Global Registration of Multiple Bone Fragments Using Statistical Atlas Models: Feasibility Experiments
Global Registration of Multiple Bone Fragments using Statistical Atlas Models: Feasibility Experiments
Mehdi Hedjazi Moghari and Purang Abolmaesumi
Abstract— We propose a new technique to automatically register multiple bone fragments of a fracture using a global registration method guided by a statistical anatomical atlas model. The atlas is created by using the principal compo nent analysis of threedimensional meshes generated from the Computed Tomography (CT) data of five left femur bone cadavers. The fracture fragments are initially aligned to the generated atlas by using a local point descriptor which can robustly identify a set of potential corresponding points among fragments and the mean atlas model. A global registration algorithm is then utilized to finetune the alignment among the fractures and the mean atlas model. Performed experimental simulations on the human femur bone cadavers verify the feasibility of the proposed registration algorithm.
I. INTRODUCTION
Global registration of multiple 3D data sets is an essential and complex problem in medical imaging and computer vision, where three or more data sets must be aligned in a common reference frame. One of the applications of such registration is in computerassisted orthopedic trauma surgery, where surgeons are interested in reassembling the bone fractures back into an original solid bone model (Figure I). Generally the registration problem is divided into two sub problems of finding corresponding points among data sets, and deriving the transformation parameters that minimize the distance among the estimated correspondences. Most prior art [1], [2], [3], [4], [5], [6], [7] concentrate on the registration task and assume that the corresponding points are either available or the data sets are close enough to each other such that the closest points among the data sets are reliable candidates for the corresponding points. However, in real applications, the corresponding points among data sets might not be known and manual preliminarily alignment of the data sets may not be an easy task. The remaining prior work, which assumes corresponding points are unknown, considers a symmetrical structure shape for the data sets to ease the automatic preliminarily alignment (see e.g., [8], [9]), which is not realistic in the majority of clinical applications.
Another point to consider in the registration of multiple data sets, is to distinct the pairwise registration (local reg istration) algorithms from the global registration algorithms. The pairwise registration algorithms only register two data
Mehdi Hedjazi Moghari is with the Department of Electrical & Computer Engineering, Queen’s University, Kingston, Ontario, Canada. Purang Abolmaesumi is with the School of Computing, Queen’s Univer sity, Kingston, Ontario, Canada. Email: purang@cs.queensu.ca.
sets at a time; however, the global registration technique simultaneously registers multiple data sets. The latter means that, as shown in Figure I, all the overlapping areas (among all the data sets) can be simultaneously considered in the global registration algorithm to estimate the transformation parameters. One may argue that pairwise registration algo rithms may also be utilized to solve the global registration problem for multiple data sets, by sequentially registering each two pairs of data sets. However, this approach will lead to an uneven spread of registration error among the data sets [10]. In this article, we propose a new algorithm to
Fig. 1. Registration of two femur bone fractures to the a femur bone mean shape (template). T1, T2 and T12 are the rigid transformations mapping Fracture 1 to the mean shape, Fracture 2 to the mean shape and Fractures 1 and 2 to each other, respectively.
simultaneously and globally register a set of multiple bone fractures (femur bone fractures) back to their correct healing positions by using the statistical shape model of fractured bone as an intermediate. We employ principal components analysis (PCA) to generate a mean shape of five human left femur bones. Then a local point descriptor is used to automatically align the bone fractures to the generated mean shape. Finally, the global registration algorithm, pre viously introduced in [7], is employed to globally register the fractures to the mean shape and therefore to each other. We also use the closedform solution presented in [11] to estimate the accuracy of performed registration. Simulation and experimental results are then presented to examine the performance of the proposed algorithm.
30th Annual International IEEE EMBS Conference Vancouver, British Columbia, Canada, August 2024, 2008
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II. STATISTICAL ANATOMICAL ATLAS GENERATION
In this work, we use the principal components analysis (PCA) to generate the anatomical atlas. Threedimensional bone models are generated from Computed Tomography (CT) images of a specific bony anatomy over a population. One of the models is randomly selected as a template, and all other models are registered to the template using similarity transformations to bring them in the same coordinate space as the template. Let us represent the template with y0 as a 1 ×3Np vector, where Np is the number of the points in the model. Also, let yi, i = 1, ..., Nm −1, represent the points in the other models which are transformed to the template coordinate frame (Nm is the total number of models). To find the corresponding points among the template and all other models, we first rigidly align all bones models and the template. Then, for each point in the template, we send a ray from that point along its normal vector. The intersections of the ray with the bone models are assumed as the corresponding points to the chosen point in the template. Now, let us define the mean shape as follows:
m = 1
Nm
Nm−1 X
i=0 yi (1)
Using the mean shape, the coordinates of the points in all models can be generated as follows:
yi = m +
Nm−1 X
k=1 αkuk, αk = uT k (yi −m), (2)
where uk is the k’th eigenvector of deformation matrix D defined as follows:
D = [y0 −m, y1 −m, ..., yNm −m]. (3)
If singular values of matrix D are represented by λi, then a new instance of the atlas can be generated from:
yc = m +
Nm−1 X
i=1 ciui, (4)
where ci is randomly chosen from a uniform distribution in the range of [− q
6λi
Nm , q
6λi
Nm ].
III. LOCAL POINT DESCRIPTOR FOR AUTOMATIC INITIAL ALIGNMENT
A number of local point descriptors which are invariant to the linear transformations, have been presented in the literature to find the potential correspondences among point data sets. Nearly all the introduced local point descriptors are based on discrete points. Here, we propose a new local point descriptor which is instead based on continuous functions, fitted to a group of neighboring points. In what follows, we explain the proposed descriptor used to find the potential matches between two data sets, y0 and y1: 1) The algorithm first randomly selects a point p0 from y0. It then finds points which are within a certain distance
d1 from p0. Let P = {p0, p1, ..., pn} be the set of chosen points from y0. 2) The translation invariance is achieved by translating the centroid of P, c, to the origin. 3) The rotation invariance is accomplished by rotating all the points in P with a rotation matrix R, which aligns the point cloud along its eigenvectors. 4) The reflection invariance is obtained by multiply ing the coordinates of points P with matrix F = diag(sign(fx), sign(fy), sign(fz)), where fx, fy and fz are the signed mean square of P along x, y, and z axes, respectively. 5) Finally, the scale invariance is provided by the scaling factor s = q
σ2 x + σ2 y + σ2 z, where σ2 x, σ2 y and σ2 z are the variances of P along x, y and z, respectively. The above procedure brings the points in P to the canon ical coordinate frame as follows:
˜ P = 1
s × F × R × (P −[
n .times z
}|
{ c, c, ..., c]) (5)
Then, a continuous surface function is fitted to the points in ˜ P such that the mean square error of the fitness quality is minimized. The order of the function depends on the complexity of the surface. In our study, we used a second order function as:
f(x, y, z) = a1x2 + a2y2 + a3z2 + a4xy (6) + a5xz + a6yz + a7x + a8y + a9z + 1,
where ap0 = [a1, ..., a9]T can be easily computed using a closedform solution minimizing the mean squares of the fitness error. Equation (6) should be satisfied for each point pi = [xpi, ypi, zpi]T in ˜ P:
f(xpi, ypi, zpi) = −1, pi ∈˜ P (7)
Equation (7) can then be written in a matrix format as: M× ap0 = b, where M is a n × 9 matrix in terms of pi and b = [−1, ..., −1]T 1×9. Therefore, ap0 can be calculated as:
ap0 = inv(MT × M) × MT × b (8)
We consider ap0 as an attribute vector for point p0. To find the potential matching points of p0 in y1, the attribute vectors for all the points in y1 are calculated and compared with ap0. A simple similarity measure such as normalized crosscorrelation can be used to find the potential matches of p0 in y1.
IV. LOCAL AND GLOBAL REGISTRATIONS
Assume there are Nf threedimensional fracture models, u0, ..., uNf , that we would like to register to the anatomical atlas model m. The registration process consists of two steps of local and global registrations. In the local registration, each fracture is registered to the atlas model as follows: 1) Calculate the attribute vector a for all the points in the mean shape m.
1In our performed experimental simulations d is empirically chosen 30mm2.
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2) Choose the largest fracture, containing the highest number of points, from the fracture set. 3) Randomly select a point in the chosen fracture and compute its attribute vector. 4) Compute the potential matches between the fracture and the atlas model. 5) For each potential matching pair, calculate a similarity transformation which locally maps the fracture to the mean shape. 6) Select the similarity transformation leading to the least mean surface registration error between the fracture and the mean shape, and go back to Step 2. To compute the similarity transformation, we use the neigh boring points of the two potential matching points. First, the centroid of the point cloud selected on the fracture is mapped to the centroid of the point cloud selected from the mean shape. Then, the eigenvectors and eigenvalues of the two clouds are used to rotate and scale fracture cloud to the latter one, respectively. After locally register ing each fracture to the atlas model, a global registration algorithm is required to globally and simultaneously fine tune the registration parameters among all the fractures and the template for the final, accurate alignment. We have used the Unscented Kalman Filterbased registration algorithm introduced in [7], to globally register all the fractures to each other and to the atlas model. Finally, the closedform solution presented in [11] was utilized to estimate the accuracy of the performed registration. The estimated accuracies are then used to generate a color map of the registered bone fractures.
V. RESULTS
CT data was captured from the femur bone of five human cadavers. The data had 0.3 mm inplane resolution with slice distance of 0.625 mm. The data was then segmented using simple thresholding techniques and threedimensional surface meshes were generated using a Marching Cubes algorithm. We used similarity transformations to align all the bone meshes in a reference coordinate frame. Equations (1) and (2) were used respectively to generate the mean shape m, and the eigenvalues λi. Figure 2 displays the cumulative eigenvalues λi computed with PCA. In the first simulation, the performance of the proposed point matching algorithm is verified. First, using Equation (4) we generated different femur bone instances by using 0, 20,..., and 100 percent of the atlas variances λi, respectively. A point was randomly chosen from the mean shape m, and its corresponding point in the new generated bone instance was found using the proposed algorithm in Section III. Figure 3 displays the success rate of the proposed algorithm in finding the correct corresponding points between the mean shape and the new generated instances of the femur bone over 840 trials. Furthermore, to examine the sensitivity of the proposed algorithm to noise, we performed a similar simulation. In this case, instead of choosing atlas variances, noise with different variances was added to the mean shape to generate a new instance of the femur bone. Then, the corresponding points between the noisy instances of the bone and the mean shape
were computed. Table I illustrates the success rate of the proposed algorithm in finding the accurate correspondences between the mean shape and noisy femur bone instances with different variances of noise added in each of the 840 trials. In the last experiment, we registered a set of fractures from a femur bone to the mean shape. We artificially divided one of the femur bone meshes into three pieces and applied a random transformation to each of them. Then, we registered the top and bottom pieces back to the mean shape using the proposed algorithm in Section IV. This procedure was repeated with five different ranges of transformation. Target registration error (TRE) was calculated as a measure for accuracy of the performed registrations. The computed TRE in each of the trials is shown in Table II. The result of one of the registration trials is shown in Figure 4.
0 1 2 3 4 5 0
20
40
60
80
100
Number of modes
Cumulative variance
Fig. 2. Cumulative variance explained by five models of the variation of mean shape.
0 0.2 0.4 0.6 0.8 1 50
60
70
80
90
100
Percentage of variances
Rate of correct matches (%)
λi λi λi λi λi
Fig. 3. Sensitivity of the proposed point matching algorithm to the variations in mean shape.
TABLE I SENSITIVITY OF THE PROPOSED POINT MATCHING ALGORITHM TO THE
NOISE ADDED TO THE MEAN SHAPE.
Variance of the noise (mm2)
0
0.5
1
1.5
2
Success rate (%)
100
86
77
68
58
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TABLE II
ACCURACY OF REGISTERING TWO FRACTURES TO THE MEAN SHAPE.
Range of Transformation
±24
±28
±32
±36
±40
Registration Error(mm)
1.6
1.5
1.6
1.5
1.6
Fig. 4. Mean target registration error for the registration of two fractures to the mean shape; lighter colors show more registration error. Overall mean error is 1.5mm.
VI. DISCUSSION
As shown in Figure 2, there are noticeable variations among the mean shape and five femur bone data sets, causing the computed eigenvalues λi to be large and the slope of the cumulative eigenvalues to be small. The large eigenvalues may be due to the low number of input femur bone data sets. By increasing the number of data sets, we should be able to generate a more reliable statistical atlas model of a femur bone. The effect of large variances, i.e. λi, on the performance of the proposed algorithm is shown in Figure 3. The success rate of the proposed algorithm which finds the matching points between the mean shape and a new instance of the femur bone, drops monotonically as the amount of variations between the mean shape and the new instance increases. By increasing the amount of variations, the local invariant geometry features among the mean shape and new instances may not be preserved and therefore, the chance of finding the correct matching points reduces. Table I shows the sensitivity of the proposed algorithm to the variance of the noise. As expected from the previous case, the performance of the proposed point matching algorithm reduces as the variance of the noise perturbing the mean shape increases. Finally, Figure 4 displays the accuracy of performed registration. The mean registration error is 1.5 mm which is below the usual clinical acceptable error of 2 mm.
VII. CONCLUSIONS AND FUTURE WORK
We presented a new registration technique to optimally and simultaneously align multiple bone fractures to a statistical anatomical atlas model. The proposed algorithm uses a statistical anatomical atlas as an intermediary to register bone fractures back to their healing positions. A new local point descriptor was also introduced to automatically align the
fractures with the atlas for initialization of the registration process. The performed simulations verify the accuracy of the proposed approach in registering femur bone fractures to a corresponding bone atlas. The limited number of bone models used in the current study may not be enough to capture all variations of the anatomy within the statistical representation of the femur bone. Therefore, as future work, we would like to increase the number of femur bone data sets to increase the accuracy of the generated atlas. In addition, we will examine the per formance of the proposed algorithm on anatomical instances which are not used in the atlas generation procedure. We would also like to use a groupwise registration algorithm to simultaneously register all the bone data sets in a reference coordinate frame. Groupwise registration decreases the bias of the generated atlas towards a specific femur bone mesh model. Furthermore, the sensitivity of the proposed algorithm to distance d, and the order of function f is another issue to investigate. Finally, it would be of interest to compare the performance of our proposed local point descriptor with that of other descriptors defined in the literature.
VIII. ACKNOWLEDGMENTS
We would like to thank Dr. M. Kunz at the Human Mobility Research Centre, Queen’s University, for helping us with the generation of the CT meshes of human femur bones. We also would like to thank the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Canadian Institutes of Health Research (CIHR) for supporting this project.
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