SAMPLED-DATA H FILTERING FOR ROBUST KINEMATICS ESTIMATION: APPLICATIONS TO BIOMECHANICS-BASED CARDIAC IMAGE ANALYSIS

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1 SAMPLED-DATA H FILTERING FOR ROBUST KINEMATICS ESTIMATION: APPLICATIONS TO BIOMECHANICS-BASED CARDIAC IMAGE ANALYSIS Shan Tong 1, Albert Sinusas 2, and Pengcheng Shi 1,3 1 Biomedical Research Laboratory, Department of Electrical and Electronic Engineering Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong 2 Section of Cardiology, Yale University School of Medicine, New Haven, CT 06520, U.S.A. 3 College of Biomedical Engineering, Southern Medical University, Guangzhou, China ABSTRACT A sampled-data H filtering strategy is proposed for the estimation of cardiac kinematic functions from periodic medical image sequences. Given the biomechanics-based myocardial dynamics, stochastic multi-frame filtering frameworks are constructed to deal with the parameter uncertainty of the biomechanical constraining model and the noisy nature of the imaging data in a coordinated fashion. As robustness is of paramount importance in cardiac wall motion estimation, especially for clinical applications, this mini-max H strategy is particulary powerful for real-world problems where the types and levels of model uncertainties and data disturbances are not available a priori. For the hybrid cardiac analysis system with continuous dynamics and discrete measurements, the state estimates are predicted according to the continuous-time state equation between observation time points, and then updated with the new measurements obtained at discrete time instants, yielding physically more meaningful and more accurate estimation results for the continuously evolving cardiac dynamics. The strategy is validated through synthetic data experiments to illustrate its advantages and on canine MR phase contrast images to show its clinical relevance. Index Terms H filtering Robust kinematics estimation, sampled-data are dealt with in such a coordinated effort, that patient-specific kinematics/material estimates can be obtained in some optimal senses [4, 5]. In [4, 6], Gaussian assumption with known statistics is made on system and data uncertainties, so that Kalman filter can be applied to obtain the minimum-mean-squared-error estimate for cardiac kinematics. In practical situations, however, the Gaussian assumption is typically unrealistic and the noise statistics is usually not available a priori. In order to relax such restrictions, H -based strategy has been proposed for the motion recovery problem [5]. However, as cardiac dynamics is a physiological process evolving continuously in time, the state estimation should be performed on the hybrid cardiac analysis system with continuous-time dynamics and discrete-time measurements. In [4, 5], the system dynamics is first converted to discrete time to obtain the predicted state estimates. Such conversion requires the assumptions of piecewise constant system input and time-invariant system, which are unrealistic in most real cases, especially for large sampling interval T. A transition matrix also needs to be calculated for the discretizing conversion, which is computationally very expensive for such systems with large degrees of freedom as in cardiac image analysis. 1. INTRODUCTION Quantitative and noninvasive estimation of cardiac kinematic functions has significant physiological and clinical implications, and accordingly there have been abundant efforts devoted to cardiac motion and deformation recovery from medical image sequences [1]. Given a set of imaging-derived, sparse, noisy measurements on cardiac kinematics, additional constraining models of mathematical or biomechanical nature are needed to constrain the ill-posed problem for a unique solution [1, 2, 3]. However, as all these constraining models are required to be known exactly as prior information, which is almost impossible in practical situations and especially for pathological cases, system uncertainties should be incorporated into the constraining models to obtain the flexibility of dealing with subject-dependent data sets. On the other hand, the imaging/imaging-derived measurements are corrupted by noises of various sources, so data uncertainties also need to be properly considered in the estimation process. Filtering strategies from control and estimation literature have been applied to couple the constraining model with the noisy observations, in which the system uncertainty of a priori constraining models and the data uncertainty of a posteriori noisy measurements In this paper, we present a sampled-data H filtering strategy for kinematics estimation of the hybrid cardiac analysis system. Instead of converting system dynamics to discrete time, state estimates are predicted according to the original continuous-time state equation and then updated with new measurements at discrete observation time points, which is physically more meaningful for the continuously evolving cardiac dynamics. As approximation errors in the discretization are avoided, estimates of higher accuracy are obtained, and such advantages of sampled-data filtering have been validated in its H 2 filtering counterpart [6]. Meanwhile, sampled-data H filtering differs from H 2-based estimation as follows. 1) No a priori knowledge of noise statistics is required. 2) The mini-max estimation criterion is to minimize the worst possible effects of the disturbances (process and measurement noises) on the state estimation errors, which will ensure that if the disturbances are small (in energy), the estimation errors will be as small as possible (in energy), regardless the noise types. These two aspects make H filtering more appropriate for such real-world problems as cardiac image analysis, where the noise statistics is unknown and robustness of the estimation is highly important. Experiments have been conducted on synthetic data to validate the advantages of this strategy, and also on canine MR images to show its clinical relevance.

Method 20dB 30dB 20dB 30dB (Gaussian) (Gaussian) (Poisson) (Poisson) KF 2.91±0.65 2.74±0.49 5.76±0.82 4.62±0.61 H 2.92±0.63 2.87±0.

2 Method 20dB 30dB 20dB 30dB (Gaussian) (Gaussian) (Poisson) (Poisson) KF 2.91± ± ± ±0.61 H 2.92± ± ± ±0.51 (a) Original configuration and material composition of the object frames #4, #8, #12, (b) Deformed geometry at selected #16 Fig. 1. Generation of the 16-frame synthetic data sequence. 2. METHODOLOGY 2.1. Biomechanics-Based Myocardial Dynamics For computational simplicity, our current implementation adopts the linear isotropic continuum material for the myocardium in order to illustrate the basic ideas of the sampled-data H filtering strategy. More complicated material models and deformation formulation can also be incorporated into this filtering framework [7]. Under spatio-temporal biomechanical constrains, the governing dynamic equation for the myocardium is derived by applying the principle of minimum potential energy [4]: MÜ + C U + KU = R (1) where M, C and K are the mass, damping and stiffness matrices respectively, R is the load vector, and U is the displacement vector. M is a known function of the material density and is temporally and spatially constant. K is a function of the material constitutive law, and is related to the material-specific Young s modulus and Poisson s ratio. C is frequency dependent, and we assume small proportional Rayleigh damping with C = αm + βk in our implementation State Space Representation for Hybrid Cardiac Analysis System In order to apply filtering strategies to our estimation problem, which are typically seen in estimation and control literature [8, 9], the above dynamic equation (1) is transformed into a state-space representation of a continuous-time system: ẋ(t) = A(t)x(t) + B(t)w(t) + ṽ(t) (2) where the state vector x(t), the input vector w(t), the system matrix A(t), and the input gain B(t) are as follows: A(t) =» x(t) = [U(t), U(t)], w(t) = [0, R(t)],, B(t) = 0 I M 1 K M 1 C» M 1, and ṽ(t) is the process noise describing the disturbances/uncertainties in cardiac dynamics. As the material model parameters and the geometry of heart will change over time, this state-space equation represents a time-varying system. Although cardiac dynamics has a continuously evolving nature, in medical image analysis and other real life systems, the system states can be observed only at discrete points in time, yielding available measurements only at discrete observation time instants. As a result, the observations provided by the imaging/imaging-derived data are expressed in a discrete-time measurement equation: y(k) = Dx(k) + e(k) (3) Table 1. Differences between the ground truth and the KF/H estimated nodal positions, (mean error ± standard deviation) Noise Type (20dB) Average Positional Error ( 10 2 ) Gaussian 2.92±0.63 Poisson 2.93±0.52 Uniform 2.91±0.55 Rayleigh 2.88±0.58 Exponential 2.89±0.62 Table 2. Comparison of average nodal positional errors from sampled-data H filter results under various types of noise. where D is the measurement matrix, e(k) is the discrete-time measurement noise accounting for the imaging data uncertainties, and k is used to denote the observation time instant t = kt. In summary, the realistic system model for cardiac image analysis is a hybrid system with continuous-time dynamics and discretetime measurements, which is represented by equations (2) and (3) Sampled-Data H Filtering for Hybrid Cardiac Analysis System In order to achieve robustness in kinematics estimation, sampleddata H filtering is applied to the hybrid system described by equations (2) and (3), which aims to estimate the states of a continuoustime system given sampled measurements of the output. The performance of the sampled-data H filter is measured by the size of the estimation error relative to the sizes of the process noise, the measurement noise, and the uncertainty in the initial state, which is defined as follows: J = x(t) ˆx(t) 2 ṽ(t) 2 + e(k) 2 + (x o ˆx o) R 0(x 0 ˆx o) Unlike the discrete H filter in [5], the performance measure of the sampled-data H filter is defined directly in terms of the continuous-time system state x(t) and disturbance ṽ(t), and intersample behavior of the system is thus taken into account. The denominator of J can be regarded as mixed L 2/l 2 norm on the uncertain disturbances affecting the system. The weighting matrix R 0 is a measure of the relative importance of the uncertainty due to the unknown initial state x o. A larger R 0 reflects smaller uncertainty in x o relative to the uncertainty in ṽ(t) and e(k). Given a prescribed noise attenuation level γ > 0, the sampleddata H filter will search ˆx(t) such that the optimal estimate of x(t) among all possible ˆx(t) should satisfy (4) sup J γ 2 (5) where the supremum is taken over all possible disturbances and initial states. The sampled-data H filter can be interpreted as a minimax problem where the estimator strategy plays against the exogenous disturbances ṽ(t), e(k) and the uncertainty in the initial state x o. The problem formulation in equation (5) guarantees the bounded estimation error over all possible disturbances of finite energy, regardless of the noise statistics. As a result, the filter achieves greater

Fig. 2. Comparison of displacement magnitude maps. From top to bottom: ground truth, estimation from H filter, estimation from KF. From left to right, frames #4, #8, #12 and the color scale.

uncertainties are not available a priori.

3 Fig. 2. Comparison of displacement magnitude maps. From top to bottom: ground truth, estimation from H filter, estimation from KF. From left to right, frames #4, #8, #12 and the color scale. (Poisson noise, SNR = 20dB) robustness to disturbance variations and is well suited to such realworld problems as in cardiac analysis, where the types and levels of system disturbances and data uncertainties are not available a priori. The sampled-data H filtering algorithm for the hybrid cardiac analysis system of equations (2) and (3) is given as follows [9]: ˆx(t) = A(t)ˆx(t) + B(t)w(t) (6) ˆx(kT ) = ˆx(kT ) + P (kt )D [y(k) Dˆx(kT )] (7) where ˆx(kT ) = lim ε 0 ˆx(kT ε), and P (kt ) is the stabilizing solution to the following Riccati equation with jumps: P (t) = A(t)P (t) + P (t)a(t) + P (t)2 + I (8) γ 2 P (kt ) = P (kt )[I + D DP (kt )] 1 (9) with the initial condition P (0 ) = R 1 0. The filter given above is a linear system with finite jumps at discrete instants of time, which also has an intuitively appealing structure. In between the sampling instants, the state estimate evolves according to the continuous-time system dynamics, and the predicted state ˆx(kT ) is obtained by solving equation (6) on the time interval [(k 1)T, kt ], with the previous state estimate ˆx((k 1)T ) as the initial condition of the differential equation. Then at the observation time t = kt, the new measurement y(k) is used to update the estimate with the filter gain being P (kt )D. The filtering Riccati equation (8) is a Riccati differential equation quite similar to the continuous filtering Riccati differential equation in [10] with the measurement matrix equal to zero, as there is no measurement between observation time instants. P (kt ) is also obtained by solving the differential equation (8) with P ((k 1)T ) as the initial condition. It is necessary to point out that solving equation (8) is quite nontrivial and numerical integration is usually required. We adopt the Möbius schemes proposed in [11] for its ability of dealing with numerical instability, and the details are omitted here Discussions In [4, 5], the continuous-time state equation (2) in the hybrid system is first converted to discrete time and discrete Kalman or H filters are applied for kinematics estimation. This approach is reviewed briefly here for comparison purpose. After discretization, the state equation becomes: x(k + 1) = F (k + 1, k)x(k) + G(k)w(k) + v(k) (10) where the transition matrix F (k+1, k) = e AT, G(k) = A 1 (e AT I)B, and v(k) is the discrete-time process noise. Fig. 3. Comparison of strain maps at frame #8. From top to bottom: ground truth, estimation from H, estimation from KF and the color scales. From left to right: x-strain, y-strain, xy-strain. (Poisson noise, SNR = 20dB) It is necessary to point out that there are several implicit approximations/assumptions in the above converting process. First, the assumption that the input is piecewise constant (i.e. constant during each sampling interval) is needed for the conversion, which is not satisfied for large sampling interval T. Second, evaluation of the transition matrix F (k + 1, k) is nontrivial as it in general has no explicit form. If and only if the commutativity property and the time-invariant system assumption are satisfied, F (t + T, t) = e AT. From the above analysis, we can see that the accuracy of the estimate from discrete Kalman and H filters largely depends on the extent to which the discretized state equation (10) approximates the true continuous-time dynamics of equation (2). For large sampling interval T, the assumptions of piecewise constant input and timeinvariant system become unrealistic, and thus equation (10) is not a faithful representation of the original continuous-time state equation. In contrast, the sampled-data H filtering strategy evolves the state estimate and the filter gain according to the original continuoustime system dynamics through equations (6) and (8), and thus has several advantages over the system discretization approach. As cardiac dynamics has a continuously evolving nature, predicting in continuous time is closer to the underlying physical/physiological process, and thus physically more meaningful estimation results could be obtained. Without discretizing the state equation, the assumptions of piecewise constant input and time-invariant system are not required. In this way, the approximation errors introduced in the conversion process and propagated with the inaccurate transition matrix no longer exist, and more accurate estimates could be obtained. Such advantages of sampled-data H filtering over the discretization approach have been validated on its H 2 filtering counterpart [6], and we will focus on demonstrating the robustness of this sampled-data H strategy in next section due to space limit. And from a computational point of view, although the matrix exponential is only an approximation of the transition matrix in the system discretization approach, its calculation is nontrivial and computationally very expensive. In our strategy, such calculation is naturally avoided. Meanwhile, the sampled-data H filtering scheme can also deal with the nonuniform sampling case, offering more flexibility in the estimation process.

highlighted. 3. EXPERIMENTS AND VALIDATION 3.1.

4 Fig. 4. Top row: Canine MRI phase contrast data (from left to right): MR intensity, x-velocity, y-velocity. Bottom row (from left to right): displacement constraints on the endo- and epi-cardial boundaries, phase contrast velocity field, and TTC-stained post mortem myocardium with infarcted tissue highlighted. 3. EXPERIMENTS AND VALIDATION 3.1. Validation with Synthetic Data To validate the accuracy and robustness of this sampled-data H filtering strategy, synthetic experiments have been performed on an object undergoing deformation in the vertical direction with the bottom being fixed. As shown in Fig.1(a), the object consists of two different materials with Young s Modulus E inside = 105, E outside = 75, and Poisson s ratio ν = sampling frames of the cyclic motion are acquired (Fig.1(b)), with displacements for all nodal points as the ground truth data set. Then only the displacements at the boundary nodes are selected and added with different types and levels of noise to generate the partial and noisy measurements. Both sampled-data H filter and its H 2 counterpart continuousdiscrete Kalman filter (KF) are implemented for the kinematics estimation. The material parameters are fixed as E = 75, ν = 0.47 in the prior constraining model. For data frames #4, #8, #12, Fig.2 shows the displacement magnitude maps of the ground truth, and the estimation results of H filter and KF from the noisy data (Poisson noise, SNR=20dB). The strain maps at frame #8 are presented in Fig.3. Point-by-point positional errors under different levels of Gaussian and Poisson noises are computed as shown in Table.1 for quantitative assessment and comparison of the H and KF results. Overall, the KF results for Poisson-corrupted data are not satisfactory, which indicates that if the Gaussian assumptions on the noise statistics are violated, small noise errors may lead to large estimation errors for KF. On the other hand, H filter produces accurate and very similar results for two sets of data contaminated by different levels of Gaussian and Poisson noises, showing its desired robustness for real-world problems. Table.2 further demonstrates the robustness of sampled-data H filter, where essentially the same results are obtained for data with various types of additive noise Canine Image Application Fig.4 demonstrates the MR phase contrast image data set. Myocardial boundaries and frame-to-frame boundary displacements are extracted using a unified active region model strategy [12], and the displacements are used as our system input data along with the midwall phase contrast velocities. The infarcted tissue is highlighted in the triphenyl tetrazolium chloride (TTC) stained post mortem myocardium (Fig.4), which provides the clinical gold standard for the assessment of the image analysis results. Fig. 5. Estimated displacement magnitude, radial, circumferential, and RC shear strain maps for frame #9 (with respect to frame #1). The estimated radial (R), circumferential (C), and R-C shear strain maps are shown in Fig.5, with the material parameters in the model set to Young s modulus E = 75000P ascal, Poisson s ratio ν = Abnormality at the infarct region can be identified from the strain maps, and the most obvious difference is observed in the RC shear strain map, where the lower-right quarter of the myocardium has much larger strain than other normal tissues. These patterns are in good agreement with the highlighted histological results of TTC stained postmortem myocardium in Fig.4, demonstrating the clinical relevance of the filtering strategy. Acknowledgment Thanks to IBM PhD fellowship for supporting Shan Tong. This work is supported in part by Hong Kong Research Grants Council CERG- HKUST6151/03E and the 973 Program of China 2003CB REFERENCES [1] A.J. Frangi, W.J. Niessen, and M.A. Viergever, Threedimensional modeling for functional analysis of cardiac images: A review, IEEE Transactions on Medical Imaging, vol. 20(1), pp. 2 25, [2] J. Park, D.N. Metaxas, and L. Axel, Analysis of left ventricular wall motion based on volumetric deformable models and mri-spamm, Medical Image Analysis, vol. 1(1), pp , [3] P. Shi, A.J. Sinusas, R.T. Constable, and J.S. Duncan, Volumetric deformation analysis using mechanics-based data fusion: Applications in cardiac motion recovery, International Journal of Computer Vision, vol. 35(1), pp , [4] P. Shi and H. Liu, Stochastic finite element framework for simultaneous estimation of cardiac kinematic functions and material parameters, Medical Image Analysis, vol. 7, pp , [5] E.W.B. Lo, H. Liu, and P. Shi, H filtering and physical modeling for robust kinematics estimation, in IEEE International Conference on Image Processing, 2003, p [6] S. Tong, A. Sinusas, and P. Shi, Continuous-discrete filtering for cardiac kinematics estimation under spatio-temporal biomechanical constrains, to appear in International Conference on Pattern Recognition, 2006.

5 [7] C.L.K. Wong and P. Shi, Finite deformation guided nonlinear filtering for multiframe cardiac motion analysis, in MICCAI, 2004, pp [8] Y. Bar-Sharlom, X.R. Li, and T. Kirubarajan, Estimation with applications to tracking and navigation, Wiley, [9] W. Sun, K. M. Nagpal, and P. Khargonekar, H control and filtering for sampled-data systems, IEEE Transactions on Automatic Control, vol. 38, pp , [10] K. M. Nagpal and P. Khargonekar, Filtering and smoothing in an H setting, IEEE Transactions on Automatic Control, vol. 36, pp , [11] J. Schiff and S. Shnider, A natural approach to the numerical integration of riccati differential equations, SIAM Journal on Numerical Analysis, vol. 36(5), pp , [12] L.N. Wong, H. Liu, A. Sinusas, and P. Shi, Spatio-temporal active region model for simultaneous segmentation and motion estimation of the whole heart, in IEEE Workshop on Variational, Geometric and Level Set Methods in Computer Vision, 2003, pp

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