A System Identification Approach to Estimating a Dynamic Model of

A System Identification Approach to Estimating a Dynamic Model of Head Motion for MRI Motion Correction Burak Erem1 , Onur Afacan1 , Ali Gholipour1 , Sanjay P. Prabhu2 , and Simon K. Warfield1 1

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Computational Radiology Laboratory, Radiology Department, Boston Children’s Hospital Advanced Image Analysis Lab, Radiology Department, Boston Children’s Hospital

Abstract. Motion-compensated MRI is a promising technique to mitigate the effects of motion on MRI. This works best when accurate realtime motion measurements are available. Many measurement techniques track motion with a delay and produce noisy measurements. We propose an estimator that uses a dynamic system identification approach to estimate rigid body head motion from concurrent measurements of position and orientation, which can be used to predict motion shortly into the future. We compare our method to static estimates and a Kalman filter-based method in our experiments, in which we evaluate the effects of motion using real and simulated tracking data.

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Introduction

Magnetic resonance imaging (MRI) is highly sensitive to patient motion, resulting in image artifacts that can severely limit the diagnostic quality of images. In neuroimaging, uncontrolled motion of the patient’s head can be a particularly problematic source of image artifacts, and a common clinical solution is to perform imaging after the patient has been sedated or anesthetized, both of which are associated with various health risks. An alternative solution – motioncompensated MRI – is possible when measurements of the patient’s head motion during scanning are available. Given motion measurements, a number of different approaches have been proposed to compensate for motion that occurs during a scan (e.g., [1, 2]). These approaches model the motion of the head as rigid body motion, and can work well when motion measurements are accurate and available in real time. In practice, however, the effective precision of any motion measurement technology will be limited because measurements will invariably contain some amount of noise, which in turn will adversely affect the estimates of rigid body motion parameters. Moreover, many such technologies are limited in the rate at which they can make motion measurements, and, crucially, they measure the motion that occurs before (and not during) the stages of a pulse sequence when the field-of-view (FOV) needs to be adjusted to acquire the correct data. These limitations prevent current methods from fully realizing the potential of motion-compensated MRI.

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In this paper, we propose an algorithmic method that addresses some of the aforementioned limitations of motion measurements for motion-compensated MRI. Given concurrent measurements of multiple correspondence points and orientations from the head, we are able to simultaneously estimate rigid body motion parameters and their dynamics, which can be used to predict parameters that occur shortly after measurement times. The method is based on recent work on fast methods for robust dynamical system identification, published in the control theory literature, that allows for simultaneously denoising and estimating the dynamics of a noisy sequence of data [3]. The present work extends that work by incorporating a model of rigid body motion that directly relates motion parameters to measurements, and otherwise paring down the optimization problem to suit the present application. We evaluate the proposed method using experiments on real and simulated data, and compare the results of our approach to previous approaches. The dynamic estimates resulting from this algorithm outperform static estimates of rigid body motion parameters, as well as an approach based on a previously published Kalman filtering method that was proposed for this purpose [4].

2 2.1

Background Measurements of Positions and Orientations

Several technologies exist that offer the possibility of measuring motion in an MRI scanner, such as image registration of navigators acquired between repetitions of a pulse sequence, optical trackers, and electromagnetic trackers. It would be useful to have a common framework to use the data measured from them. One possibility, which is the one we will adopt in the sequel, is that each tracker returns one or more of two basic types of tracking data: (1) corresponding points (i.e., time-varying points that move with the object of interest), or (2) corresponding orientations (i.e., time-varying vectors that move with the orientation of the object). 2.2

Model of Rigid Body Motion

In MRI motion compensation for neuroimaging, the head and brain are typically modeled as a single rigid body. This is an approximation, since the brain itself is able to move within the head, and thus the true transformation of the geometry is not strictly limited to rotation and translation. However, as an MRI scanner is also limited to adjusting the FOV using only rotation and translation, it is also an adequate approximation for the purposes of motion compensation. Mathematically, rigid body motion of a set of points at any given time can be and translation of a set of reference points. Let P (t) = stated as the rotation  p1 (t) · · · pN (t) be a 3 × N matrix containing a set of position vectors (one in each column) in R3 at time t. Also let P (0) be the matrix containing the set of reference points (which we always take to be the initial positions measured from

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the trackers). Then rigid body motion of the object containing these reference points can be described by rotation, R(t) ∈ SO(3) ⊂ R3×3 , and translation, ∆p(t) ∈ R3 , as P (t) = R(t)P (0) + ∆p(t)11×N , (1) where 11×N is a constant 1 × N matrix (i.e., row vector) containing N ones. Note that SO(3) here denotes the rotation group, or equivalently the special orthogonal group in 3 dimensions. One could also perform a similar transformation on orientation vectors, which can simply be regarded as correspondence points at the origin   which never undergo translation. Therefore, if we let V (t) = v1 (t) · · · vM (t) be a 3 × M matrix containing set of orientation vectors (one in each column) in R3 at time t, one can describe the effect of rigid body motion on the reference orientation vectors, V (0) (again, taken as the initial orientations measured from the trackers), as V (t) = R(t)V (0). 2.3

(2)

Dynamic System Identification from the Hankel Matrix

The problem of dynamic system identification from noisy measurements is a classic problem in control theory as well as one that is the focus of ongoing research (e.g., fast methods [3]). A discrete-time linear time invariant system with states, x(t) ∈ Rn , and observations, y(t) ∈ Rm , that has no inputs can be described using state space equations x(t + 1) = Ax(t)

(3)

y(t) = Bx(t) + η(t),

(4)

where η(t) is noise that perturbs the observations. The problem of system identification in this case is to estimate A and x(t) from a sequence of noisy measurements, y(t), and given the observation matrix B. Note that, in general, the model order n is also unknown and must either be estimated or assumed a priori. A well-known result in the control theory literature is that the rank of a Hankel matrix   x(t − w) · · · x(t − w/2)   .. .. .. (5) Hx,w (t) =   . . . x(t − w/2) · · ·

x(t)

constructed from the sequence of states x(t − τ ), . . . , x(t) is n. Such a matrix can be used to solve for the matrix A using the Ho-Kalman realization theorem [3], which can be used to predict future states of the system from known previous states. In theory, when the system is observable, and in the absence of noise, one can construct a similar Hankel matrix from the sequence of observations y(t − τ ), . . . , y(t) whose rank should also be n, but in practice noise prevents this from being the case. To overcome this, the problem can been recast as an optimization problem that aims to approximate the sequence of observations within a specified error bound, , while minimizing the nuclear norm (k·k∗ , as a

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surrogate for the rank) of the Hankel matrix constructed from them. Formally, this is stated as min yˆ(t−w),...,ˆ y (t)

kHyˆ,w (t)k∗

subject to kˆ y (t0 ) − y(t0 )k∞ ≤ , ∀t0 = t − w, . . . , t.

(6)

This allows one to simultaneously filter y(t), while estimating a low-rank Hankel matrix that can be factored to find the matrix A. Note that this problem can also be used for prediction of y(t + 1), without needing to explicitly solve for A, by adding yˆ(t + 1) as an optimization variable, substituting kHyˆ,w+1 (t + 1)k∗ as the objective function, but only imposing the bound constraints for t0 = t − w, . . . , t. This is a so-called “inpainting” approach to prediction, which works by forcing yˆ(t + 1) to obey the dynamics in yˆ(t − w), . . . , yˆ(t) due to nuclear norm minimization. A fast algorithm was proposed by Ayazoglu et al. to solve problems of this form, based on the alternating directions method of multipliers (ADMM) [3]. For our work, we have implemented a pared-down version of the more general solver, which enables our problem to be solved on the order of tens of milliseconds on commonly available computing hardware.

3

Methods

In (1) & (2), we specified the relationship between correspondence points (and orientations) and rigid body motion parameters, R(t) and ∆p(t). However, these equations are not in the expected form for the observation equations of a dynamical system, as written in (4). In this section, we first transform the equations into the appropriate form for use in system identification, i.e., such that they can be adapted to the estimation framework presented in Sec. 2.3. Then we present the complete dynamical parameter estimation problem, combining the rigid body motion model and the optimization problem for system identification. 3.1

Observation Model from Rigid Body Motion Parameters

Let vec(·) denote the vectorization operator that takes a matrix as input and returns a vector by linearly indexing the matrix, effectively stacking the columns of the matrix on top of each other, and ⊗ denote the Kronecker product. A wellknown result is vec(ABC) = (C T ⊗A) vec(B). We use this relationship to convert (1) & (2) into the form of (4). Thus (1) and (2) become vec(P (t)) = (P (0)T ⊗ I) vec(R(t)) + (1N ×1 ⊗ I)∆p(t), T

vec(V (t)) = (V (0) ⊗ I) vec(R(t)),

(7) (8)

where I is a 3 × 3 identity matrix. These equations can be consolidated into a block matrix-vector system, y(t) = Bx(t), where       vec(P (t)) (P (0)T ⊗ I) (1N ×1 ⊗ I) vec(R(t)) y(t) = , B= , x(t) = , (9) vec(V (t)) ∆p(t) (V (0)T ⊗ I) (03M ×3 )

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and 03M ×3 denotes a 3M × 3 constant matrix of zeros. Note that, although the rotation matrices R(t) belong to a nonlinear manifold, namely SO(3), this result shows that our observation model is linear (i.e., it linearly maps between nonlinear manifolds). In the sequel, we will use these matrices in an estimation problem, similar to one presented in (6), to predict rigid body motion from a sequence of measured correspondence points and orientations. 3.2

Simultaneous Estimation of Rigid Body Motion Parameters and their Dynamics

Here we combine the system identification framework in Sec. 2.3 with the observation equation, (9), by imposing additional constraints on the optimization problem in (6). This results in simultaneous predictions of R(t + 1) ∈ R3×3 and ∆p(t + 1) ∈ R3 , as well as their dynamics. Note that, rather than constraining solutions such that R(t + 1) ∈ SO(3) during the optimization problem, we first solve a relaxed version of the problem in R3×3 and then project the solution to SO(3). Thus the dynamic prediction problem can be stated as min x ˆ(t−w),...,ˆ x(t+1)

kHyˆ,w+1 (t + 1)k∗

subject to kˆ y (t0 ) − y(t0 )k∞ ≤ , ∀t0 = t − w, . . . , t yˆ(t0 ) = B x ˆ(t0 ), ∀t0 = t − w, . . . , t + 1,

(10)

where y, B, and x are as defined in (9). We adapted our solver for (6) to also solve ˆ + 1) is extracted from x problem (10). After optimization, the matrix R(t ˆ(t + 1) and then projected to the nearest matrix (in the Frobenius norm sense) in SO(3). The projection is performed using the Kabsch algorithm, first by computing the ˆ + 1), and then replacing R(t ˆ + 1) by singular value decomposition U SV 0 = R(t 0 0 U DV , where D = diag([1 1 sign(det(U V ))]). The parameters ∆p(t + 1) can be extracted directly from x ˆ(t + 1) without further processing. An important parameter of this algorithm is w, which is an upper bound on the order of the dynamical system that can be identified, and thus we study the sensitivity of the algorithm to choice of w in the experiments.

4

Experiments

We evaluate our methods using experiments on both simulated and measured motion and image data. In the first experiment, we analyze the effect of parameter choices on the performance of the proposed estimator on motion measurements from an electromagnetic tracker placed on an adult volunteer undergoing MRI. In the next experiment, we simulate motion on an image of a phantom as well as noisy motion measurements, and show how the proposed estimator affects motion correction results when motion correction is applied prospectively (i.e., when FOV adjustments are made in real time). For a comparison, we implemented an unscented Kalman filter (UKF) using the state update equation x(t + 1) = x(t) + γ(t) (i.e., purely noise-driven

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Fig. 1. Sensitivity of our proposed estimator and the unscented Kalman filter (UKF) to their respective parameter choices that control the tracking dynamics. Top-left shows tracking results during motion and motion-free time (MFT). Top-right shows the performance tradeoff of both estimators during MFT. Topleft and bottom-left show that it is difficult to tune the UKF to converge to the measured data after motion ceases and MFT commences, whereas our proposed method converges quickly in both cases.

dynamics as in [4]). Here γ is random process noise, assumed to be normally distributed with zero mean and covariance νI. We also used our observation model, but with a nonlinear parameterization of the rotation matrix using the matrix exponential, R(t) = exp(S(t)), of a skew symmetric matrix, S(t), whose only free parameters are its three upper triangular elements s1,2 (t), s1,3 (t), and s2,3 (t). In all experiments, in both our method and the UKF, we assumed that the observation noise was normally distributed with zero mean and standard deviation  = 0.1mm. When appropriate, we also compared our results to static estimates of rigid body motion parameters, obtained by projecting the least squares solution of the observation equation at one time instant to SO(3). The purpose of these comparisons was to evaluate the effect of incorporating system identification as part of the estimation procedure, as opposed to assuming a priori that there are only noise-driven dynamics (e.g., UKF) or that the dynamics can be ignored during estimation (e.g., static estimate). 4.1

Parameter Sensitivity

In the method we proposed, the parameter w controls the maximum order of the dynamic system that can be identified. In the UKF, since A = I, the dynamics are assumed to be solely influenced by the noise variance parameter, ν. In this

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Fig. 2. Simulated prospective motion correction results using: ground truth motion (top-left), the proposed approach (top-right, NRMSE=0.08), static estimate (bottomleft, NRMSE=0.10), and UKF estimate (bottom-right, NRMSE=0.13). A zoomed in view of the dots in the phantom show the effects of each approach on image details.

experiment, we obtained real motion measurements from an electromagnetic tracker placed on the head of an adult volunteer that was instructed when to move. The scan contained ample motion-free time (MFT) (Fig. 1 top-left), and it is critical to correctly estimate patient motion during this time in order to efficiently utilize as much uncorrupted data as possible. To test the sensitivity of each algorithm to parameter choices, we applied our method using a large range of w parameters, from 10 to 100, and the UKF for a large range of ν parameters, from 1e-10 to 1. The performance of each estimate was evaluated using two criteria: (1) the mean absolute error between the predicted observations and the measured observations during MFT, and (2) the standard deviation of the estimates about their own means during MFT. In Fig. 1 top-right we show a plot of how the performance of each estimator varies according to these two criteria

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as the parameters are varied. This plot shows that there is a tradeoff between the two performance criteria, and that our estimator outperforms the UKF. The two plots on the left show the tracking results for two different parameter choices of both our method and the UKF. The top-left plot shows how choosing the parameter for the UKF that corresponds to the corner of the tradeoff curve can yield poor tracking results, and the bottom-left plot shows that even when it is fined-tuned for this specific example, it is slower to converge than our method during MFT. Each tracking sample corresponds to one k-space line, which means that even in the better case (bottom-left), the slow convergence of the UKF means that it can corrupt several hundred k-space lines, which will result in lower quality images. 4.2

Simulated Prospective Motion Correction

For our second experiment, we simulated prospective motion correction using static motion estimates, UKF motion estimates, and our proposed method. It was necessary for this experiment to use simulated data to be able to identically repeat the same experiment for each of the estimators. Thus we used a 2D image of the American College of Radiology (ACR) phantom “acquired” in simulations with 2D radial sampling. During radial sampling, simulated translations and rotations were applied according to a predetermined ground truth trajectory of motions that were applied for each experiment identically. The same motion transformations were also applied to correspondence points and vectors, and noise was added to simulate noisy measurements from an electromagnetic tracker (as though it were placed on the phantom to track its motion). Each of the estimation procedures were applied to the same noisy sequence of data, and our method was used to predict one discrete time step into the future. Motion correction was applied “prospectively” in the simulated radial sampling for each of the three estimators. In addition, a gold standard radial reconstruction was produced by performing “prospective” correction using the ground truth motion parameters as input, which is essentially a radial reconstruction with no motion. The resulting reconstructed images are shown in Fig. 2. We also show zoomed in views of the dots in each image, which demonstrate the effects of the different types of estimation errors on the image details. The gold standard radial reconstruction was used to calculate normalized root mean squared errors (NRMSE) for each of the reconstructions based on motion estimates. Our method performed the best (NRMSE=0.08), the static estimate was second-best (NRMSE=0.10), and the UKF estimate performed the worst (NRMSE=0.13).

5

Conclusion

In this work we show that a dynamic system identification approach can be used to improve motion correction of MRI. In our experiments, we found that the performance of our proposed method was less sensitive to the parameter choice (of maximum allowed system order) than the UKF’s parameter choice (of the

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amplitude of the noise assumed to be driving the dynamics). Our results suggest that learning the dynamics from the data is better than ignoring them with a static estimate, and that both are better than assuming an incorrect model of the dynamics a priori.

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