Landmark-driven Parameter Optimization for non-linear Image Registration Alexander Schmidt-Richberg, Ren´e Werner, Jan Ehrhardt, Jan-Christoph Wolf and Heinz Handels Institute of Medical Informatics, University of L¨ ubeck, L¨ ubeck, Germany ABSTRACT Image registration is one of the most common research areas in medical image processing. It is required for example for image fusion, motion estimation, patient positioning, or generation of medical atlases. In most intensity-based registration approaches, parameters have to be determined, most commonly a parameter indicating to which extend the transformation is required to be smooth. Its optimal value depends on multiple factors like the application and the occurrence of noise in the images, and may therefore vary from case to case. Moreover, multi-scale approaches are commonly applied on registration problems and demand for further adjustment of the parameters. In this paper, we present a landmark-based approach for automatic parameter optimization in non-linear intensity-based image registration. In a first step, corresponding landmarks are automatically detected in the images to match. The landmark-based target registration error (TRE), which is shown to be a valid metric for quantifying registration accuracy, is then used to optimize the parameter choice during the registration process. The approach is evaluated for the registration of lungs based on 22 thoracic 4D CT data sets. Experiments show that the TRE can be reduced on average by 0.07 mm using automatic parameter optimization. Keywords: registration, landmark detection, parameter optimization
1. PURPOSE Registration of anatomical images is a key problem in medical image analysis and prerequisite for numerous applications such as estimation of heart or lung motion,1 image reconstruction,2 image fusion, or the generation of shape and motion atlases.3 Due to the wide range of possible applications, various registration algorithms have been proposed in recent years. Typically, registration methods include a smoothing (regularization) of the transformation. The amount of regularization is determined by a parameter α that has to be optimized with respect to the specific application and image data, which is often very labor-intensive. In practice, optimal values may also differ from case to case. Moreover, multi-scale registration strategies are commonly applied and experiments suggest a variation of the amount of regularization depending on the scale. Parameter optimization is most commonly done by minimizing the target registration error (TRE), i.e. the mean distance of corresponding landmarks after registration. As landmarks serve equally distributed characteristic points like bifurcations in vessel trees. Manually detecting landmarks in the images is very time-consuming. Therefore, approaches for automatic landmark detection have been recently proposed.4 Since inaccuracies of single landmark positions cannot be obviated due to full automation, direct consideration of these landmarks in the registration approach by matching corresponding points on each other5, 6 is quite error-prone. However, the mean distance of automatically detected landmarks TREA is an excellent metric for evaluation registration accuracy. This is illustrated in Fig. 1. To analyze the significance of TREA for evaluating registration results, registration was performed for multiple values of α. The accuracy of the results is then quantified on the one hand with respect to TREM calculated using manually detected landmarks, which serve as gold standard. On the other hand, TREA for landmarks yielded by an automatic detection is examined. As seen in the graph, the metrics Further author information: (Send correspondence to Alexander Schmidt-Richberg) Alexander Schmidt-Richberg: E-mail:
[email protected], Telephone: +49 (0) 451 500 5645
Medical Imaging 2011: Image Processing, edited by Benoit M. Dawant, David R. Haynor, Proc. of SPIE Vol. 7962, 79620T · © 2011 SPIE · CCC code: 1605-7422/11/$18 · doi: 10.1117/12.877059
Proc. of SPIE Vol. 7962 79620T-1
TRE^A TRE^M RMS
130
12 120 10 110 8 100 6 90 4
Root Mean Square
Target Registration Error
14
140
Figure 1. The metrics RMS, TREA and TREM after registration with several values of α. The TREM is computed using manually defined landmarks and therefore considered as gold standard. As seen, TREA and TREM are highly correlated, while the RMS does not provide reliable information about registration accuracy.
80
2
70 0
0.5
1
1.5
2
2.5
3
Regularization weight
feature a very high correlation, indicated by a correlation coefficient of on average 0.993 (tested on 22 thoracic CT data sets, correlation coefficients between 0.943 and 0.998). For comparison, the correlation coefficient between TREM and the root mean square (RMS) of the registered images is only 0.833. Motivated by this observation, we use automatically detected landmarks for a case-specific optimization of the registration procedure, in particular to find an optimal regularization parameter α. The approach is evaluated based on thoracic CT data sets for the estimation of respiratory lung motion using a non-linear intensity-based registration approach. However, the principle idea of this contribution is general and can be used independently of a specific clinical application or registration method.
2. METHODS The aim of image registration is finding a transformation ϕ : Ω → Ω that matches a moving image IM : Ω → R to a fixed image IF : Ω → R, with Ω ⊂ R3 denoting the image domain. In most intensity-based non-linear registration approaches, this is done by minimizing the energy functional JReg [ϕ] := D[IM ◦ ϕ, IF ] + αS[ϕ] .
(1)
Here, D denotes a distance measure quantifying the dissimilarity between fixed image IF and transformed moving image IM ◦ ϕ and S is a regularizing term restricting the transformations to be physically plausible. The parameter α weights this smoothness constraint and usually has to be optimized in repeated runs. The method presented in this paper consists of two steps. First, a set of landmarks is automatically detected in moving and fixed image and correspondences are established. These landmarks are then used in the actual registration approach to determine an optimal value for the parameter α. Depending on the application, moving and fixed image may be from different modalities, a medical atlas and a patient scan, or image frames of a temporal image sequence. In this work, we focus on the registration of the lung on the base of thoracic CT images, which is of high clinical relevance (see for example Werner et al.7 ). Therefore, the methods for landmark detection (Sect. 2.1) and non-linear registration (Sect. 2.2) presented in the following are specific for this field of application. However, the optimization approach introduced in section 2.2.1 is generally independent of a particular technique for landmark detection or image registration.
2.1 Automatic landmark detection Anatomical landmarks represent distinct anatomical points with uniqueness in their vicinity. The algorithm to automatically detect a set of L corresponding landmarks in fixed image IF and moving image IM follows to a large extent the approach described by Murphy et al.8 and is detailed in Ehrhardt et al.4 It consists of two steps: identification of appropriate landmark candidates in IF and a robust transfer of the candidates to IM .
Proc. of SPIE Vol. 7962 79620T-2
Figure 2. Positions of the manually (left) and automatically (right) detected landmarks.
2.1.1 Landmark identification To identify landmark candidates, so-called distinctiveness values8 are computed for the voxels within a region of interest. The computation of distinctiveness values consists of two steps: First, the dissimilarity of the intensity values of a voxel and its neighboring voxels is determined. In a second step, this dissimilarity term is weighted by a normalized feature-based term. Appropriate landmark candidates feature high distinctiveness values. To force the landmarks to be well distributed throughout the region of interest, a minimum Euclidean distance is postulated between landmark candidates. In this paper, we focus on registration of thoracic CT images. Thus, appropriate landmark candidates are bifurcations of the vessels or the bronchial tree inside the lungs. Bifurcations show specific curvature characteristics, and so the curvature based F¨orstner operator9 is applied for computation of the distinctiveness values. 2.1.2 Landmark transfer Landmark candidates of IF are transfered to IM using a template matching method. This is done in two runs: a matching based on intensity values of the images, followed by a second run based on the answer of the applied feature based operator (here: the F¨ orstner operator). For both runs, the correlation coefficient was maximized to establish an optimal matching. To improve robustness, landmark candidates of IF are discarded if the correlation value of the first run is below a correlation threshold or if the transferred landmark positions differ by more than a prescribed distance between the two template matching runs.
2.2 Intensity-based non-linear registration Goal of the registration is to minimize the energy functional (1). In this work, the transformation is given by a displacement field u : Ω → R3 with ϕ(x) := x + u(x). A common choice for S is the diffusion regularization10 3
S[ϕ] :=
1 2 l=1
Ω
∇ul (x)2 dx ,
(2)
where ul is the l-th component of u. A smoothing is achieved by penalizing large gradients in the vector field. Providing results similar to those of a linear-elastic model,1 this term is much more efficient with respect to computation time.10 For the minimization of the energy functional (1), a variational approach is employed. We end up with the semi-implicit iterative update scheme: u(k+1) = (I − ατ A)−1 u(k) + τ fD (u(k) ) ,
Proc. of SPIE Vol. 7962 79620T-3
where A is a linear differential operator corresponding to the regularizer S and τ is the step width. The force term fD (u) is related to the distance measure D and chosen as fD (ϕ) :=
IF (x) − IM ◦ ϕ(x) ∇IM ◦ ϕ(x) , κ2 + ∇IM ◦ ϕ(x)2
where κ2 is the reciprocal of the mean squared spacing. This term is closely related to Thirion’s demon forces and while the corresponding energy is not known, it provides better results than the standard Sum of Squared Differences (SSD) in regions with low image contrast, like the vessel tree of the lung.11 To improve the results as well as computation time, a multi-scale strategy is applied. 2.2.1 Adaptive parameter optimization i in the moving image and The target registration error is defined as the mean distance between a landmark lM the corresponding landmark in the fixed image after registration:
TREA (α) :=
L 1 i lM − ϕα (lFi ) . L i=1
Here, ϕα denotes the transformation obtained by registration with a regularization weight α. Optimization aims at finding the value α ˆ that minimizes TREA , accordingly α ˆ := argmin TREA (α) . α
A
In our experiments, the function TRE was convex w.r.t. α (cf. Fig. 3). Therefore, optimization is performed in a straight-forward empiric manner using an interval-based search. Given a starting guess α and a step width − δα , α0 := α and α+1 := α + δα , and TREA is calculated δα , registration is run with three values α−1 := α for each result. If it is smallest for α0 , this choice is considered optimal, otherwise smaller or bigger values are recursively tested. While converging slower than a standard gradient descent approach, this method has the advantage that for each step only one new value of TREA has to be computed. In comparison, the computation of dTREA /dα would require at least two values of TREA . In a multi-scale setting, α ˆ s denotes the optimal value for scale s. The transformation ϕαˆ s is used as initialization for the next scale s + 1 and the corresponding α ˆ s is chosen as initial guess α s+1 . 2.2.2 Estimation of the Target Registration Error ˆ as In principle, computing TREA (α) requires registering the images with a given α. Finding an optimal value α described in section 2.2.1 can therefore be extremely time-consuming. To considerably attenuate this drawback, only the first 100 iterations are executed and a curve fitting on the TRE values of each iteration is employed to estimate TREA . Let tA i denote the TRE of the registration after iteration i. A model curve tp1 ,p2 ,p3 (i) := p1 exp(−p2 i) + p3 is fitted to the TRE values by minimizing Φ(p1 , p2 , p3 ) =
N 2 1 A ti − tp1 ,p2 ,p3 (i) 2 i=1
with respect to the model parameters p1 , p2 and p3 . The TRE then converges against p3 ≈ TREA . The non-linear least-squares-fitting is solved using the Levenberg-Marquardt algorithm.12 2.2.3 Stop criterion As stop criterion for the registration, usually intensity-based metrics like the mean squared distance (MSD) or root mean square (RMS) are considered. However, as seen in Figure 1, these metrics are not very reliable to quantify registration accuracy. This becomes apparent when examining the convergence behavior of both metrics during the registration process (cf. Figure 4, right). While the RMS suggests convergence of the algorithm, the TRE values continue to decrease. Therefore, in this work a joint stop criterion is applied that halts the registration if both TREA and RMS have converged.
Proc. of SPIE Vol. 7962 79620T-4
Pat01 Pat02 Pat03 Pat04 Pat05 Pat06 Pat07 Pat08 Pat09 Pat10 Pat11 Pat12
1.70
1.60
TRE
1.50
1.40
2.40 2.20 2.00 1.80
1.30
1.60
1.20
1.40
1.10
1.20
0.1
0.3
0.5
0.7
0.9
1.1
1.3
1.5
1.7
Case01 Case02 Case03 Case04 Case05 Case06 Case07 Case08 Case09 Case10
2.60
TRE
1.80
0.1
0.3
Regularization Weight
0.5
0.7
0.9
1.1
1.3
1.5
1.7
Regularization Weight
Figure 3. Target registration error TREM after registration of the proprietary (left) and the DIR-lab (right) data sets with different values for the regularization weight α.
2.3 Experiments For evaluating the landmark-driven parameter optimization, experiments are carried out using thoracic 4D CT data sets from two different sources: • 12 image pairs of our own fund. The images were acquired during free breathing and reconstructed using an optical flow based method.2 Spatial resolution is 0.98 × 0.98 × 1.5 mm. • 10 publicly available data sets hosted by DIR-lab, University of Texas, USA.13 The spatial resolution is between 0.97 × 0.97 × 2.5 mm and 1.16 × 1.16 × 2.5 mm. The first 5 data sets were sub-sampled in-plane to 256 × 256 voxels and therefore feature a lower signal-to-noise ratio than cases 6 to 10. Registration accuracy is quantified by the TREM , calculated using different sets of landmarks as for the optimization process. These sets were manually detected by clinical experts and are therefore suitable for usage as gold standard. For the DIR-lab data, the publicly available sets of 300 landmarks per image are used for evaluation. First, registration is performed with several fixed α values to determine the optimal weight α ˆ for each case and a value α ¯ that is optimal on average for the whole pool of considered data sets. We then focus on comparing three ¯ B) The TREM after registration with the measures: A) The TREM after registration with the fixed value α. fixed value α ˆ , i.e. the optimal value for the specific data set. C) The TREM after registration using the proposed automatic parameter optimization. Since only algorithm C) requires landmarks in the images, the TRE-based stop criterion is applied just in this case.
3. RESULTS In Figure 3, the target registration errors are visualized for registration with different regularization weights α. The results are listed in detail in table 1. As seen, the optimal choice of the regularization weight highly differs between the cases (ˆ α between 0.3 and 1.7). Moreover, the different sources of the images demand for different weights: while the optimal choice is around 0.55 for the proprietary data, it is on average 1.38 for the first 5 DIR-lab images (with sub-sampling) and 0.46 for the last 5 (without sub-sampling). Averaged about all data sets, α ¯ = 0.5 is an appropriate choice. Using the automatic landmark-based optimization, registration accuracy was improved for all 22 data sets. In comparison to registration with α ¯ , the registration error is reduced by up to 0.578 mm, on average by 0.066 mm. In all cases, the adaptive approach also performed better then the registration with the fixed case-specific optimum α ˆ (0.033 mm on average). The optimal choice of α ˆ s for each scale varies strongly depending on the data source. For the proprietary data, it is clearly advisable to reduce the amount of regularization on each scale from approximately α ˆ 0 ≈ 0.9 on the coarsest to α ˆ 2 ≈ 0.5 on the finest level. For the DIR-lab data, no clear statement can be made because the
Proc. of SPIE Vol. 7962 79620T-5
4
3.5
3
TRE RMS
1.9
200
1.7 1.6
150 1.5 100
1.4
2.5 0
50
100
150
200
250
1.8
Root Mean Square
4.5
Target Registration Error
Target Registration Error
300 Measured TRE values Curve fitted to values
0
Iteration
250
1000
1500
2000
Iteration
A A Figure 4. On the left, the TRE values during the iterative registration process tA i and the curve fitted to t31 to t100 are displayed. The graph on the right demonstrates the convergence behavior of the metrics TRE and RMS. While the RMS converges around iteration 250, the TRE continues to decrease and thus indicates further improvement of the registration result.
sample size of only 5 cases with the same preprocessing applied is too small to draw a statistically meaningful conclusion. The curve fitting performed well and stable in all cases. Figure 4 (left) exemplarily shows the values tA i measured A over all iterations and the curve fitted on the values tA 31 to t100 . The values of the first 30 iterations are not regarded to emphasize the influence of later iterations. Generally, the estimated value for TREM is always slightly higher than the actually reached value after all iterations. However, this has no significant influence on the estimation of α ˆ s : Estimated and effective optimum differ by only approximately 0.14, averaged about all cases and scales. Most of the divergences occur for the coarsest scale. Consideration of the TREM in the stop criterion results in a higher number of iterations for most of the cases. On average, the number of iterations is increased by approximately 29%. The computation time of the registration using the automatic parameter optimization (on average 83 min) is considerably higher than with a fixed weight (12 min). The overhead can be divided into landmark detection (≈ 5%), dismissed trial registrations for parameter optimization (73%) and additional iterations due to the TRE-based stop criterion (22%).
4. DISCUSSION The results show that optimal values for the regularization weight differ considerably from case to case. While these differences are relatively small if the images originate from the same source and if the same preprocessing is applied, they are much bigger between images from different origins. Some images are also very sensitive to the choice of α (e.g. Patient12, Case07 and Case08). However, in a clinical setting, determining α ˆ by computing the registration for multiple values of α is not feasible due to the required processing time. Moreover, this procedure would also demand landmarks in the images to calculate the TRE. With the proposed approach for landmark-based parameter optimization, no prior determination of the regularization parameter is required. Still, the results were improved in all cases, not only compared to the registration with α ¯ but also with α ˆ , which is in praxis unknown. Naturally, the improvement is especially apparent if α ˆ considerably differs from α ¯ or if the data is very sensitive to the choice of the weight. For the proprietary data, the experiments indicate that high regularization is advisable for the coarse scales, while D should be weighted higher on the last levels for a fine adjustment of the transformation. However, this observation does not hold for the DIR-lab data due to different imaging and preprocessing procedures. Consideration of the TRE in the stop criterion further improves the registration accuracy. This is especially apparent for Case07 and Case08, where the MSD suggests convergence of the registration while the TRE keeps to decrease (cf. Figure 4, right). As a drawback of the presented approach, computation time is increased by approx. 400% to 800%. While this is still faster than performing the whole registration for multiple weights, the automatic parameter optimization can be considered as necessary primarily when registering images from different sources. For comparably homogeneous data sets like our proprietary data, the improvement of the registration accuracy may not justify the additional computation time in a clinical setting. However, in this case the presented approach can still be
Proc. of SPIE Vol. 7962 79620T-6
DIR-lab data
Proprietary data
Data set
w/o Reg. TREM
Fixed Weight TREM (¯ α = 0.5)
Optimal Weight TREM (ˆ α)
Adaptive Weight TREM (ˆ α0 , α ˆ1, α ˆ2 )
Pat01 Pat02 Pat03 Pat04 Pat05 Pat06 Pat07 Pat08 Pat09 Pat10 Pat11 Pat12 Mean
4.254 ± 2.56 6.264 ± 4.77 5.450 ± 2.73 6.206 ± 2.15 6.791 ± 3.75 6.437 ± 3.09 4.314 ± 3.13 10.76 ± 7.80 6.244 ± 6.81 6.062 ± 5.04 7.979 ± 5.29 8.315 ± 6.40 6.590 ± 4.46
1.054 ± 0.71 1.087 ± 0.55 1.201 ± 0.77 1.117 ± 0.64 1.354 ± 0.90 1.114 ± 0.48 1.227 ± 0.64 1.124 ± 0.67 1.208 ± 1.00 1.070 ± 0.65 1.102 ± 0.59 1.342 ± 1.04 1.167 ± 0.72
1.047 ± 0.49 1.087 ± 0.56 1.201 ± 0.78 1.117 ± 0.63 1.338 ± 0.72 1.114 ± 0.48 1.227 ± 0.63 1.123 ± 0.63 1.188 ± 0.67 1.066 ± 0.43 1.102 ± 0.63 1.299 ± 0.91 1.159 ± 0.63
(0.9) (0.5) (0.5) (0.5) (1.1) (0.5) (0.5) (0.3) (0.3) (0.7) (0.5) (0.3) (0.55)
1.043 ± 0.70 1.073 ± 0.54 1.186 ± 0.74 1.110 ± 0.61 1.324 ± 0.84 1.111 ± 0.47 1.223 ± 0.63 1.113 ± 0.65 1.166 ± 0.88 1.062 ± 0.66 1.087 ± 0.58 1.272 ± 0.83 1.148 ± 0.68
(1.1, 1.1, 0.7) (0.7, 0.3, 0.5) (0.9, 0.5, 0.3) (1.1, 0.7, 0.3) (1.1, 0.9, 0.5) (1.1, 0.9, 0.5) (1.3, 0.7, 0.5) (0.5, 0.3, 0.5) (0.7, 0.3, 0.3) (1.1, 0.5, 0.9) (0.9, 0.9, 0.5) (0.5, 0.3, 0.3) (0.91, 0.62, 0.48)
Case01 Case02 Case03 Case04 Case05 Mean
3.892 ± 2.78 4.338 ± 3.90 6.943 ± 4.04 9.830 ± 4.85 7.477 ± 5.50 6.496 ± 4.21
1.106 ± 0.52 1.056 ± 0.49 1.215 ± 0.66 1.495 ± 1.04 1.470 ± 1.22 1.268 ± 0.79
1.030 ± 0.49 1.004 ± 0.47 1.158 ± 0.65 1.423 ± 1.02 1.458 ± 1.22 1.214 ± 0.77
(1.7) (1.7) (1.3) (1.5) (0.7) (1.38)
1.023 ± 0.49 1.001 ± 0.47 1.152 ± 0.64 1.415 ± 1.01 1.451 ± 1.22 1.208 ± 0.77
(2.1, (1.3, (0.7, (0.9, (1.1, (1.22,
1.7, 1.5, 0.9, 1.7, 0.7, 1.3,
2.1) 2.1) 1.3) 1.5) 0.7) 1.54)
Case06 Case07 Case08 Case09 Case10 Mean
10.89 ± 6.96 11.03 ± 7.42 14.99 ± 9.00 7.918 ± 3.97 7.301 ± 6.34 10.42 ± 6.74
1.404 ± 0.84 1.656 ± 1.60 1.994 ± 2.36 1.395 ± 0.81 1.370 ± 1.03 1.564 ± 1.33
1.375 ± 0.80 1.591 ± 1.31 1.736 ± 1.73 1.395 ± 0.81 1.363 ± 0.95 1.492 ± 1.12
(0.9) (0.3) (0.3) (0.5) (0.3) (0.46)
1.353 ± 0.76 1.431 ± 0, 86 1.416 ± 1.09 1.353 ± 0.75 1.346 ± 0.90 1.380 ± 0.87
(0.5, 0.7, (0.3, 0.3, (0.5, 0.3, (0.7, 0.3, (0.5, 0.3, (0.5, 0.38,
0.7) 0.3) 0.3) 0.5) 0.3) 0.42)
Table 1. Target registration errors after registration with fixed regularization weight α, ¯ the case-specific optimum α ˆ and the proposed automatic landmark-based optimization approach. All TRE values are in mm.
Proc. of SPIE Vol. 7962 79620T-7
applied for optimizing weights for an unseen data pool and for determining optimal weights for each scale in a multi-scale setting.
5. CONCLUSION Registration of medical images is a key problem in many applications. This study shows that automatically detected landmarks can be used for parameter optimization to considerably improve registration accuracy. The improvement is especially apparent if the data pool consists of images of different sources, with different resolution or with different preprocessing steps applied. In this paper, only one possible use of automatically detected landmark sets is demonstrated. Other parameters could be optimized as well in a straight-forward manner. In future work, we will also examine more sophisticated approaches for optimizing the weights while iterating.
ACKNOWLEDGMENTS This work is supported by the German Research Foundation DFG (EH 224/3-1). We further thank D. Low and W. Lu from the Washington University School of Medicine in St. Louis for providing the CT data.
REFERENCES [1] Werner, R., Ehrhardt, J., Schmidt-Richberg, A., and Handels, H., “Validation and comparison of a biophysical modeling approach and non-linear registration for estimation of lung motion fields in thoracic 4D CT data,” in [Image Processing, SPIE Medical Imaging 2009], 7259, 0U1– 0U8 (2009). [2] Ehrhardt, J., Werner, R., S¨aring, D., Lu, W., Low, D. A., and Handels, H., “An optical flow based method for improved reconstruction of 4D CT data sets acquired during free breathing,” Med Phys 34(2), 711–21 (2007). [3] Ehrhardt, J., Werner, R., Schmidt-Richberg, A., and Handels, H., “Prediction of Respiratory Motion Using A Statistical 4D Mean Motion Model,” in [The Second International Workshop on Pulmonary Image Analysis, Medical Image Computing and Computer-Assisted Intervention - MICCAI 2009], 3–14 (2009). [4] Ehrhardt, J., Werner, R., Schmidt-Richberg, A., and Handels, H., “Automatic Landmark Detection and Non-linear Landmark- and Surface-based Registration of Lung CT Images,” in [Medical Image Analysis for the Clinic - A Grand Challenge, MICCAI 2010], 165–174 (2010). [5] Rohr, K., Stiehl, H. S., Sprengel, R., Buzug, T. M., Weese, J., and Kuhn, M. H., “Landmark-based elastic registration using approximating thin-plate splines.,” IEEE Trans Med Imaging 20(6), 526–34 (2001). [6] Olesch, J., Papenberg, N., Lange, T., Conrad, M., and Fischer, B., “Matching CT and ultrasound data of the liver by landmark constrained image registration,” in [Image Processing, SPIE Medical Imaging 2009], 7261 (2009). [7] Werner, R., Ehrhardt, J., Schmidt-Richberg, A., Bodmann, B., Cremers, F., and Handels, H., “Dose Accumulation based on Optimized Motion Field Estimation using Non-Linear Registration in Thoracic 4D CT Image Data,” in [World Congress on Medical Physics and Biomedical Engineering], 25/IV, 950–953 (2009). [8] Murphy, K., van Ginneken, B., Pluim, J. P. W., Klein, S., and Staring, M., “Semi-automatic reference standard construction for quantitative evaluation of lung ct registration.,” in [Int Conf Med Image Comput Comput Assist Interv], 11(Pt 2), 1006–1013 (2008). [9] Hartkens, T., Rohr, K., and Stiehl, H., “Evaluation of 3D operators for the detection of anatomical point landmarks in MR and CT images,” Comput Vis Image Underst 86, 118–36 (2002). [10] Modersitzki, J., [Numerical Methods for Image Registration ], Oxford Univ Press (2004). [11] Schmidt-Richberg, A., Ehrhardt, J., Werner, R., and Handels, H., “Diffeomorphic Diffusion Registration of Lung CT Images,” in [Medical Image Analysis for the Clinic: A Grand Challenge, MICCAI 2010], 55–62 (2010). [12] Watson, G. and Mor´e, J., [The Levenberg-Marquardt algorithm: Implementation and theory], vol. 630, 105–116, Springer Berlin / Heidelberg (1978). [13] Castillo, R., Castillo, E., Guerra, R., Johnson, V. E., McPhail, T., Garg, A. K., and Guerrero, T., “A framework for evaluation of deformable image registration spatial accuracy using large landmark point sets,” Phys Med Biol 54, 1849–1870 (Jan 2009).
Proc. of SPIE Vol. 7962 79620T-8
