Geometry Calibration Phantom Design for 3D Imaging

Transcription

1 Geometry Calibration Phantom Design for 3D Imaging Bernhard E.H. Claus General Electric Company, Global Research Center ABSTRACT Most approaches to 3D x-ray imaging geometry calibration use some well-defined calibration phantom containing point markers. The calibration aims at minimizing the so-called re-projection error, i.e., the error between the detected marker locations in the acquired projection image and the projected marker locations based on the phantom model and the current estimate of the imaging geometry. The phantoms that are being employed consist usually of spherical markers arranged in some spatial pattern. One widely used phantom type consists of spherical markers in a helical arrangement. We present a framework that establishes a good intuitive understanding of the calibration problem, and allows to evaluate the performance of different phantom designs. It is based on a linear approximation of the error propagation between parameters of the imaging geometry, a projection alignment error (which is not identical to the re-projection error), and the backprojection misalignment, which ultimately dictates 3D image quality. This methodology enables us to characterize the statistics of the parameters describing the imaging geometry, based on simple assumptions on measurement noise, i.e., phantom and pre-processing accuracy. We also characterize the 3D misalignment in the backprojection (which is used in the 3D reconstruction), which directly impacts 3D image quality. In a comparison of different phantom designs using backprojection misalignment as a metric a candy cane" phantom was found to give superior performance. The presented approach gives many useful intuitive insights into the calibration problem and its key properties. It can also be leveraged, e.g., for an easy implementation of a fast and robust calibration algorithm. Keywords: ALG, DX, METR, PHT, SIM. INTRODUCTION Geometry calibration using appropriate phantoms has been widely used in order to enable 3D imaging when the geometry of the imaging system is not well-characterized in advance [],[3-6]. Calibration is especially relevant when systems that were originally designed for flexibility in interventional 2D imaging, like C-arm systems, are now being utilized for 3D imaging. Imaging geometry calibration consists generally of a strategy to minimize the re-projection error. That is, for a welldefined phantom, finding the imaging geometry such that the predicted marker projections (based on that geometry and a model of the phantom) match best the locations of the markers identified in the image. The term imaging geometry, as used here, encompasses the detector position and orientation, as well as the x-ray source position (relative to the phantom). In practice, the general approach to (off-line) calibrating an imaging system consists of first performing a spin acquisition of a specific calibration phantom containing markers. The phantom remains stationary during the spin. The projection images are then evaluated in a preprocessing step in order to extract marker shadow locations and correspondences from the images. This is followed by the calibration step itself, which establishes the optimal estimate of the imaging geometry for each projection. The error metric that is commonly used in the calibration is the RMS (root mean-square) error between the detected and the predicted marker shadow locations, based on the current estimate of the imaging geometry. Phantom design for geometric calibration has so far been mostly a compromise between ease of pre-processing (avoiding potential overlap of marker shadows in the projection image, and easy solution of the correspondence problem), and expected calibration accuracy. One widely used phantom design consists of spherical markers in a helical arrangement [4],[5]. The purpose of this study is the establishment of a framework for a systematic evaluation of predicted phantom performance, specifically its impact on 3D image quality. Our approach uses a linear model of error claus@research.ge.com; phone ; fax ; General Electric Global Research Center, One Research Circle, Niskayuna, NY 239

2 propagation to link a) a projection alignment error, b) the statistics of the parameters describing the imaging geometry, and c) the 3D backprojection misalignment. Several phantom design options are evaluated with this approach. 2. THE PROJECTION ALIGNMENT ERROR For a given suitable calibration phantom consisting of a set of markers in some spatial arrangement, a (small) deviation of the estimated imaging geometry from the true geometry will lead to an error in the projected marker shadow locations. (We use marker shadow to refer to the projection of the marker). This error when collected in a single error vector- can be written as a linear combination of several linearly independent basis vectors, where each of the basis vectors is associated with a specific parameter of the imaging geometry. Specifically, let us assume that the considered calibration phantom consists of eight markers, arranged at the corners of a slightly deformed cube (as illustrated in Figure ). One can see that in this example for the true imaging geometrypairs of markers are projected onto the same location on the detector. I.e., in that configuration, the marker shadows will be found at four different locations on the detector. However, if the hypothesized imaging geometry differs from the true geometry, then the marker shadows will be at different locations on the detector, and the deviation between the two sets of marker shadows will have different characteristics, depending on what parameter of the imaging geometry causes that deviation. Some errors in different specific imaging geometry parameters as well as the associated error components in the marker shadow locations are summarized in Figure where the circles denote the nominal marker shadows, and the solid dots denote the true marker shadows for the misaligned geometry /geometry with errors. The first error type that is shown is associated with a translational error of the detector position in a plane parallel to the detector itself, i.e., in the u -direction. There is a similar error (not shown) for a translation of the detector in the v -direction. The second error type that is illustrated is a scaling of the projection image which is associated with a translation of the detector in a direction orthogonal to the detector plane, in the w -direction. The third error type is associated with a translation of the source position parallel to the detector plane, in the u -direction. Moving the source in this manner will lead to two different effects. First, this translation of the source will cause an overall translation in the marker shadows; and second, it will lead to a relative translation of the marker shadows associated with the top layer of the phantom vs. the marker shadows associated with the bottom layer of the phantom. Only the second component of the error, which is specific to this translation of the source, is illustrated here. There is a corresponding error component (not shown) for a translation of the source position in the v -direction. Source Position Nominal marker shadow locations Marker shadows (solid) w/ translational error (along u-axis) in detector position Marker shadows (solid) w/ scaling error (along w-axis) in detector position Model of Calibration Phantom Marker s hadows (s olid) w/ error (along u-axis) in source position (only specific component) Marker shadows (solid) w/ error (along w-axis) in source position (only specific component) Dete ctor w Marker shadows (solid) w/ rotational error in detector orientation v Marker shadows (solid) w/ error in detector tilt u (only specific component) Figure : Error components associated with errors in specific parameters of the imaging geometry.

3 The fourth error type is associated with a translation of the source orthogonal to the detector plane, in the w - direction. This leads to an overall scaling of the projection image (not shown) as well as a change in the relative scaling between the images of the top layer of the phantom, and the bottom layer of the phantom. This second component of the error is specific to the vertical translation of the source, and is shown here. The fifth error type is associated with a rotation of the detector in the detector ( u / v ) plane. The last type of error is associated with a tilt of the detector. This leads in effect to a spatially varying magnification factor (due to the spatially varying detector to object distance). As is illustrated in Figure, the magnification factor at the top of the detector is smaller than the magnification factor at the bottom of the detector. Again, this error component has a counterpart for a detector tilt in the other direction (not shown). All degrees of freedom of the imaging system geometry have therefore a different associated specific error component. There is an error component associated with each of the parameters defining the 3D source position (translation along all three axes), the 3D detector position (translation along all three axes) and the detector orientation (in-plane rotation, as well as tilt in two directions). Since all of these error types are due to a deviation in the imaging geometry, we refer to this type of error as projection alignment error. One can show that these different specific error components, as illustrated in Figure, are in fact independent in the sense that the associated error vectors are linearly independent. For example, when formally writing down the specific errors discussed above in row vector format, where for each marker we have two vector elements, one for the u - and one for the v -direction, one obtains the matrix that is shown in Figure 2 an easy check shows that these projection alignment error vectors are indeed linearly independent. In this example the vectors corresponding to the specific errors are even orthogonal! Note that for more general phantoms, and/or other imaging geometries relative to the imaged phantom, these error vectors will generally be at least linearly independent (unless the phantom is singular ). Figure 2: Matrix containing the row vectors associated with the alignment errors that correspond to specific parameters of the imaging geometry, as illustrated in Figure. Only the error component specific to a variation in the considered parameter is shown. It can be easily seen that these vectors are linearly independent. For a given phantom and imaging geometry, the set of error vectors can be obtained by computing the partial derivatives of the ( u, v ) position of the marker shadows (again collected in a single vector) on the detector with respect to the imaging geometry parameters, e.g., by using finite differences. That is, the alignment error vectors are given by the columns of the Jacobian matrix. With the Jacobian matrix A, the relationship between variations in the imaging geometry and the associated variations in the marker shadow locations can be described by the equation r x A = () Alignment errors, expressed as 6-element row vectors (2D error each, 8 markers): Detector translation (2 vectors, u and v) Scaling / detector position error along w Horizontal source position error (2 vectors, u/v) Vertical source position error (along w) Detector rotation Detector tilt (2 vectors, tilt in 2 directions) Error component (u/v) due to marker, marker2,, w ithin alignment error vector Alignment errors, expressed as 6-element row vectors (2D error each, 8 markers): Detector translation (2 vectors, u and v) Scaling / detector position error along w Horizontal source position error (2 vectors, u/v) Vertical source position error (along w) Detector rotation Detector tilt (2 vectors, tilt in 2 directions) Error component (u/v) due to marker, marker2,, w ithin alignment error vector

4 which holds for small deviations from the operating point (i.e., a given phantom and imaging geometry), where x is a 9-dimensional vector denoting deviations in the imaging geometry, and r is a 2N -dimensional vector of deviations in the coordinates for the marker shadows corresponding to the set of N markers. Using the singular value decomposition (SVD) one can rewrite the matrix A in the form T A = U Λ V, (2) where U is a (2Nx9) orthogonal matrix, V is a (9x9) orthogonal matrix, and Λ is a (9x9) diagonal matrix. As was shown above, the matrix A is generally of full rank, i.e., all diagonal elements of Λ are non-zero. Only if singular sets of 3D marker locations are considered (relative to the imaging geometry), the matrix A can be singular. For example, if all markers are located in a plane that is parallel to the detector, then both a vertical translation of the detector and a vertical translation of the source will result in a change of the magnification factor however, using only marker shadow position information, that change in magnification cannot be uniquely attributed to a variation in either of these parameters. This non-uniqueness translates into the matrix A being singular. 3. IMAGING GEOMETRY PARAMETER STATISTICS Denoting the deviation in marker shadow coordinates (collected in a vector) by r, it follows from (), (2) that we have the relationship x = V Λ U T r, (3) which holds for small deviations r from the operating point. I.e., for a small error (or deviation) r in the projected marker coordinates, equation (3) tells us exactly what the corresponding deviation x in the imaging geometry is. This framework can be leveraged to directly implement an algorithm for computing the optimal estimate of the imaging geometry (which minimizes the RMS re-projection error). Starting out with a model of the known phantom and an initial estimate of the imaging geometry, one compares the predicted marker shadows from the current geometry estimate to the true (observed) marker shadows. One then finds the optimal estimate of the imaging geometry by, for example, iteratively applying Newton steps, i.e., computing the local derivative / Jacobian matrix (which represents the set of basis vectors, based on the current estimate), approximating the observed re-projection error r by a linear combination of these basis vectors, and updating the imaging geometry correspondingly. For a noise analysis in the calibration, one can see that there are two error (or noise) sources in the geometry calibration that enter into a measurement error r : First, the error in accurately identifying the locations of the marker shadows in the collected projection images (due to pixelation, image noise, etc.), and second, inaccuracies in the calibration phantom itself (e.g., due to manufacturing tolerances), which cause it to differ from the (perfect) phantom model that is used internally in the imaging geometry estimation. From (3), one can see that only the component of r that lies in span (U ), and which is given by UU T r, has any impact on the estimated geometry. We call this component UU T r the projection alignment error. It represents the deviation between the ideal marker shadow locations (i.e., without noise) for the true imaging geometry and the predicted marker shadows based on the best estimate of the imaging geometry. T The remaining component of r (which is in the nullspace of U ) corresponds to a noise component that cannot be explained by a deviation in the imaging geometry, and has therefore no impact on the calibration result. This remaining error component is exactly the re-projection error that is minimized in the calibration process. To analyze the impact of the noise on the calibration result, let us now assume that the components of the noise vector r are zero-mean iid Gaussian with StdDev γ. For a system with M degrees of freedom (typically we have M=9, unless we have some independent measurement of a subset of the geometry parameters; we further assume that there is no distortion, and that the pixel pitch and pixel grid is well-defined) we have the sum of variances of all elements in the vector UU T r, namely T T T T T 2 ( r UU UU r) = E( r UU r) = M γ E (where the equality holds since the columns of U are orthonormal). Assuming all vector components have the same variance, we have thus an approximate standard deviation of γ M 2N for each component of the projection

5 alignment error vector UU T r. Note that this derivation also shows that the StdDev of the imaging geometry parameters is approximately proportional to N. Source Position Calibration Phantom (at Isocenter) Detector Spin Trajectory Parameters describing deviations in imaging geometry: () Source Position: Along tangential direction of spin (src_pos_tangent) Along direction orthogonal to plane of rotation (src_pos_orthogonal) Along radial direction (src_pos_radial) (2) Detector orientation: Tilt (det_tilt_in-plane, det_tilt_orthogonal) Rotation (det_rot) (3) Detector position: Principal point, i.e., coordinates of projection of source position onto detector plane (u, v) Source-to-detector distance (SDD) Considered Approx. Imaging System Geometry Source-to-detector distance (SDD) =.cm Source-to-Isocenter distance (SID) = 66.cm Detector size = 3.x3. cm Figure 3: Illustration of the assumed imaging geometry and the associated parameters. The different parameters describing the imaging geometry as determined through the calibration process- are obviously not uncorrelated. Each parameter is related to a component of the projection alignment error vector r that is independent from the error components associated with all other geometry parameters. However, the geometry parameters are correlated through non-independent components of the error. The variances and correlations of the different geometry parameters can be easily analyzed with the following argument. Using the same noise model as above, the covariance matrix of the vector of geometry parameter deviations x has the following form T T T T 2 T 2 ( x x ) = E( V Λ U r r U Λ V ) = V Λ V γ E. (4) This expression explicitly yields both the variances of the individual geometry parameters, as well as the covariances and correlations between different parameters. Analysis of the variances of the geometry parameters through (4) shows that some parameters have a fairly significant variance. As it turns out, however, these large variances are due to highly correlated geometry parameters, the effects of which cancel each other out. For example, for a cylindrical phantom of 6 cm length, and 6 cm diameter, containing 25 markers in a helical configuration, Table shows the correlations between the different geometry parameters. The assumed imaging geometry and its parametrization are illustrated in Figure 3. One can see from that table that in particular the detector tilt and the position of the principal point ( u, v ) are highly correlated, and also the radial distance of the source position from the center of rotation and the SDD. Indeed, these strong correlations are intuitively obvious tilting the detector, while leaving the detector center at the same position causes only a small change in the observed image, while causing the principal point ( u, v ) to move substantially. Similarly, moving both the focal spot and the detector (radially) away from the phantom (thus increasing the SDD), while maintaining the average magnification factor will only have a small impact on the observed image. Consequently, small noise components may have a significant impact on these parameters. The correlation coefficients in Table confirm this insight. Note that, while the numbers shown in Table were derived for a specific phantom and imaging geometry, the correlations for different phantoms and imaging geometries will generally deviate only very little from the values observed here.

6 Correlation Src_pos Src_pos Src_pos Det_tilt Det_tilt Det rot U V SID Coefficient tangent orthog radial in-plane orthog Src_pos tangential Src_pos orthogonal Src_pos radial Det_tilt in-plane Det_tilt orthogonal Det rotation U V..2 SID. Table : Correlation coefficients between different parameters of the imaging geometry, using a helical phantom containing 25 markers. 4. PHANTOM DESIGN CRITERIA Deriving calibration accuracy criteria. The discussion of the correlations of the estimated imaging geometry parameters shows that a direct naive evaluation of the deviations in these parameters will yield only a poor indicator of goodness of calibration, since obviously large deviations in these parameters do not necessarily imply a large misalignment error. Note that this observation does not depend on the specific parametrization chosen: any other intuitive parametrization of the geometry would produce similar strong correlations. However, realizing that the correlation between estimated geometry parameters is actually driven by the calibration criterion (which minimizes the re-projection error, the RMS error between predicted and observed marker shadows), one may be tempted to conclude that the projection alignment error may represent a good aggregate measure of calibration accuracy. In fact, UU T r is the difference between the marker shadow locations (for the ideal phantom) associated with the true imaging geometry, and the corresponding locations for the estimated imaging geometry. Therefore, UU T r, with each component adjusted by the appropriate magnification factor, even characterizes the misalignment in the backprojection: Since the true and the calibrated marker shadow are a distance δ apart, the backprojection of a marker using the estimated system geometry is approximately δ / κ from the true 3D location, where κ is the magnification factor associated with the marker position (in 3D). Therefore, the projection alignment error (combined with the appropriate magnification factor) actually measures the 3D misalignment in the backprojection, thereby serving as an indicator for the expected 3D image quality that can be achieved in a reconstructed volume using this 3D calibration method. One significant caveat, however, to using UU T r (with a magnification factor correction) directly as a measure for calibration accuracy is that this metric measures only the 3D backprojection misalignment at the 3D location of the markers, and is approximately independent of the phantom size, or the specific 3D distribution of markers (as we have seen above). Only the magnification factor κ may vary depending on the specific marker location for reasonable phantom sizes, κ varies only very little, however. Therefore, the estimation for the 3D backprojection misalignment is a good estimate only at the true location of the markers whereas (as we will see below) the 3D backprojection misalignment generally increases, for example, with increasing distance to the center of the phantom. Thus, the 3D backprojection misalignment (or UU T r ) should not be trusted as an indicator for calibration accuracy. However, using instead the 3D backprojection misalignment for a suitable selection of points distributed in a suitable volume of interest (VOI) allows for a meaningful evaluation and comparison of phantom designs. Such a criterion will be derived in more detail further below. Other phantom design criteria. In addition to enabling a highly accurate calibration, the phantom design should be such that the markers are easily identifiable in the images, easy to segment with (sub-pixel) accuracy, and possibly symmetric (such that no pose estimation of markers is required). For example, spheres, cylindrical rods, plates w/

7 regular shape, or polyhedra (straight edges) may be good candidates as markers for use in a calibration phantom. The spatial arrangement of the markers should enable easy solution of the correspondence problem (i.e., which marker in the image corresponds to which marker in the phantom?), and provide minimal occlusion (full or partial superimposition of marker shadows, in particular when the phantom is seen from many different view angles). Generally, many combinations of markers and spatial arrangements can be imagined. However, one needs to consider that the solution of the correspondence problem is generally more complex when occlusion is present. On the other hand, attempting to avoid occlusion completely may represent a significant restriction of the phantom design choices. Also, different marker shapes/sizes as well as spatially regular arrangements may be required to enable an efficient solution to the correspondence problem. Choosing marker types. The naive form of the calibration accuracy metric (which relies only on 3D marker locations) can already be used to compare the expected performance of phantoms containing different types of markers (with a similar spatial distribution). Specifically, the StdDev for the 3D backprojection misalignment at the marker locations (i.e., UU T r, with an appropriate magnification factor correction) can be estimated as γ M 2 2 M = ( κ αm) + αs (5) κ 2N κ 2N where κ represents the approximate magnification factor of the considered imaging system, α m represents the StdDev of the marker placement accuracy in the phantom (in x / y / z ), and α represents the StdDev of the marker shadow s 2 2 ( κ α m ) αs segmentation/detection (in u / v ). Note that the term γ = + describes the StdDev of the components of the noise vector r. The number of degrees of freedom M is not variable generally we have M = 9, in cases where some parameters of the gantry can assumed to be fixed and known, smaller values of M may be possible, realistically probably up to M = 6. The variable N denotes the number of markers (more specifically, 2 N denotes the number of constraints derived from the used markers for spherical BB markers we have generally 2 constraints per marker, one for each coordinate axis. For other markers the error terms may be different, and one may have more error equations per marker). For a phantom containing BB markers, realistic numbers of markers may range from 2-2, the expected detection accuracy will range from.-.2 pixels (for a large marker shadow size of about pixels in diameter or more), and the expected marker placement accuracy within the phantom is about.mm (all values given are considered StdDevs). When using different markers, e.g., long rods, one would expect a comparable marker placement accuracy in the phantom. However, the expected location detection accuracy may be improved due to the longer (and straight) edges, by a factor of approximately 5-x. Also, in this case one may use a different error term, e.g., the edge location of the rod at both edges (in a direction orthogonal to the rod), near both ends of the rod, for a total of 4 error equations per marker. Due to the superimposition and correspondence problem, the number of markers will realistically be limited at about 2-2. For an example of such a phantom, see Figure 4. Considering the naive calibration accuracy metric presented in equation (5), one can now compare phantoms of the same size, and with roughly comparable marker distributions, but which differ in the marker type they use. The main variables impacted by the marker type are the segmentation accuracy α, and the number of constraints s 2 N. Using κ =.5, and α m =. mm, and an expected segmentation accuracy of α s =. 6 mm, one can see that by increasing the segmentation accuracy (i.e., decreasing α s ) one can at best achieve an improvement by about %, for a fixed number of constraints (i.e., 2N). A similar improvement can be achieved by increasing the number of constraints by about 2%. That is, even taking into account that rods may allow for better segmentation accuracy and more constraints per marker than BBs, a phantom containing 5 or more spherical markers will yield better accuracy than a phantom (of a similar size and spatial distribution) containing rods (2 or fewer). This derivation can also be used to show that a reduction of the number of degrees of freedom of the calibration problem (e.g., from 9 to 6, which can be achieved by assuming a perfectly rigid gantry with known intrinsic parameters) has only a comparatively small effect on the calibration accuracy, as measured by the backprojection misalignment. Since the improvement enters only as a square-root of the factor of reduction in the number of degrees of freedom, a reduction of M=9 to M=6 leads only to a 8% improvement in the backprojection misalignment. Note that a similar improvement can be achieved, e.g., by increasing the number of markers in the phantom by 5%.

8 2 3 Figure 4: Illustration of helix type phantom, rod phantom, candy cane type phantom. (Figure will be updated!) 5. DETAILED NOISE IMPACT ANALYSIS We will now derive an expression that allows to compute the alignment error for a set of sample point locations in a volume of interest (VOI). Specifically, by choosing a selection of 3D locations that are representative of the VOI (e.g., by choosing points located in a regular grid within the VOI) one obtains an equation A Vol x = r Vol, (6) which is the equivalent of equation (), only for a different set of point locations in space. This equation (6) puts the deviation in geometry parameters x into relation with the alignment error r Vol for the considered set of points, where A Vol is the Jacobian associated with this set of points (and this imaging geometry). Using now the statistics of the vector x (as estimated though the calibration process), the statistics of the alignment error r Vol can be derived, which is directly related (through the respective magnification factors of the set of considered points) to the 3D backprojection misalignment at these 3D point locations. However, this analysis cannot be performed directly, since the elements of x are highly correlated, as was shown above in equation (4). A workaround to this problem consists of rewriting equation (6) as T A Vol V V x = rvol, (7) using the orthogonal matrix V from the SVD factorization (2) of the Jacobian A associated with the calibration phantom. From (3) we see that T T V x = Λ U r, (8) which is a vector of independent random variables with known StdDev (since r is a vector of iid elements with known StdDev γ, U is an orthogonal matrix, and Λ is diagonal, therefore Λ U T r is a vector of independent random variables, with component i having a StdDev of γ / λi, where λ i is the i th diagonal element of Λ ). By inserting Λ U T r for V T x into equation (7) the StdDev of each element of r Vol can be immediately determined, thereby defining the statistics of both the projection alignment error and (with the help of the associated magnification factors) the 3D backprojection misalignment at each point of the considered volume of interest (VOI). Interestingly, the results derived in this work parallel results from rigid-body point based registration (see [2], as well as the references given therein). This does not come as a surprise, since the calibration problem discussed here is essentially formulated as a registration problem. In [2], it is shown that the registration error at the marker locations is approximately independent of the marker (or fiducial) configuration, and is therefore only a poor measure of quality of the registration. Also, it was shown that the error is approximately proportional to N, where N is the number of fiducial markers. Both results are mirrored in the derivation in the present paper.

9 Since we are considering a circular spin acquisition, where the 3D volume of interest that can be imaged and reconstructed is essentially defined as a cylindrical region, it is natural to define a grid of sample points within that volume on a cylindrical coordinate system. Specifically, we consider points that are located on cylindrical surfaces with a radius of., 2.5, 5., 7.5,. cm respectively, with a cylinder length of 2cm (as illustrated in Figure 5), thereby sampling a cylindrical volume 2.cm long, with a diameter of 2.cm. When considering again a helix phantom of 6cm length and 6cm diameter, containing 25 BBs, and a StdDev of.3mm in the positional accuracy of the observed marker shadow positions (which includes segmentation noise and phantom manufacturing accuracy), one obtains now the StdDev for the 3D backprojection misalignment at the considered sample points that is plotted in Figure 5: From left to right, the StdDev for points at increasing radii is shown, and within each radius the points are arranged from one end of the cylindrical volume to the opposite end. It is obvious that the 3D misalignment is smallest at the center of the volume (i.e., near the center of the phantom), and it increases towards the ends of the cylindrical volume, and with increasing distance from the central axis of the volume. For this phantom, and with the assumed StdDev of.3mm in the marker shadow position measurements, the StdDev of the backprojection misalignment is less than.mm at the center of the reconstructed volume, increasing to slightly more than.3mm at the points farthest from the center of the volume. Figure 5: Illustration of sample points in VOI on cylindrical coordinate system (left). Plot of backprojection misalignment (right) as a function of sample point index (ordered with increasing radial distance from center of VOI, within each radius from one end of VOI to other end of VOI). StdDev in backprojection misalignment is in.mm units. 6. PHANTOM DESIGN COMPARISON Helix Phantom Design Options. In a comparison of different helix configurations with the approach outlined above, one can see that the calibration accuracy (as measured with the backprojection misalignment for sample points in a VOI) improves with increasing number of markers, and increasing diameter and length of the helix (as long as all marker shadows fall onto the detector). A better 3D distribution of the markers (i.e., by using a 3 turn helix vs. a 2 turn helix) has only a small impact on the calibration accuracy. Note that in this first stage, the analysis did not take superimposition of BB shadows in the projection image into account. However, when increasing the diameter of the helix in order to achieve improved accuracy (while maintaining at least 2 turns of the helix for good 3D distribution of the marker points), superimposition of markers becomes a concern. This is particularly true when also taking into account that the phantom may not always be perfectly positioned when acquiring a calibration run. This observation implies that, if one wants to maintain easy pre-processing without marker superimpositions, one may have to settle for less than optimal calibration accuracy. If, on the other hand, more complex pre-processing may be considered (with the ability to handle superimposition of markers), then the basic helical structure of the phantom may be modified, since some of the inherent advantages (in terms of ease of pre-processing) of the helix phantom configuration have now disappeared. Comparing phantoms containing spherical markers. Even when superimposition of marker shadows is present, the helix represents an attractive design, since it exhibits a good 3D distribution of markers, as well as an (hopefully) easy

10 solution of the correspondence problem (although the pre-processing will become more complex). However, based on the previous discussions, as well as on the previously considered rod phantom, there is another class of phantoms that could potentially be very attractive: the candy cane phantom design (see Figure 4) which includes some of the advantages of the rod phantom in a phantom consisting of spherical markers. In particular, the rotationally shifted copies of strings that are oriented at an angle relative to the axis of rotation minimize the region of intersection between strings (in the projection images), and they provide for an easy correspondence problem. This design is potentially superior to the helix design due to the better spatial distribution of markers (at every point along the axis of the phantom we have a balanced distribution of markers), and the occlusions are not clustered (whereas for the helix design the occlusions are preferentially located at the projections of the edges of phantom, and at edges of detector, and they may involve clusters of significant numbers of markers) We evaluate these two phantom design families in terms of their 3D backprojection misalignment for a VOI. The parameters of the considered specific phantom designs are summarized in Table 2. Note that, due to the characteristic locations of the occlusions for a helix type phantom, we also consider helix type phantoms that only partially project onto the detector. That way, the clusters or markers that are superimposed do not project onto the detector, while the benefit of having a bigger phantom may offset the reduced number of markers that are now usable. For the candy cane type phantom, we only evaluate configurations that always fully project onto the detector. Helix Phantom Parameters Candy Cane Phantom Parameters Number of BBs 24, 36, 48, 6, 72, 84, 96, 8 Number of Strings 4, 5, 6, 7, 8, 9 Diameter of Helix., 5., 2., 25., 3., Number of BBs per string 5, 2, , 4. cm Number of turns 2, 3,4 Diameter of Phantom 2., 5., 8. cm Turning angle for each string 3, 6, 9 degrees Table 2: Parameters of helix and candy cane phantom designs that are being evaluated. As accuracy metric we use, as defined before, the StdDev of 3D backprojection misalignment (generated by a.3mm StdDev marker shadow localization error), but now as the average StdDev over only the sample points located in the volume of interest at a radius of -5 cm from the center of rotation (thereby taking into account the region within the VOI at a moderate distance to the center of the VOI, but disregarding the center of the VOI, where the accuracy is highest and not as sensitive, and the periphery of the ROI, where the accuracy may be less critical). In particular, in order to achieve a comprehensive evaluation of the phantom performance, the worst-case misalignment was chosen for a number of projections from view angles ranging across 8 degrees, with a.33 degree separation; with a tilt of the phantom (corresponding to positioning inaccuracy) of.,.5, 3., 4.5, and 6. degrees; where all markers that are projected close to other marker shadows or close to the detector edge (< 9mm) are discarded, since they would not provide a high-accuracy marker shadow location measurement (due to full or partial superimposition of marker shadows). 7. RESULTS Occlusion/Superimposition of markers When analyzing the (worst-case) superimposition problem for markers, by counting only the clean marker shadows, i.e., the ones which are projected onto the detector, not close to to the detector edge, and not close to (or overlapping) any other marker shadow, one can see that the superimposition of markers is a significantly smaller problem for candy cane type phantoms than for helix type phantoms. This is illustrated in Figure 6, where the number of clean BBs is plotted against the total number of BBs in the phantom, across all considered configurations. Since the number of BBs has a fairly strong impact on calibration accuracy, this observation may imply that (for the same number of BBs in the phantom) a candy cane type phantom may offer superior performance as compared to a helix type phantom.

11 # of clean BBs Helix - Nominal # of BBs vs. # of clean BBs # of BBs (nominal) Helix # of clean BBs 5 5 CandyCane - Nominal # of BBs vs # of clean BBs # of BBs (nominal) CandyCane Figure 6: Number of clean markers vs. total number of markers in phantom, for helix (left) and candy cane (right) type phantom designs. 3D misalignment as function of number of markers When analyzing the worst case 3D backprojection misalignment (as described above), but now only for a fixed phantom diameter of 5cm, one can see that it is significantly smaller for candy cane type phantoms than for helix type phantoms, as shown in Figure 7. In addition, helix type phantoms seem to be less stable (mostly due to BB superimposition), since helix type phantoms exhibit a larger variability in the backprojection misalignment. StdDev cm Phantoms - # of BBs (nominal) vs. misalignment # of BBs (nominal) CandyCane Helix Figure 7: StdDev (.mm units) of backprojection misalignment as a function of total number of BBs in phantom, for different helix and candy cane configurations. StdDev CandyCane - # of BBs vs. Misalignment 2 3 # of BBs (nominal) 2 cm diameter 5 cm diameter 8 cm diameter Figure 8: StdDev (.mm units) of backprojection misalignment as a function of total number of BBs, for candy cane phantom configurations of different diameter. In an additional comparison of helix type configurations (not shown) it was observed that increasing the phantom diameter improves the accuracy only little, and only as long as the phantom is fully projected onto the detector. For even larger phantom diameters, the accuracy declined. With these sets of results it has been shown that for a similar phantom cost (as measured by the phantom diameter and the number of markers), a candy cane type phantom will offer superior performance than a helix type phantom. Further analysis of the candy cane type phantoms shows that this type of phantom is well-behaved in the sense that the measured performance varies according to expectations. In particular we can see in Figure 8 (which contains data points for all considered candy cane configurations in a plot of worst case 3D backprojection misalignment against nominal number of markers) that bigger phantom diameter and larger number of BBs lead to a better performance. Independent of the specific configuration, the phantoms exhibit relatively small variability in their performance (for a given diameter and number of markers). This observation is confirmed in the results shown in Figure 9. These results show the impact of the number of strings and the turn-angle on the 3D misalignment, by comparing different configurations of a 5cm diameter candy cane

12 phantom. It can be seen that the phantom accuracy is essentially a function only of the number of BBs. The number of strings has no significant impact on the backprojection misalignment, and the turn angle of the strings also has no significant impact on the accuracy. 5 cm CandyCane - # of BBs (nominal) vs. misalignment 5cm CandyCane - # of BBs (nominal) vs. misalignment StdDev # of BBs (nominal) 4 strings 5 strings 6 strings 7 strings 8 strings 9 strings StdDev degree turn angle 6 degree turn angle 9 degree turn angle # of BBs (nominal) Figure 9: StdDev (.mm units) of backprojection misalignment as a function of total number of BBs, for candy cane phantom configurations with different numbers of strings (left) or with different string turn angles (right). 8. CONCLUSIONS/SUMMARY In this work, an intuitive decomposition of the alignment error is given. It is directly linked to the SVD of the Jacobian matrix of the projection operator (for the considered set of points and the considered imaging geometry). This formalism is used to quantify the statistics of the parameters describing the imaging geometry as well as phantom performance for different phantom configurations. The used performance criterion for calibration accuracy is the geometrical misalignment in the backprojection within a volume of interest (VOI). Measuring deviations in imaging geometry parameters is not well-suited for this purpose, since these parameters are highly correlated. Furthermore, the backprojection misalignment at the phantom marker locations is also a poor measure of calibration accuracy, since it was shown to be approximately independent of the marker configuration. These observations mirror similar results from point-based registration [2]. Comparative analysis of phantom design options shows that a phantom containing rods (instead of spherical markers / BBs), while improving marker shadow segmentation accuracy, does not result in an improved calibration accuracy since the number of markers in such a phantom will be relatively small. For phantoms containing BBs, a comparison between helix and candy cane designs shows superior results for the candy cane design. The candy cane phantoms exhibit the expected improvement in accuracy for phantoms with a larger diameter, as long as it is fully projected onto the detector, and for phantoms containing more markers, while being insensitive to other design choices. REFERENCES. Y. Cho, D.J. Moseley, J.H. Siewerdsen, D.A. Jaffray, Accurate technique for complete geometric calibration of cone-beam computed tomography systems, Med. Phys. 32, pp , J.M. Fitzpatrick, J.B. West, C.R. Maurer, Jr., Predicting Error in Rigid-Body Point-Based Registration, IEEE Trans. Med. Imaging, Vol. 7, No. 5, pp , N. Navab, A. Bani-Hashemi, M. Mitschke, A. J. Fox, D.W. Holdsworth, R. Fahrig, and R. Graumann, Dynamic geometrical calibration for 3-D cerebral angiography, SPIE, Vol. 278, pp , J. Pouliot, A. Bani-Hashemi, J. Chen, et al., Low-dose Mega-Voltage Cone-Beam CT for Radiation Therapy, Int. J. Radiation Oncology Biol. Phys., Vol. 6, No. 2, pp , A. Rougée, C. Picard, Y. Trousset, and C. Ponchut, Geometrical calibration for 3D X-ray imaging, SPIE, Vol. 897, Image Capture, Formatting, and Display, pp. 6 69, K. Wiesent, K. Barth, N. Navab, P. Durlak, T. Brunner, O. Schuetz, and W. Seissler, Enhanced 3D-reconstruction algorithm for C-arm systems suitable for interventional procedures, IEEE Trans. Med. Imaging, Vol. 9, No. 5, pp.39 43, 2.