Decoupling Respiratory and Angular Variation in Rotational X-ray Scans Using a
Prior Bilinear Model
Data-driven respiratory signal extraction from rotational X-ray scans has always been challenging due to angular effects overlapping with respiration-induced change in the scene. In the context of motion modelling their main drawback is the fact that most of these methods only extract a 1D signal, that, at best, can be decomposed into amplitude and phase. In this paper, we use the linearity of the Radon operator to propose a bilinear model based on a prior 4D scan to separate angular and respiratory variation. Out-of-sample extension is enhanced by a B-spline interpolation using prior knowledge about the trajectory angle to extract respiratory feature weights independent of the acquisition angle. Though the prerequisite of a prior 4D scan seems steep, our proposed application of respiratory motion estimation in radiation therapy usually meets this requirement. We test our approach on DRRs of a patient’s 4D CT in a leave-one-out manner and achieve a mean estimation error of in the gray values for previously unseen viewing angles.
Extracting information about a patient’s breathing state from X-ray projections is important for many time-resolved application, such as CT reconstruction or motion compensation in radiation therapy . Data-driven approaches have been proposed ranging from the established Amsterdam Shroud  to more sophisticated approaches based on epipolar consistency . In the context of motion models , their main drawback is the fact that most of these methods only extract a 1D signal, that, at best, can be decomposed into amplitude and phase . When the motion representation is covered by an active shape model , it is highly desirable to extract more features resulting in a lower reconstruction error.
A mathematical foundation to decompose multiple sources of variation was given by De Lathauwer et al.  who formulated Higher-order Singular Value Decomposition (HOSVD) on a tensor of arbitrary dimensionality. Meanwhile, bilinear models have seen use in many fields including medical applications. Among others, Tenenbaum et al.  used a bilinear model to separate pose and identity from face images, while Çimen et al.  constructed a spatio-temporal bilinear model of coronary artery centerlines.
In this work, we show that due to the linearity of the radon transform, a bilinear decomposition of respiratory and angular variation exists (Sec. 2.1). Subsequently, Sec. 2.2 and 2.3 will cover both model training and its application in feature estimation. With out-of-sample extension being an ill-posed problem, we propose a B-spline interpolation of rotational weights based on prior knowledge about the trajectory angle. We validated our model in a leave-one-out manner using digitally reconstructed radiographs (DRRs) of a patient’s 4D CT.
2 Material & Methods
2.1 Radon Transform under Respiratory and Angular Variation
An X-ray projection at rotation angle and respiratory phase is given by the Radon transform applied to the volume
where N indicates the arbitrary dimension of the volume and projection image. It has been shown that the respiratory-induced changes in the anatomy can be described by an active shape model of the internal anatomy :
where is the data mean of the mode, contains the eigenvectors corresponding to the first principal components, and are the model weights corresponding to phase . Thus, Eq. 1 can be further processed
Now, represents a model (with mean ) for describing the projection image under the fixed angle given the respiratory weights . The inversion of the Radon transform itself is ill-posed for a single projection. However, can be inverted more easily as mostly encodes variation in superior-inferior direction observable in the projection. As a result, respiratory model weights can be estimated from a single projection image if the angle-dependent model matrix is known .
Furthermore, we propose a Radon transform to be approximated by a linear combination of other Radon transforms, such that
The resulting scalar factors form the weight vector . Note that in this formulation, we implicitly assume a continuous trajectory and that the breathing motion is observable from each view. This gives rise to a bilinear formulation to any given projection image . However, bilinear models typically do not operate on mean-normalized data. Therefore, we use the decomposition described in Eq. 2 without mean subtraction:
where is a model tensor with respiratory and rotational feature dimensionality and . Here, denotes the mode product along the given mode . For more details on tensor notation please refer to .
2.2 Model Training
For model training, a prior 4D CT scan is required yielding phase-binned volumes . DRRs are computed at angles along a circular trajectory. The resulting projection images form the data tensor . Using HOSVD  we perform dimensionality reduction on the data tensor. First, is unfolded along mode :
Fig. 1 illustrates the unfolding process. Second, SVD is performed on each unfolded matrix
which yields the tensor basis to project onto (see Fig. 2):
Finally, can be described by a model tensor :
where and carry the low-dimensional () model weights for respiratory and angular variation, respectively.
2.3 Weight Estimation
Given an observed projection image at unknown respiratory phase and angle , our objective is to find coefficients , for respiration and rotation to best represent the observation in terms of the model:
However, as and need to be optimized simultaneously, this task is highly ill-posed. Tenenbaum et al.  used an Expectation-Maximization algorithm to cope with this problem. However, they benefit from the fact that only the pose is a continuous variable whereas identity is a discrete state, drastically simplifying their EM-approach. In our case, both respiratory and angular variation have to be considered continuous. Fortunately, we can incorporate prior knowledge about the trajectory into the estimation process. From the trajectory, the angle of each projection image is known even though the corresponding weights are only given for particular angles within the training samples.
2.3.1 Content B-spline Interpolation.
Using the rotation angle as prior knowledge, we propose extending the bilinear model with a B-spline curve fitted to the rotational weights from training:
with being the B-spline basis functions. Using a uniform parametrization with respect to the training angles, of new angle is given as
2.3.2 Style Weight Computation.
With the rotational weight interpolated, multiplying and , first removes the angular variation:
Collapsing the 1-dimension results in the angle-dependent model matrix for the new angle . Closing the loop to Eq. 3 without mean, computation of the respiratory weights simplifies to a linear problem solved via the pseudo-inverse of :
2.4 Data & Experiments
Evaluation was performed on a patient’s 4D CT consisting of eight phase-binned volumes at respiratory states , , , , inhale, and , , exhale. Using CONRAD , DRRs of size with an isotropic pixel spacing were created by forward projecting each of the volumes at angles over a circular trajectory of . Consequently, the final data tensor featured a dimensionality of .
A dense bilinear model was trained on the entire data tensor to assess the variance explained by the respiratory domain. For comparison, we investigated a linear PCA with mean normalization of the 4D CT volumes to assess weights and variance prior to the influence of the Radon transform. The dense bilinear model also provided a reference for the features in the subsequent leave-one-out evaluation where each phase and every 6th angle were removed one after another from the data tensor prior to training. Subsequently, the corresponding projection image and its respective angle were fed to the model and the rotational and respiratory weights were estimated. Accuracy was assessed with respect to the mean gray-value error between the reconstructed image and the original.
3 Results & Discussion
Fig. 3 shows the weights of the first three principal components for the linear PCA of the volumes as well as each phase and angle in the projection images. Notably, both first bilinear components are near constant. Unlike linear PCA (Eq. 2), data in the bilinear model does not have zero-mean (Eq. 5). Consequently, the first component points to the data mean while variation in the respective domain is encoded starting from the second component . Appropriately, the -th bilinear respiratory component corresponds to the -th linear component of the 4D CT indicating that separation of respiratory and angular variance in the projections is in fact achieved. Additionally, the respiratory variance explained by the principal components is plotted (top right). Most of the variance in the bilinear case is already explained by movement towards the mean. Thus, the linear components better reflect that four components accurately describe over of the data variance. Fig. 4 shows the eigenimages corresponding to angular and respiratory variation, respectively. Noticeably, the rotational eigenimages contain mostly low-frequent variation inherent to the moving gantry whereas the respiratory direction encodes comparably high-frequent changes.
For evaluation, Tab. 1 lists the mean gray value error in the reconstructed projection images for each test-angle averaged over all estimated phases. The mean gray value in the original image is listed for reference. The average error was () compared to a reference mean value of . Exemplarily for one phase-angle combination, Fig. 5 shows the leave-one-out estimation result for and . The proposed B-spline interpolation yields rotational weights close to the dense bilinear model up to the 10th component. Since both sets of weights correspond to slightly different eigenvectors, due to one model being trained on less data, deviation especially in the lower components is to be expected. Still, four to five respiratory weights are estimated accurately which, as shown previously, is sufficient to recover over respiratory variance. As such, we believe they contain much more information than just the respiratory phase. In future work, we will investigate their use to predict entire dense deformation fields.
|gray value error|
Our current leave-one-out evaluation assumed two simplifications, that will pose additional challenges. First, a perfect baseline registration of the training CT to the projection images may not be the case in every scenario. However, for the case of radiation therapy accurate alignment of patient and system is a prerequisite for optimal treatment. Second, no anatomical changes between the 4D CT and the rotational scan are taken into account. Further investigation on how these effects interfere with the decomposition are subject to future work.
In this paper, we demonstrate that the Radon transform under respiratory and angular variation can be expressed in terms of a bilinear model given a continuous trajectory and that motion is observable in every projection. Using a prior 4D CT, we show that projection images on the trajectory can be bilinearly decomposed into rotational and respiratory components. Prior knowledge about the gantry angle is used to solve this ill-posed out-of-sample problem. Results demonstrate that up to five components of the respiratory variance are recovered independent of the view-angle. This explains more than of the volumetric variation. As such, recovery of 3D motion seems possible. Currently our study is limited by two simplifications, namely perfect alignment and no inter-acquisition changes. Their investigation is subject to future work.
This work was conducted at the ACRF Image X Institute as part of a visiting research scholar program. The authors gratefully acknowledge funding of this stay by the Erlangen Graduate School in Advanced Optical Technologies (SAOT).
-  Aichert, A., Berger, M., Wang, J., Maass, N., Doerfler, A., Hornegger, J., Maier, A.K.: Epipolar Consistency in Transmission Imaging. IEEE Trans Med Imag 34(11), 2205–2219 (2015)
-  Çimen, S., Hoogendoorn, C., Morris, P.D., Gunn, J., Frangi, A.F.: Reconstruction of Coronary Trees from 3DRA Using a 3D+t Statistical Cardiac Prior. In: Med Image Comput Comput Assist Interv (MICCAI). pp. 619–626 (2014)
-  De Lathauwer, L., De Moor, B., Vandewalle, J.: A Multilinear Singular Value Decomposition. SIAM J Matrix Anal Appl 21(4), 1253–1278 (Jan 2000)
-  Fassi, A., Seregni, M., Riboldi, M., Cerveri, P., Sarrut, D., Ivaldi, G.B., de Fatis, P.T., Liotta, M., Baroni, G.: Surrogate-driven deformable motion model for organ motion tracking in particle radiation therapy. Phys Med Biol 60(4), 1565–1582 (2015)
-  Fischer, P., Pohl, T., Faranesh, A., Maier, A., Hornegger, J.: Unsupervised Learning for Robust Respiratory Signal Estimation from X-Ray Fluoroscopy. IEEE Trans Med Imag 36(4), 865–877 (2017)
-  Keall, P.J., Aun Ng, J., O’Brien, R., Colvill, E., Huang, C.Y., Rugaard Poulsen, P., Fledelius, W., Juneja, P., Simpson, E., Bell, L., Alfieri, F., Eade, T., Kneebone, A., Booth, J.T.: The first clinical treatment with kilovoltage intrafraction monitoring (KIM): a real-time image guidance method. Med Phys 42(1), 354–358 (2015)
-  Kolda, T.G., Bader, B.W.: Tensor Decompositions and Applications. SIAM Rev 51(3), 455–500 (2009)
-  Maier, A., Hofmann, H.G., Berger, M., Fischer, P., Schwemmer, C., Wu, H., Müller, K., Hornegger, J., Choi, J.H., Riess, C., Keil, A., Fahrig, R.: CONRAD–a software framework for cone-beam imaging in radiology. Med Phys 40(11), 111914 (2013)
-  McClelland, J.R., Hawkes, D.J., Schaeffter, T., King, A.P.: Respiratory motion models: A review. Med Image Anal 17(1), 19–42 (2013)
-  Tenenbaum, J.B., Freeman, W.T.: Separating Style And Content With Bilinear Models. Neural Comput 12(6), 1247–1283 (2000)
-  Yan, H., Wang, X., Yin, W., Pan, T., Ahmad, M., Mou, X., Cerviño, L., Jia, X., Jiang, S.B.: Extracting respiratory signals from thoracic cone beam CT projections. Phys Med Biol 58(5), 1447–64 (2013)