms are compared and discussed in the final section. MATERIALS AND METHODS The calibration phantoms Figure 2 shows the four calibration styli and the cone phantom. Figure 2, a and b are standard Polaris (Northern Digital Inc., Waterloo, ON, Canada) styli with a sharp and a spherical tip. Figure 2c shows a rod stylus similar to the one used by Khamene and Sauer (2005). Figure 2, d and e show the cone phantom and the cone stylus that we have designed. The cone phantom is an improvement on one of our previous phantoms (Hsu et al. 2007). Although this phantom is not a stylus physically, it works on the same principle as a stylus by locating points with known 3-D locations in the scan plane. These are the four styli and the cone phantom that we will use for probe calibration comparison. During
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Fig. 2. The calibration styli and phantom.
Freehand 3-D US calibration ● P.-W. HSU et al.
probe calibration, we will scan a point, x, that is on the styli/phantom. This point xP⬘ ⫽ (u, v, 0)t can be segmented in the US image, where u and v are the column and row indices of the cropped image in pixels. This point is changed to metric units by an appropriate scaling factor
冢 冣 su 0 0
r2
L
xW ⫽ TW¢STS¢PxP,
(1)
where TW¢S is given by the Polaris (Northern Digital Inc.) readings. Polaris styli Figure 3 shows the coordinate systems involved when calibrating with either of the Polaris (Northern Digital Inc.) styli. During pointer calibration, the location of the stylus’ tip, xL, is found in the stylus’ coordinate system, L. This point can be mapped to 3-D space by: xW ⫽ TW¢LxL,
(2)
where the transformation TW¢L can be read off the Polaris. If we place the stylus’ tip in the scan plane, its location in space is also given by eqn (1). This means that only the calibration TS¢P and possibly the scale factors Ts are unknown in the two expressions for xW. Calibration can therefore be found by equating eqns (1) and (2), i.e., minimizing
f1 ⫽ f2 ⫽
兺 ⱍT
ⱍ
TS¢PTsxiP⬘ ⫺ TW¢LixL ,
W¢si
where | · | denotes the usual Euclidean norm on ⺢3. The above equation is minimized by using the iterative Levenberg-Marquardt algorithm (More 1977). Rod stylus Figure 2c shows the rod stylus that is used to follow the approach by Khamene and Sauer (2005). The corresponding coordinates are shown in Fig. 4. We have attached a 15-cm-long, 1.5-mm-thick rod to a Polaris (Northern Digital Inc.) active marker. Both ends of the rod are sharpened for accurate pointer calibration. Again, the end points of the rod rL1 and rL2 are found by two separate pointer calibrations, and their coordinates in 3-D space are TW¢LrL1 and TW¢LrL2 , respectively. Now, the point of intersection of the rod and the scan plane is given by eqn (1). Furthermore, this point lies on the line formed by the two end points of the stylus. This means that the distance from the point xW to the line r1Wr2W is zero, i.e.,
ⱍ共r
W 2
ⱍ
⫺ r1W兲 ⫻ 共r1W ⫺ xW兲 ⫽0 r2W ⫺ r1W
ⱍ
ⱍ共T
ⱍ
ⱍ
r ⫺ TW¢Lr1L兲 ⫻ 共TW¢Lr1L ⫺ TW¢STS¢PxP兲 ⫽ 0. TW¢Lr2L ⫺ TW¢Lr1L
L W¢L 2
ⱍ
ⱍ
The ⫻ in the above equations denotes the cross product of two vectors in ⺢3. Because |TW¢Lr2L ⫺ TW¢Lr1L| is a constant, calibration can be found by minimizing
S
P
L
i
)
TS
TW
r1
Fig. 4. The coordinate systems involved when calibrating with a rod stylus.
where su and sv are the scales in millimeters per pixel to give xP ⫽ TsxP⬘. If the probe calibration TS¢P is known, then the segmented point can be mapped to world space by:
S
x
World coordinate system W
Ts ⫽ 0 sv 0 , 0 0 0
TW
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L
P
f3 ⫽
兺 ⱍ共T
r ⫺ TW¢Lir1L兲
L W¢Li 2
i
World coordinate system
x
TW
L
W
Fig. 3. The coordinate systems involved when calibrating with a Polaris stylus.
ⱍ
⫻ 共TW¢Lir1L ⫺ TW¢SiTS¢PxiP兲 . However, it is practically impossible to minimize f3 without knowing the scale factors. Therefore, the scale factors need to be determined explicitly and fixed before minimizing f3.
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Cone phantom The cone phantom (Fig. 2d) is primarily comprised of a polypropylene block machined into the shape of two hollow cones with 1-mm wall thickness. The center of the circle where the two cones join is the fiducial point, x, that we will use for probe calibration. The cones have outer diameters varying between 12-mm and 52-mm. They are supported by an aluminium frame structure consisting of two 135 ⫻ 135-cm square plates held together by four bolts. There are three cone-shaped divots, di, in the two blocks to fit a 3-mm ball-pointed Polaris (Northern Digital Inc.) stylus. The position of the divots, dW i , are therefore known in 3-D space. These divots serve as the principal axes of the phantom and are used to determine the phantom position in 3-D space. The location of x is measured with a Mitutoyo (Mitutoyo Corporation, Kawasaki, Japan) coordinate measuring machine (CMM) relative to the divots, i.e., xA ⫽ fCMM(dAi ), where the function fCMM is determined by the CMM and is valid for arbitrary coordinate system A. In particular, xW ⫽ fCMM(dW i ) in world space. All dimensions are precision manufactured by our workshop, with a tolerance of ⫾0.1 mm. If we align the scan plane with the circle where the two cones join, we get a circle with a 12-mm diameter. This circle can be segmented with its center xP⬘ automatically. As we shall see later, the image scales are necessary for reliable segmentation. Let us assume for the moment that the scales, Ts, are known; hence xP ⫽ TsxP⬘ can be computed. Now, recall that we have already found the coordinates of this centre in 3-D space. We can transform this point to the sensor’s coordinate system by using the inverse of the position sensor’s readings: ⫺1 xS ⫽ TW¢S xW ⫺1 ⫽ TW¢S f CMM(dW i ).
For probe calibration, we need to find the single transformation TS¢P that best transforms {xPi} to {xSi} ⫽ ⫺1 兵TW¢S f 共dW i 兲其. This can be found (Arun et al. 1987) by i CMM minimizing f4 ⫽
兺 ⱍT i
共dWi 兲 ⫺ TS¢PxP ⱍ.
⫺1 W¢S CMM
f
i
Cone stylus Figure 2e shows a photograph of our new cone stylus. It consists of a stainless steel shaft, 120 mm in length and 13 mm in diameter. On one end of the shaft, a platform is ground to fit a PassTrax (Traxtal Technologies, Toronto, ON, Canada) position sensor in such a way that the z axis of the sensor is parallel to the shaft. The other end of the shaft is sharpened for accurate
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pointer calibration. As previously, this means that the point rL is found by a pointer calibration. The main feature of this stylus is that part is thinned in the shape of two cones, to meet at the critical point x. This point is precisely 20 mm from the stylus’ tip. Here, the diameter is 1.5 mm. The stylus was precision manufactured by our workshop with a tolerance of ⫾0.1 mm. From the known geometric model of the stylus, xL L ⫽ r ⫺ (0, 0, 20)t mm. If the position sensor cannot be placed with its z axis parallel to the shaft, the same technique can still be used provided that both ends of the shaft are pointer calibrated so that the orientation of the shaft is known. We now place fiducial marks by aligning this critical point with the scan plane. From here, the calibration process is identical to that for a Polaris (Northern Digital Inc.) stylus, the only difference being that we are aligning the critical point, and not the stylus’ tip. The calibration parameters are found by minimizing f5 ⫽
兺 ⱍT i
ⱍ
TS¢PTsxiP ⫺ TW¢Li共rL ⫺ 共0, 0, 20兲t兲 . ⬘
W¢Si
Segmentation Figure 5 shows typical US images of the various styli/phantom. The rod stylus does not need to be aligned with the scan plane, and the calibration is performed by poking the rod into the scan plane in different places. The sharp stylus is aligned visually so that the stylus’ tip just appears in the US image. The cone phantom is also aligned visually by moving the probe up and down its length until we get a circle with the smallest radius. The spherical and cone stylus are aligned until we see a strong reflection, as shown in Fig. 5, b and c. This happens when the stylus is perpendicular to the scan plane and a strong reflection is obtained. Our cone stylus has a small flat region between the two inverted cones to ensure a strong reflection once the stylus is aligned. A similar point spread function is seen in Fig. 5c; this occurs if the rod is perpendicular to the scan plane. Because we need to scan the rod at different angles, an image similar to Fig. 5a is seen when the rod is not perpendicular to the scan plane. Because the styli are made from stainless steel, the US images are pointspread functions and not a circle. The radius of the styli cannot be determined from the US images. However, as the cone stylus is moved through the scan plane, we can see the varying radius of the stylus and this assists the user to align the stylus with the scan plane. The cone phantom is designed so that we can use the radius of the phantom to assist segmentation. When segmenting the image of the four styli, we first search through the entire image to find the pixel with maximum intensity. This pixel serves as a starting point for the segmentation algorithm. We are anticipating a
Freehand 3-D US calibration ● P.-W. HSU et al.
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Fig. 5. Typical images of the different styli and phantom with the segmentation result imposed.
strong reflection and an image of the point-spread function, which will be symmetrical about the direction of propagation of the US waves. This means that for a linear probe, there should be a vertical line of symmetry. To
allow for curvilinear probes, we iterate this line through all possible angles and across a horizontal distance about 10% of the image width, near the pixel with maximum intensity. The angles are incremented in steps of 1°.
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RESULTS
Group 1 Probe Calibration 1-1
Pointer Calibration 2
Probe Calibration 1-2
Probe Calibration 1-3
Probe Calibration 1-4
Probe Calibration 1-5
Probe Calibration 5-4
Probe Calibration 5-5
Group 2
...
... Pointer Calibration 5
Volume 34, Number 10, 2008
Group 5 Probe Calibration 5-1
Probe Calibration 5-2
Probe Calibration 5-3
Fig. 6. The calibration protocol. Five pointer calibrations were performed. For each pointer calibration, five probe calibrations were performed.
During each iteration, the correlation of the image on either side of the line is computed. Once we have found the line of symmetry corresponding to the maximum symmetric autocorrelation, the required point is assumed to be on this line. We sum the intensity across each depth on this line and find the depth that has the highest intensity. The point on the line of symmetry corresponding to the brightest depth is segmented. This algorithm is used to segment the image of each stylus, and the results are imposed on Fig. 5a– c and e. The line of symmetry is shown as well. The near-vertical line of symmetry verifies that the point has been segmented correctly. Because the top, rather than the center of the surface is segmented, a correction is performed in the optimization process to shift the segmented point downwards by half the stylus thickness. We have measured the spherical tip of the Polaris (Northern Digital Inc.) stylus with a digital micrometer to be 3.0 mm in diameter. The thickness of the rod and cone stylus is 1.5 mm. To calibrate our linear probes in room temperature water so that the temperature is more stable, we now post-process the segmented points by shifting them upwards towards the probe face. Assuming that the user will measure the water temperature, the speed of sound in water at this temperature can be computed (Bilaniuk and Wong 1993). We then shift the pixels upwards by the sound speed in water temperature correction factor . 1540 For the cone phantom, we first place a user-defined threshold on the B-scan to remove any unwanted scatters, resulting in a binary image consisting mainly of the circle that is to be segmented. We now shift each pixel upwards by the temperature correction factor. Edge detection is performed by applying a Sobel filter to overlapping 3 ⫻ 3 blocks of the image. If the image scales are known, we can apply the Hough transform (Hough 1959) on the resulting edges to detect a circle with a 12-mm diameter.
To measure the calibration quality of the different styli and the phantom, we calibrated a Diasus (Dynamic Imaging Ltd., Birmingham, UK) 5–10 MHz linear array probe. The analog radiofrequency (RF) US data, after receive focusing and time– gain compensation, but before log compression and envelope detection, was digitized using a Gage CompuScope (GaGe Applied Technologies Inc., Lockport, IL, USA) 14200 PCI 14-bit analog to digital converter, and transferred at 25 frames per second to a Pentium® 4 2.80-GHz personal computer running Microsoft Windows XP (Microsoft Corp., Redmond, WA, USA). Because we have access to the RF data, the image scales could be set at our discretion. The B-scans were displayed at 0.1 mm/pixel, with a cropped size of 30.0 ⫻ 38.1 mm. The probe was tracked using a PassTrax (Traxtal Technologies) target for the Polaris (Northern Digital Inc.) optical tracking system. A pointer calibration was initially performed on each stylus. Five probe calibrations were followed. The five probe calibrations are used to evaluate the repeatability of the calibration. During each probe calibration, 20 points were located throughout the image with the stylus. To evaluate the impact of the pointer calibration on probe calibration, this whole routine was again repeated five times, each time with a different pointer calibration, as shown in Fig. 6. For probe calibrations with the cone phantom, a pointer-calibrated stylus (Fig. 2b) was used to locate the phantom in space. Ten readings were taken at each divot, and the mean was used for probe calibration. Five calibrations, each with 20 images of the cone spread throughout the B-scan, were performed. Again, this process was repeated five times, each time recalibrating the stylus and relocating the phantom by its divots. Thus, a total of 25 calibrations were computed for each stylus/ phantom. Precision One way to assess the calibration quality is by computing its precision. A point, pP, in the scan plane
1.0
Exclude pointer calibration variation
2.1 5.0
Include pointer calibration variation
0.8
Precision (mm)
Pointer Calibration 1
Ultrasound in Medicine and Biology
0.6
0.4
0.2
Sharp Stylus
Spherical Stylus
Rod Stylus
Cone Phantom
Cone Stylus
Fig. 7. Precision of the probe calibrations.
Freehand 3-D US calibration ● P.-W. HSU et al. Lateral 1.4
Axial
5.0
Elevational
1.2
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ponents in the lateral, axial and elevational directions. Figure 8 shows the mean variation of the four corners and the center of the B-scan in these three principle axes.
1.0
0.6
0.4
0.2
Sharp Stylus
Spherical Stylus
Cone Phantom
Rod Stylus
Cone Stylus
Fig. 8. Mean variation of the four corners and the center of the B-scan in the three principle axes of the scan plane.
can be mapped to the sensor’s coordinate system after probe calibration, because pS ⫽ TS¢PpP. If all the calibrations are identical, the point, pS, should remain stationary. Precision is therefore measured by calculating 1 ⫽ N
N
兺 共p
Si
⫺ pSi兲,
|TW¢LiwL ⫺ TW¢STS¢PTswP⬘|.
i⫽1
where pSi is the point in the sensor’s coordinate system mapped by the ith calibration, pSi is used to denote the arithmetic mean of (pSi) and N is the number of calibrations. It is obvious that the above measure is dependent on the point chosen in the B-scan. We will therefore measure the variation of the four corners and the center of the B-scan. To investigate whether the pointer calibration is sufficiently accurate and measure its impact on the calibration result, we compute two precision measures. First, the precision for each group of the five calibrations specific to the same pointer calibration is calculated. These five precisions for the five groups of calibrations are then averaged. Specifically, 1 1 ⫽ 25
Accuracy Although precision measures calibration reproducibility, this does not reflect the accuracy of the calibration, because there may be a systematic error. We assess the accuracy of our calibrations by measuring the point reconstruction accuracy of a point target. We scan the tip of a wire that is 1.5 mm thick in water at room temperature. The image of the wire tip, w, is reconstructed in space by using the obtained calibrations. The 3-D location of the wire tip is found by using an independent stylus. The root-mean-square error of the pointer calibration is 0.8 mm. Ten readings of the wire tip were obtained to eliminate errors from the stylus’ readings. The true location of the wire tip is taken to be the mean of the stylus’ readings. Accuracy is measured by the amount of mismatch between the reconstructed image and the true location, i.e.,
5
Point reconstruction accuracy may depend on the position of the probe and the position of the wire tip in the B-scan. We therefore capture images of the wire tip at five different locations in the B-scans—near the four corners and the center of the B-scan. The wire tip is aligned with the scan plane by hand, and segmented manually from the B-scans. The probe is held perpendicular to the wire and rotated through a full revolution about the lateral axis at six different locations. Furthermore, five images of the wire are taken at each probe position and at each location in the B-scans. A total of 150 images of the wire tip are captured. The point reconstruction accuracy is shown in Fig. 9, where the graph shows the mean of the 30 repetitions at each position in the B-scan.
5
兺 兺 共p
Si j
⫺ pSj i兲,
j⫽1 i⫽1
th
Si j
where p denotes the point mapped by the i probe calibration associated with the jth pointer calibration, and pSj i is the mean of 共pSj 1, pSj 2, ···, pSj 5兲, for a given, j. The results are shown in Table A1. Table A2 shows the precision without differentiating the 25 calibrations, i.e.,
8.0
Point Reconstruction Accuracy
7.0
Bias Spread
6.0
Accuracy (mm)
Precision (mm)
0.8
5.0
4.0
3.0
2 ⫽
1 25
25
兺 共p
Si
⫺ pSi兲.
2.0
1.0
i⫽1
The mean variations of the four corners and the center of the B-scan are also given in Fig. 7. Each of these variations can be broken down into their corresponding com-
Sharp Stylus
Spherical Stylus
Rod Stylus
Cone Phantom
Cone Stylus
Fig. 9. Point reconstruction accuracy, bias and spread of the point reconstructions.
Ultrasound in Medicine and Biology
s5
1
true location bia s2
spread
centroid
ad
bias 3
s4
re sp
bia
s bia
spread
bia
bias 6
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Fig. 10. Calculation of bias and spread of the point reconstruction accuracy.
Point reconstruction accuracy consists of a bias and a spread. The bias consists of calibration error and other systematic errors, such as systematic misalignment, systematic segmentation, sound speed errors and an incorrect position of the true location. Because we have placed our probe at six different positions, any systematic error will cause the reconstructed points to be clustered into six groups, as shown in Fig. 10. For each probe position, we compute the centroid of the points belonging to that cluster. The bias is measured to be the distance from the centroid to the true location. Figure 9 also shows the mean of the biases and their spread. The spread is measured by the root-mean-square distance from each reconstructed point to the corresponding centroid, i.e., the standard deviation of the points within each cluster. Table A3 shows the biases and spread in the form bias ⫾ spread. DISCUSSION Figure 7 shows the precision of the probe calibrations with a fixed pointer calibration. These precisions can be achieved if the pointer calibration is supplied by the manufacturer of the stylus. Probe calibration precision that includes variations as a result of pointer calibration imprecision is also shown in the same figure. The slight increase in error of the sharp and cone stylus is caused by pointer calibration imprecision. The precision of the spherical stylus improved slightly. This means that the pointer calibration of the spherical stylus is very precise and the main source of error is caused by probe calibration imprecision. The precision of the cone phantom remains unchanged; thus phantom positioning in 3-D space with the spherical stylus is repeatable. From the two precision measures, we see that the rod stylus performs considerably worse than the other styli/phantom. There is also a discrepancy between the two precision measures. This means that, although for a particular pointer calibration a probe calibration precision of about 2 mm can be achieved, a significantly
Volume 34, Number 10, 2008
worse precision of 5 mm should be expected when given an arbitrary probe calibration with an uncertain pointer calibration. The variations in the three principle axes as a result of calibration imprecision are given in Fig. 8. If the calibration is correct, then variations in the lateral direction will be primarily segmentation errors. The variation in the axial direction will be caused by a combination of segmentation and sound speed errors, and the elevational direction variations caused by alignment errors. In all calibrations, the variation is greatest in the elevational direction. This agrees with our knowledge of such techniques, where aligning the scan plane with the stylus/ phantom is difficult. The cone phantom gives the best precision in the lateral direction. This is because a circle inherently contains much more information than an isolated point. This makes segmentation easier and more precise. The relatively high elevational error signifies that aligning the cones phantom is harder than the spherical and the cone stylus. This is because aligning such styli is aided by the strong reflection when the styli are aligned perfectly. The rod stylus performs poorly, as previously discussed. The biases are only slightly less than the point reconstruction accuracy shown in Fig. 9. The spreads of the point reconstruction error are much less than the bias. This means that the majority of the point reconstruction errors lies in systematic errors, such as calibration, systematic misalignment, systematic segmentation and sound speed errors. The spreads given in the table are slightly larger, but follow closely to the precision, including pointer calibration variation given in Fig. 7. This spread is a combination of variations as a result of calibration, position sensor, segmentation and stylus/ phantom misalignment. The relatively high error in the point reconstruction accuracy using the calibrations from the rod stylus further suggests that such a stylus is unreliable for probe calibration. Both the sharp and spherical stylus have similar point reconstruction accuracies, being worse than the cone phantom and cone stylus. This difference is mainly attributed to the fact that it is difficult to align the scan plane with the tip of the sharp and spherical styli. Although the spherical stylus produces a better reflection, which can suggest whether we have aligned its tip correctly, this information is not found to be helpful because the calibration accuracies have not been improved. In fact, the accuracy of a spherical stylus is worse than the sharp stylus. This is because of the 3-mm-thick spherical tip of the stylus. Even with a good visual feedback, it is still difficult to align the tip of the spherical stylus with the scan plane. The cone stylus produces better accuracies than other styli, because any misalignment of the stylus with the scan plane is readily
Freehand 3-D US calibration ● P.-W. HSU et al.
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Table 1. Probe calibration factors for the different styli and the cone phantom Factor
Sharp
Spherical
Rod
Cone phantom
Cone stylus
Precision Accuracy Image scales Easy to use Rapid calibration Easy to make
0.61 3.21 Not required Easy Rapid Very easy
0.44 3.63 Not required Easy Rapid Very easy
5.04 7.75 Required Very easy Very rapid Very easy
0.59 1.77 Required Moderate Moderate Moderate
0.61 2.18 Not required Easy Rapid Easy
visible in the B-scans. The cone phantom produces the best accuracy. It is easier to segment a circle accurately, and we are thus able to locate accurate fiducial points in the scan plane. The poor performance of the rod stylus is a result of its design and calibration routine, which requires us to poke the scan plane in different orientations with the rod. If the scan plane is poked with the rod parallel to the normal of the scan plane, then the scan plane can be anywhere along the length of the rod. Thus, ideally the scan plane should be poked from different directions, each as oblique as possible to the normal of the scan plane. However, the image of the rod degrades as its inclination with the normal increases, resulting in segmentation errors. Therefore, for a good calibration, we need both a good image of the rod and a high inclination of the rod relative to the normal of the scan plane. These are two incompatible conditions. As a result, an imprecise and inaccurate calibration is obtained. Although precision and accuracy may be the most important measures to judge a calibration technique, there are other factors that should be noted. First, the image scales need to be known to segment the circle for the cone phantom. We have this information because we have access to the RF data. This may not be the case for other research groups, in which case the scales may need to be determined by another technique, such as using the US machine’s distance measurement tool (Hsu et al. 2006). The US settings, such as transmitter and receiver gain and time gain compensator, need to be set manually by the user for optimal segmentation. The user also needs to ensure that the correct circle has been segmented by the algorithm and replace any obvious incorrect segmentations. The stylus has the advantage that segmentation is not highly dependent on the US machine settings. The image scales can also form part of the optimization process, when functions f1 ⫽ f2, f3, f5 are minimized. However, it appears that function f3, for the rod, has multiple local minima. In our calibrations, it is necessary to fix the image scales and choose a starting point sufficiently near the solution for the optimization to converge to the correct minimum. Nevertheless, be-
cause the stylus does not need to be aligned with the scan plane, calibration is easy and rapid to perform. A summary of the various factors that should be taken into account when choosing a calibration technique is given in Table 1. This table is drawn up based on our experiences working with these styli/phantoms. Precision and accuracy are drawn from our experiments. All other factors are graded into five classes: very poor, poor, moderate, good and very good, except the need for the image scales, which is simply classed as required or not required. From the table it can be deduced that the rod stylus lies at one end of the scale, offering speed and simplicity as its main advantages, but failing to ensure a reliable, precise and accurate calibration. The cone stylus clearly surpasses the other styli, offering better accuracy and retaining calibration simplicity. The cone phantom produces the most accurate calibration, but the phantom is considerably larger and requires more skill to use. Table 2 shows the same factors for other calibration phantoms that our group has used in the past. This allows us to compare our cone stylus and the cone phantom with other phantoms we have used. The precision and accuracy achievable with the cone stylus is similar to those achievable with the plane and the Cambridge phantom, whereas the cone stylus is much more simple to manufacture and use. The mechanical instrument (Gee et al. 2005) remains the most accurate calibration technique, but it is difficult and expensive to manufacture. Calibration is most rapid when the Z-phantom is used, because the phantom does not need to be aligned with the scan plane. However, the Zphantom can only produce calibrations with a reasonable accuracy. CONCLUSION We have compared different techniques to calibrate freehand 3-D US probes using a stylus. The rod stylus is simple and quick to use, but produces poor precision and accuracy. This stylus may be useful to obtain a quick estimate of the calibration, although this may be needed in the first place for the optimiza-
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Table 2. Probe calibration factors for the other phantoms Factor
Plane
Cambridge
Instrument Gee et al. (2005)
Z-phantom Hsu et al. (2008)
Precision Accuracy Image scales Easy to use Rapid calibration Easy to make
0.57 2.28 Not required Very difficult Moderate Very easy
0.88 1.67 Not required Very difficult Moderate Moderate
0.19 0.35* Not Required Moderate Moderate Very difficult
0.78 2.5 Required Easy Very rapid Difficult
* This is not point reconstruction accuracy; hence it is not directly comparable with the other accuracies. tion when using such a stylus. The cone stylus is clearly an improvement on both the sharp and the spherical stylus, with a small modification to the design. Better accuracies are achieved, while maintaining its ease of use and calibration simplicity. The cone phantom produces the best calibration accuracy at the expense of a more sophisticated phantom, segmentation and calibration protocol. We have shown that the point reconstruction accuracy achievable with the cone stylus and cone phantom are 2.2 mm and 1.8 mm, respectively, surpassing the 3.1 mm and 3.6 mm achievable with the sharp and spherical stylus. The performance of our cone stylus and phantom lie between the state-of-the-art Z-phantom and Cambridge phantom, with which accuracies of 2.5 mm and 1.7 mm are achieved. Another advantage of the cone phantom over the stylus approaches is that calibration does not require two targets to be tracked simultaneously. REFERENCES Amin DV, Kanade T, Jaramaz B, et al. Calibration method for determining the physical location of the ultrasound image plane. In: Med Image Comput Comput Assist Interv Int Conf Med Image Comput Comput Assist Interv, Lecture Notes in Computer Science. Springer. 2001. Vol. 2208, pp. 940 –947. Arun KS, Huang TS, Blostein SD. Least-squares fitting of two 3-D point sets. IEEE Trans Pattern Analys Mach Intel 1987;9:698 –700. Barratt DC, Penney GP, Chan CSK, et al. Self-calibrating 3D-ultrasound-based bone registration for minimally invasive orthopedic surgery. IEEE Trans Med Imaging 2006;25:312–323. Bilaniuk N, Wong GSK. Speed of sound in pure water as a function of temperature. J Acoust Soc Am 1993;93:1609 –1612. Boctor M, Jain A, Choti MA, Taylor RH, Fichtinger G. A rapid calibration method for registration and 3D tracking of ultrasound images using spatial localizer. In: Proceedings of SPIE. 2003. Vol. 5035, pp. 521–532. Bouchet LG, Meeks SL, Goodchild G, Bova FJ, Buatti JM, Friedman WA. Calibration of three-dimensional ultrasound images for image-guided radiation therapy. Phys Med Biol 2001;46:559 –577. Chen TK, Abolmaesumi P, Thurston AD, Ellis RE. Automated 3D freehand ultrasound calibration with real-time accuracy control. In: Med Image Comput Comput Assist Interv Int Conf Med Image Comput Comput Assist Interv. Lecture Notes in Computer Science. Springer. 2006. Vol. 4190, pp. 899 –906. Comean RM, Fenster A, Peters TM, Integrated MR and ultrasound imaging for improved image guidance in neurosurgery. In: Proceedings of SPIE. 1998. Vol. 3338, pp. 747–754.
Dandekar S, Li Y, Molloy J, Hossack J. A phantom with reduced complexity for spatial 3-D ultrasound calibration. Ultrasound Med Biol 2005;31:1083–1093. Detmer PR, Bashein G, Hodges T, Detmer PR, Bashein G, Hodges T, Beach KW, Filer EP, Burns DH, Stradness DE. 3D ultrasonic image feature localization based on magnetic scanhead tracking: In vitro calibration and validation. Ultrasound Med Biol 1994;20:923– 936. Fenster A, Downey DB, Cardinal HN. Three-dimensional ultrasound imaging. Phys Med Biol 2001;46:R67–R99. Gee AH, Houghton NE, Treece GM, Prager RW. A mechanical instrument for 3D ultrasound probe calibration. Ultrasound Med Biol 2005;31:505–518. Gee AH, Prager RW, Treece GH, Berman LH. Engineering a freehand 3D ultrasound system. Pattern Recogn Lett 2003;24:757–777. Gooding MJ, Kennedy SH, Noble JA. Temporal calibration of freehand three-dimensional ultrasound using image alignment. Ultrasound Med Biol 2005;31:919 –927. Hough PVC. Machine analysis bubble chamber pictures. In: International Conference on High Energy Accelerators and Instrumentation. CERN. 1959, pp. 554 –556. Hsu PW, Prager RW, Gee AH, Treece GM. Rapid, easy and reliable calibration for freehand 3D ultrasound. Ultrasound Med Biol 2006; 32:823– 835. Hsu PW, Prager RW, Gee AH, Treece GM. Real-time freehand 3D ultrasound. Ultrasound Med Biol 2008;34:239 –251. Hsu PW, Prager RW, Houghton NE, Gee AH, Treece GM. Accurate fiducial location for freehand 3D ultrasound calibration. In: Proceedings of SPIE. 2007. Vol. 6513. Khamene A, Sauer F. A novel phantom-less spatial and temporal ultrasound calibration method. In: Med Image Comput Comput Assist Interv Int Conf Med Image Comput Comput Assist Interv, Lecture Notes in Computer Science. Springer-Verlag, 2005, Vol. 3750, pp. 65–72. Leotta DF, Detmer PR, Martin RW. Performance of a miniature magnetic position sensor for three-dimensional ultrasound imaging. Ultrasound Med Biol 1997;23:597– 609. Lindseth F, Tangen GA, Lango T, Bang J. Probe calibration for freehand 3-D ultrasound. Ultrasound Med Biol 2003;29:1607– 1623. Mercier L, Lango T, Lindsesth F, Collins DL. A review of calibration techniques for freehand 3-D ultrasound systems. Ultrasound Med Biol 2005;31:449 – 471. More JJ. The Levenberg-Marquardt algorithm: Implementation and theory. In: Numerical Analysis, Lecture Notes in Mathematics. Springer-Verlag, 1977, Vol. 630, pp. 105–116. Muratore DM, Galloway RL Jr. Beam calibration without a phantom for creating a 3-D freehand ultrasound system. Ultrasound Med Biol 2001;27:1557–1566. Pagoulatos N, Haynor DR, Kim Y. A fast calibration method for 3-D tracking of ultrasound images using a spatial localizer. Ultrasound Med Biol 2001;27:1219 –1229. Prager RW, Rohling RN, Gee AH, Berman L. Rapid calibration for 3-D freehand ultrasound. Ultrasound Med Biol 1998;24:855– 869.
Freehand 3-D US calibration ● P.-W. HSU et al. Rousseau F, Hellier P, Barillot C. Confhusius: A robust and fully automatic calibration method for 3D freehand ultrasound. Med Image Analyst 2005;9:25–38. Rousseau F, Hellier P, Letteboer MMJ, Niessen WJ, Barillot C. Quantitative evaluation of three calibration methods for 3-D freehand ultrasound. IEEE Trans Med Imaging 2006;25: 1492–1501.
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Sato Y, Nakamoto M, Tamaki Y, Sasama T, Sakita I, Nakajima Y, Monden M, Tamura S. Image guidance of breast cancer surgery using 3-D ultrasound images and augmented reality visualization. IEEE Trans Med Imaging 1998;17:681– 693. State A, Chen DT, Tector C, Brandt A, Chen H, Ohbuchi R, Bajura M, Fuchs H. Case study: Observing a volume rendered fetus within a pregnant patient. In: Proceedings of the Conference on Visualization ’94, IEEE Visualization. IEEE Computer Society Press California, 1994, pp. 364 –368.
APPENDIX Table A1. Precision of the probe calibrations specific to each stylus pointer calibration in millimeters. The value assumes that the pointer calibration is known and excludes any variations due to pointer calibration imprecision Point
Sharp
Spherical
Rod
Cone phantom
Cone stylus
Top left Top right Bottom left Bottom right Centre Mean
0.52 0.53 0.53 0.47 0.51 0.51
0.53 0.51 0.54 0.47 0.51 0.51
4.35 1.31 1.61 1.55 1.46 2.05
0.33 0.54 1.08 0.71 0.30 0.59
0.50 0.54 0.55 0.47 0.51 0.51
Table A2. Precision of undifferentiated probe calibrations in millimeters. The precision includes variations due to pointer calibration imprecision and probe calibration imprecision Point
Sharp
Spherical
Rod
Cone phantom
Cone stylus
Top left Top right Bottom left Bottom right Centre Mean
0.64 0.64 0.55 0.72 0.50 0.61
0.57 0.45 0.40 0.45 0.31 0.44
4.36 4.38 5.69 5.77 5.02 5.04
0.39 1.14 0.82 0.33 0.27 0.59
0.63 0.71 0.71 0.58 0.45 0.61
Table A3. Bias and spread of the point reconstructions. The figures are quoted as bias ⫾ spread in millimeters Point
Sharp
Spherical
Rod
Cone phantom
Cone stylus
Top left Top right Bottom left Bottom right Centre Mean
4.03 ⫾ 0.96 3.57 ⫾ 1.05 2.71 ⫾ 0.70 2.78 ⫾ 0.80 2.58 ⫾ 0.80 3.14 ⫾ 0.86
4.20 ⫾ 0.89 3.62 ⫾ 0.94 3.40 ⫾ 0.58 3.65 ⫾ 0.56 3.02 ⫾ 0.68 3.58 ⫾ 0.73
7.75 ⫾ 4.88 6.93 ⫾ 4.88 7.32 ⫾ 5.67 7.27 ⫾ 5.67 6.85 ⫾ 5.17 7.22 ⫾ 5.26
1.48 ⫾ 0.58 1.69 ⫾ 0.96 1.70 ⫾ 0.63 1.80 ⫾ 0.45 1.83 ⫾ 0.50 1.70 ⫾ 0.62
2.68 ⫾ 0.94 2.28 ⫾ 1.02 1.81 ⫾ 0.80 2.12 ⫾ 0.69 1.37 ⫾ 0.77 2.05 ⫾ 0.84