Figure 1. (a) Planar tagged image at enddiastole, (b) tagged image near end-systole, (c) tagged image near end-systole after tag planes were applied

Transcription

1 Joint Reconstruction of 2-D Left Ventricular Displacement and Contours from Tagged Magnetic Resonance Images Using Markov Random Field Edge Prior Litao Yan and Thomas S. Denney, Jr. Department of Electrical Engineering Auburn University Auburn University, Alabama 36849, USA Abstract Magnetic Resonance (MR) tagging has been shown to be a useful method for non-invasively measuring the deformation of the left ventricle (LV), during the cardiac cycle. By reconstructing a displacement eld based on the movement of the tag lines, one can compute myocardial contraction measures such as strain. Existing methods depend on user-dened LV contours, which require human intervention and are therefore the biggest bottleneck in the reconstruction process. In this paper we present a method for reconstructing 2-D LV deformation without user-dened contours. We use a compound Gauss-Markov random eld to model the 2-D vector displacement eld, which is parameterized by two closed and smooth contours. By iteratively optimizing the contours, the displacement eld, and the parameters, we obtain an estimate of the displacement eld and the contours. Experimental results on in vivo human data are presented that demonstrate the accuracy of our algorithm. 1. Introduction Magnetic resonance (MR) tagging has been shown to be a useful technique for non-invasively measuring the deformation of an in vivo heart. Planar tags [1] are at saturation planes which are applied to the LV at end-diastole. Images acquired on planes orthogonal to these tag planes show the tags as dark lines which are nearly straight in images taken shortly after enddiastole and are curved in later images, as shown in Figure 1a,b. With cardiac tagging, the deformation of the myocardium can be measured and used to compute quantitative measures of contraction such as strain. (a) (b) (c) Figure 1. (a) Planar tagged image at enddiastole, (b) tagged image near end-systole, (c) tagged image near end-systole after tag planes were applied orthogonal to those in (a). The cardiac motion reconstruction process usually consists of three steps [2, 3]. First the endocardial and epicardial contours of the LV are extracted from the image data with a semi-automated algorithm [4, 5]. Second, tag line positions are extracted from the image data. Contours are used in this step to ensure that only tags inside the myocardium are extracted. Third, the tag line positions are used to either t a parameterized model of cardiac deformation [6, 7, 8, 9, 10, 11] or reconstruct a dense displacement eld [2, 3]. These reconstruction algorithms use the endocardial and epicardial contours to prevent over-smoothing of the displacement eld near these boundaries. Extracting LV contours, however, requires a considerable amount of user intervention and, as a result, is time consuming. Typically 3-4 hours are required to process a 200 image study [12]. For MR tagging techniques to be clinically viable, the requirement of user-dened contours must be eliminated. Recently, algorithms have been developed for ex- 1

2 tracting tag lines that do not require user-dened contours [13, 14]. In [15], an algorithm was presented for reconstructing 2-D LV displacement without user-dened contours that used a truncated quadratic potential function to preserve motion discontinuities at the myocardial boundaries. While this algorithm was able to detect myocardial boundaries, it also detected spurious edges in certain regions of the myocardium, which caused the displacement eld to be under-smoothed in these regions. In this paper, we present a method for reconstructing 2-D left ventricular displacement without userdened contours. Motivated by the work of Figueiredo and Leit~ao [16], we model the the displacement eld as a compound Gauss-Markov random eld (CGMRF), which is a Gauss-Markov random eld model for the displacement vector eld together with a set of discrete binary edge variables. Typically, the state of each edge variable is estimated along with the random eld [16]. In our approach, however, the state of each edge variable is determined by two, smooth, closed contours, which we model as 1-D MRF's. The displacement eld and contours are jointly estimated using a maximum a posteriori (MAP) framework. We call this algorithm the joint displacement and contour reconstruction (JDCR) algorithm. This paper is organized as follows. In Section 2 we derive probabilistic models for the measurements, displacement eld, and contours. In Section 3, we derive the JDCR algorithm based on the models in Section 2. Experimental results are presented in Section 4 for both simulated and in vivo human data. Finally, conclusions are drawn in Section 5. 2 Model Development 2.1 Notation and Coordinate Systems Two coordinate systems are used to describe the location of points in space: a material coordinate system xed on the left ventricle (LV) with end-diastole as the reference state, and a spatial coordinate system xed in space so that the material and spatial coordinate systems coincide at end-diastole. The x and y axes are orthogonal to each other, and the x-axis is parallel to the top and bottom edges of a short-axis image. The rst step in the JDCR algorithm is the reconstruction of a dense 2-D spatial coordinate displacement eld given a collection of tag line points. The spatial coordinate displacement eld is dened on an N N regularly spaced grid called the spatial point grid. The grid spacing is equal to the pixel size in an p n d d r n undeformed tagline deformed tagline Figure 2. Tag line measurement model. The undeformed tag line position is known from the imaging protocol. The material point ~p moves to the spatial point ~r, but because motion parallel to the tag line is possible, only the projection of the displacement onto the tag line normal can be measured. MR image [2], and N is chosen so that the grid encloses all the tag line points. 2.2 Measurement Model Figure 2 illustrates our basic measurement model. The undeformed tag line positions are known from the imaging protocol. The material point ~p moves to the spatial point ~r, but because motion parallel to the tag plane is possible, only the projection of the displacement onto the tag line normal can be observed. Therefore, each point ~r on a tag line is a 1-D measurement of the spatial coordinate displacement given by m = ~n ~ d + w ; (1) where m is the 1-D displacement measurement, ~n is the tag line normal and d ~ dx = is the true displacement vector. w is additive white Gaussian noise representing measurement error. Measurements are obtained from orthogonal tag line orientations to compute the 2-D displacement of the LV (See Figure 1b,c). Our data model can be written in matrix form as d y m = Nd + w; (2) 2

3 where m is an M 1 vector containing all the onedimensional displacements obtained from the tag data. N is an M 2N 2 matrix where each row contains a 2-D normal vector. d is a 2N 2 1 vector containing all the displacement vectors dened on the spatial point grid in lexicographical order. w is a Gaussian noise vector with zero mean and covariance matrix 2 I. When a tag point is not exactly on the spatial point grid, we round its position to the nearest point on the spatial point grid. In order to use the MAP estimation framework in Section 3, we write the measurement model in Equation (3) as the conditional density function: p(mjd) = 1 p (2 2 ) M expf? (m? Nd) T (m? Nd)g: (3) 2.3 Displacement Field Model We model the spatial coordinate displacement eld of the LV as a Compound Gauss-Markov Random Field (CGMRF) where the edges are constrained to lie on two closed contours. We develop this model by rst describing a standard vector CGMRF. We then present our contour model and incorporate the contour model into the vector CGMRF Compound Gauss-Markov Random Field (CGMRF) Model A compound GMRF [17] is a Gauss-Markov random eld with the covariance matrix parameterized by a set of binary variables which denote the locations of edges in the vector eld. Such a variable with value 1 signies a discontinuity between two neighboring gridpoints. Otherwise, if the value is 0, a smoothness constraint is enforced between the neighboring grid points. In particular, we dene two sets of edges h and v, which describe horizontal and vertical discontinuities respectively and are given by h = fh ij 2 f0; 1g; i = 2; 3; ; N; j = 1; 2; ; N g v = fv ij 2 f0; 1g; i = 1; 2; ; N; j = 2; 3; ; N g where, for example, h ij = 1 breaks the smoothness constraint between (i; j) and (i? 1; j), v ij = 1 breaks the smoothness constraint between (i; j) and (i; j? 1). The conditional probability density function (PDF) of d given h and v is therefore given by p(djh; v) = expf? 2 1 Z d (h; v) i j=2! v v ij [(d i;jx? d i;j?1x ) 2 + (d i;jy? d i;j?1y ) 2 ]?! h hij [(d i;jx? d i?1;jx ) 2 2 i=2 j + (d i;jy? d i?1;jy ) 2 ]? [1? 2(! v +! h )](d 2 ij 2 x + d 2 ij y )g ; (4) i j where is a global smoothness parameter, and parameters! v and! h control the relative smoothness in the vertical and horizontal directions respectively. A shorthand v ij is used to represent 1? v ij, and same notation is used for the horizontal line variables. Z d (h; v) is the partition function that normalizes the PDF. It can be seen from Equation (4) that the rst-order difference is penalized only when no edge exists between two neighboring grid points. Hence the CGMRF model can preserve discontinuities in the displacement eld. We write Equation (4) in vector notation as p(djh; v) = 1 p (2) N 2 (h; v) expf?1 2 dt?1 (h; v)dg: (5) Note that if (1? 2(! v +! h )) > 0, then?1 (h; v) is positive denite for all possible edge maps h and v. In our problem there is no reason we should favor one of these two parameters over the other, so we use! v =! h =!. The role of! is to keep the density p(djh; v) from becoming improper. To minimize its eect on the displacement estimate, we set! = 0:2499 [16] Contour Model In our model the h and v maps are not arbitrary; the active (i.e. equal to 1) edge variables are constrained to lie on two smooth and closed contours { one for the endocardium (inner contour) and one for the epicardium (outer contour). We model each contour as a polygon whose vertices are dened by the radii along a discrete set of regularly spaced angles about a known center point of the LV as shown in Figure 3. For computational eciency, the radii are discretized so that they take on values from a nite set?. The radii for the inner contour are denoted by the vector C i = fr in (1); r in (2); ; r in (N in )g: (6) The outer contour vector C o is dened similarly with the same center point as C i. We model the contour vectors C i, C o as 1-D MRF's. The probability function for C i is given by X p(c i ) = 1 N i expf? Z in j [r in (j)? r in (j? 1)] 2 g ; (7) 3

4 r(j) v i+1, j 1 v i, j+1 h ij (i, j) r(1) O r(n) contour polygon v ij h i+1, j 1 (i+1, j 1) v i+1, j 1 Figure 3. Contour model: Each contour is modeled as a polygon with vertices defined by radii along a set of regularly spaced angles about a known center point of the LV. Figure 4. The h and v maps are defined by the contour polygon where Z in = X r in(j)2? expf? XN i j [r in (j)? r in (j? 1)] 2 g ; and is a user-dened constant. The probability function for C o is similarly dened. We assume that C i and C o are independent Combined CGMRF-MRF Model To determine the h and v maps given C i and C o, we rst form two closed polygons by sequentially connecting the radii with straight lines. If either of the contour polygons intersects with the line segment that links pixel (i; j) and pixel (i? 1; j), we set h ij = 1; similarly, if a polygon intersects with the line segment that links pixel (i; j) and pixel (i; j? 1), we set v ij = 1. Figure 4 shows the relationship between a contour polygon and the h and v maps. Since h and v are uniquely determined by C i and C o, we write the conditional density p(djh; v) in Equation (5) as p(djc i ; C o ) = 1 p (2) N 2 (h(c i ; C o ); v(c i ; C o )) expf? 1 2 dt?1 (h(c i ; C o ); v(c i ; C o ))dg: (8) 3 Reconstruction Algorithm In this section we derive the joint displacement and contour reconstruction (JDCR) algorithm based on the models derived in Section 2. First we formulate the maximum a posteriori (MAP) estimate of the spatial coordinate displacement eld, smoothness parameter, and contours, and derive a practical algorithm for computing the estimate. 3.1 MAP Estimate Based on the probabilistic models in the previous section, we are now able to derive a Maximum A Posteriori (MAP) estimate of d,, C i and C o given m. The MAP estimate is dened as where (^d; ^; ^C i ; ^C o ) MAP = arg p(mjd)p(djc i ; C o )p(c i )p(c o ) max d;;ci;co = arg min L(m; d; C i ; C o ) ; (9) L(m; d; C i ; C o ) =? ln p(mjd)? ln p(djc i ; C o )? ln p(c i )? ln p(c o ); and the probability densities are dened in Equations (3), (7), and (8). 3.2 Pseudolikelihood Approximation Computing ln p(djc i ; C o ) is a hard problem because of the diculty of computing the partition function. Motivated by the work of Figueiredo and Leit~ao [16], 4

5 we use an approximation proposed by Besag [18] to represent the PDF: p(d) Y ij p(d ij jd kl ; (k; l) 2 ij ) ; (10) where ij is the 4-point neighborhood of the point ij. The above equation states that the PDF of a CGMRF can be approximated by the products of the conditional PDF of each pixel given the neighborhood. Because d is a GMRF, all the conditional PDF's are also Gaussian. Using the Gibbs-MRF equivalence property [19], it can be shown [16] that the conditional PDF in Equation (10) is given by where p(d ij j ij ; h; v) N( ij ; 2 ij) ; (11) ij = 2 ij![ h ij (d i?1jx + d i?1jy ) + v ij (d ij?1x + d ij?1y ) and + h i+1j (d i+1jx + d i+1jy ) + v ij+1 (d ij+1x + d ij+1x )]; (12) 2 1 ij(h; v; ) = [1?!( h ij + v ij + h i+1j + v ij+1 )] : (13) The dependence of h and v on C i and C o has been suppressed for notational convenience. Note that the variance is a function of the edge maps h and v and the global smoothness parameter. After some manipulations, the overall objective function to be minimized for the MAP estimate is L(d; C i ; C o ; ) ~ L(d; ; Ci ; C o ) = 2!f + i=2 j i j=2 + (1? 4!) v ij [(d i;jx? d i;j?1x ) 2 +(d i;jy? d i;j?1y ) 2 ] h ij [(d i;jx? d i?1;jx ) 2 + (d i;jy? d i?1;jy ) 2 ]g i j + 1 X M 2 ( d ~ i ~n i? m i ) 2 i + N 2 ln 2 + X i j [d 2 ij x + d 2 ij y ] ln( 2 ij(h; v; )) N in + 4 (r in (j)? r in (j? 1)) 2 j X N out + 4 (r out (j)? r out (j? 1)) 2 (14) j The parameters, 2 and act as weights that control the relative signicance of the dierent terms in the summation given by the above equation. An estimate of 2 is obtained from the tag tracking algorithm [20]. and! are chosen by the user. is estimated along with the displacement eld and contours as described below. 3.3 Optimization Partial Optimal Solution Globally minimizing ~ L(d; ; Ci ; C o ) in Equation (14) is computationally prohibitive because it involves both real and combinatorial optimization, and ~L(d; ; C i ; C o ) is non-convex. Therefore, we obtain a partial-optimal solution [16] by the following iterative procedure: Fix and contours, update the displacement eld Fix the contours and the displacement eld, update Fix the displacement eld and, nd the best estimate of the contours based on Equation (14) The approach we take for the contour estimation is based on Besag's [18] iterated conditional modes (ICM) procedure; we use a discrete line-search based on the previous contour update. For each radius in a particular angle, the algorithm inspects the change in ~L(d; ; C i ; C o ) on a set of discrete points along the radial line and chooses whichever causes L(d; ~ ; Ci ; C o ) to be minimum with all other radii xed. This procedure is repeated for each angle. The update of is obtained analytically by solving the L(d;;Ci;Co) j =^ = 0. Finally, the displacement eld update with and the contours xed is a quadratic optimization problem. The optimal displacement eld is obtained by solving the linear equation [?1 (h; v) NT N]d = 1 2 NT m; (15) where N and m are dened in Equation (2). The matrix on the left hand side of (15) has a nearest neighbor structure. We solve (15) using a conjugate gradient based sparse system solver [21]. By repeating the above procedures, we end our algorithm when the relative change of is less than 2% and there are no further contour position changes. 5

6 3.3.2 Initial Condition for the Contours The function ~ L(d; ; Ci ; C o ) is non-convex; so a good initial guess of the contours will help avoid local minima. As a starting point, we initialize the contour radii by performing a morphological closing of the deformed tag lines followed by a boundary extraction from the resulting binary image. The LV center point used to dene the radii is obtained from the image prescription Transformation to Material Coordinates Once the spatial coordinate displacement eld is reconstructed, it is transformed to material coordinates using the interpolation based algorithm in [2, 3]. 4 Experimental Results 4.1 Methods Simulated Data To evaluate the JDCR algorithm, we performed an experiment using tag line data from a simulated LV deformation. First we generated a set of tag line data for a single image plane by deforming two orthogonal sets of parallel lines with the LV deformation model in [22] as shown in Figure 5. We then reconstructed the material coordinate displacement eld and contours using the JDCR algorithm. We used = 25 (chosen by trial and error) and 2 = 1. For in vivo data, the tag line can be identied to within 0:3mm [20], but we use a larger value of 2 to account for the error incurred in rounding tag point measurements to the nearest grid point. The root-mean-square (RMS) error between the true and JDCR displacement elds was computed using the formula s 1 X RMS Error = P ij jj ~ d JDCR ij? ~ d True ij jj 2 ; (16) where the sum is over all grid points that were both in the simulated LV and inside the reconstructed contours, and P is the number of points in the sum In Vivo Human Data We evaluated our algorithm on two sets of tag lines extracted from an imaging study of a normal human volunteer. Seven short-axis planes were imaged at ten time frames during the systolic portion of the cardiac cycle. Each plane was imaged twice per time frame using a parallel tagging protocol [1]. The tag planes in the second acquisition were rotated 90 degrees relative to the rst acquisition. The rst set of tag line data was extracted using user-dened contours [4, 5] and was veried by an expert user. The second set of tag line data was extracted without user dened contours [13]. Since user-dened contours were not used in this data set to limit the domain over which the tags were extracted, this data set contains tag points outside the myocardium. The JDCR algorithm was run for both sets of tag line data with the same set values for and 2 used for the simulated data. Validation of in vivo displacement elds is dicult because the true displacement eld is not known. To assess the accuracy of the JDCR algorithm, we compared the JDCR displacement eld to the displacement eld computed from the same tag line data with the irregular domain displacement estimate (IDDE) algorithm of Denney and Prince [2, 3]. This algorithm uses a rst order smoothness model for the displacement eld that is similar to the one used in the JDCR, but user-dened contours are used to limit the domain of the reconstruction to the myocardium. While the IDDE is not the true displacement eld, it is close to the displacement eld we would reconstruct if the JDCR contour reconstruction were perfect. The IDDE was computed for each of the two in vivo tag line data sets described above using a set of user-dened, expert edited contours. Tag points outside the user-dened contours were removed from the second (contour-free) tag line data set before the IDDE was computed. For each slice and time frame, the RMS dierence was computed between the JDCR and IDDE displacement elds over the grid points interior to both the JDCR and user-dened contours using the formula described above with d ~ IDDE ij instead of d ~ True ij. 4.2 Results The results from the simulated tag data experiment are shown in Figure 5. The reconstructed displacement eld and contours are quite close to the true values. The RMS error between the true and JDCR displacement elds was mm while the RMS value of the true displacement eld was 2.50mm. The RMS values were computed over approximately 1649 grid points. The results for the rst set of in vivo tag line data (tags extracted with user-dened contours) for a midventricular slice are shown in Figure 6. The rst row shows our initial guess of the contours for delay times of 47, 150 and 340 ms after detection of the QRS complex. The second row shows the nal JDCR contour estimates. The third row shows the material displacement eld for the three time frames, and the last row shows the dierence between the JDCR and IDDE dis- 6

7 (a) (b) (c) (d) Figure 5. Experimental results using simulated tag line data: (a) deformed tag lines, (b) true material coordinate displacement field, (c) reconstructed contours, (d) reconstructed material coordinate displacement field. The displacement fields are decimated by a factor of 3. placement elds. Note that the dierences are quite small and are fairly evenly distributed across the myocardium. The results for the second set of in vivo tag line data (tags extracted without user-dened contours) for the same mid-ventricular slice are shown in Figure 7. Note that this set of tag line data contains tag points outside the myocardium, and the JDCR does a good job of distinguishing between tag points inside versus outside the myocardium, even when the initial contour estimate is fairly rough as shown in the rst row of Figure 7. The global smoothness parameter estimated for each time frame during this reconstruction is plotted in Figure 8. The estimated decreases with time frame, which re- ects the increase in spatial change in the displacement eld (strain) as the LV contracts. The RMS dierences between the JDCR and IDDE displacement elds for the seven short-axis slices are plotted Figure (9) for the contour-based tag lines and in Figure (10) for the contour-free tag lines. The number of points over which the RMS values are computed is dierent in each slice and time frame, but is approximately 1500 points. Slice 0 is near the base of the LV and slice 6 is near the apex. In each plot, the RMS dierence is expressed as a percentage of the RMS magnitude of the IDDE displacement eld. In both plots, the dierences are on the order of 10% for all slices and all time frames except the rst two. In the rst two time frames the percent error is inated because 47 ms 150 ms 340ms Figure 6. JDCR reconstruction from tag lines tracked with user-defined contours. First row: tag lines overlaid with initial contour estimate. Second row: tag lines overlaid with final contour estimate. Third row: material coordinate displacement field reconstructed by the JDCR algorithm. Fourth row: difference between the JDCR displacement field and the IDDE displacement field. Times are measured from the QRS. Vector fields are decimated by a factor of 3. the IDDE displacement eld is small in magnitude. The dierences between the JDCR and IDDE displacement elds for the contour-based tag line data can be attributed to dierences in regularization parameters and the way measurements are modeled (the JDCR rounds tag points to the nearest grid point, the IDDE uses bilinear interpolation). The percent errors for the contour-free tag line data are slightly higher on average than the corresponding errors for the contour-based tag line data. This slight increase is due to the inuence of tag points outside the myocardium, which tend to make the JDCR contours enclose a dierent domain than the user-dened contours. This eect is more prominent in the rst two time frames because the displacement gradients across the myocardial boundaries are smaller in the rst two time frames than in later time frames. 7

8 mu Time from QRS (ms) 47 ms 150 ms 340ms Figure 8. The global smoothness parameter estimated during the JDCR reconstruction for a mid ventricular slice of the LV. Figure 7. JDCR reconstruction from tag lines tracked without user-defined contours. First row: tag lines overlaid with initial contour estimate. Second row: tag lines overlaid with final contour estimate. Third row: material coordinate displacement field reconstructed by the JDCR algorithm. Fourth row: difference between the JDCR displacement field and the IDDE displacement field. Times are measured from the QRS. Vector fields are decimated by a factor of 3. 5 Conclusions % RMS Difference slice 0 (base) slice 1 slice 2 slice 3 slice 4 slice 5 slice 6 (apex) In this paper, we presented a method called the JDCR for jointly estimating the myocardial displacement eld and contours from the positions of tag lines in planar tagged MR images of the LV. The JDCR uses a compound Gauss-Markov random eld model for the displacement eld, but the active edge variables are constrained to lie on two, smooth, closed contours, which we model as 1-D Markov random elds. Our experimental results on in vivo human data demonstrate that the JDCR can accurately reconstruct LV displacement without user-dened contours even when tag points are identied outside the myocardium. This result is important because contour-free tag tracking [13, 14] combined with the contour-free JDCR reconstruction method has the potential to eliminate the Time from QRS (ms) Figure 9. RMS difference between JDCR and IDDE material coordinate displacement fields as a percentage of the RMS magnitude of the IDDE displacement field. Both displacement fields were reconstructed from tag lines tracked with user-defined contours. 8

9 % RMS Difference slice 0 (base) slice 1 slice 2 slice 3 slice 4 slice 5 slice 6 (apex) Time from QRS (ms) Figure 10. RMS difference between JDCR and IDDE material coordinate displacement fields as a percentage of the RMS magnitude of the IDDE displacement field. Both displacement fields were reconstructed from tag lines tracked without user-defined contours. need for user intervention in the cardiac deformation and strain reconstruction process. In future work, we plan to validate the JDCR on a large number of normal and ischemic human heart imaging studies and extend our methods to 3-D LV displacement reconstruction. Acknowledgement This work was supported by a Biomedical Engineering Research Grant from the Whitaker Foundation. References [1] E.R. McVeigh and E. Atalar. Cardiac tagging with breath-hold cine MRI. Magnetic Resonance in Medicine, 28:318{327, [2] T.S. Denney Jr. and J.L. Prince. Reconstruction of 3D left ventricular motion from planar tagged cardiac MR images: an estimation theoretic approach. IEEE Transactions on Medical Imaging, 14(4):625{635, December [3] T.S. Denney Jr. and E.R. McVeigh. Modelfree reconstruction of three-dimensional myocardial strain from planar tagged MR images. Journal of Magnetic Resonance Imaging, 7(5):799{810, September/October [4] M.A. Guttman, J.L. Prince, and E.R. McVeigh. Tag and contour detection in tagged MR images of the left ventricle. IEEE Transactions on Medical Imaging, 13(1):74{88, [5] M.A. Guttman, E.A. Zerhouni, and E.R. McVeigh. Analysis and visualization of cardiac function from MR imgages. IEEE Computer Graphics and Applications, 17(1):30{38, January [6] A.A. Young, D.L. Kraitchman, L. Dougherty, and L. Axel. Tracking and nite element analysis of stripe deformation in magnetic resonance tagging. IEEE Transactions on Medical Imaging, 14(3):413{421, Septmeber [7] M. J. Moulton, L. L. Creswell, S. W. Downing, R. L. Actis, B. A. Szabo, M. W. Vannier, and M. K. Pasque. Spline surface interpolation for calculating 3-D ventricular strains from MRI tissue tagging. American Journal of Physiology, 270:H281{H297, [8] W.G. O'Dell, C.C. Moore, W.C. Hunter, E.A. Zerhouni, and E.R. McVeigh. Displacement eld tting for calculating 3D myocardial deformations from parallel-tagged MR images. Radiology, 195: , [9] A. Amini, R. Curwen, R.T. Constable, and J.C. Gore. MR physics-based snake tracking and dense deformation from tagged cardiac images. In American Association for Articial Intelligence (AAAI) Spring Symposium Series. Applications of Computer Vision in Medical Image Processing, pages 126{129. The AAAI Press, March [10] P. Radeva, A. Amini, and J. Huang. Deformable B-solids and implicit snakes for 3D localization and tracking of SPAMM MRI data. Computer Vision and Image Understanding, 66(2):163{178, May [11] A. Amini, R. Curwen, and J. Gore. Snakes and splines for tracking non-rigid heart motion. In Cipolla and Buxton, editors, Lecture Notes in Computer Science, volume Springer-Verlag, Berlin, April [12] A. Bazille, M.A. Guttman, E.R. McVeigh, and E.A. Zerhouni. Impact of semi-automated versus manual image segmentation errors on myocardial 9

10 strain calculation by MR tagging. Radiology, 29(4):427{433, Investigative [13] T.S. Denney Jr. Identication of myocardial tags in tagged MR images without prior knowledge of myocardial contours. In XVth International Conference on Information Processing in Medical Imaging, Poultney, Vermont, June [14] M.A. Guttman, E.A. Zerhouni, and E.R. McVeigh. Fast, contourless tag segmentation and displacement estimation for analysis of myocardial motion. In Proc. SMR/ESMRMB, volume 1, page 41, Nice, August SMR. [15] L. Yan and T.S. Denney. 2-D motion estimation of left ventricle from tagged MR images using edgepreserving regularization. In Proceedings of the 1998 SPIE International Symposium on Medical Imaging, San Diego, CA, February [16] M. Figueiredo and M. Leit ao. Unsupervised image restoration and edge location using compound gauss-markov random elds and mdl principle. IEEE Transactions on Image Processing, 6(8):1089{1102, August [17] F. Jeng and J. Woods. Image estimation by stochastic relaxation in the compound gaussian case. In Proc. IEEE ICASSP'88, pages 1016{1019, New York, [18] J. Besag. On the statistical analysis of dirty pictures. J. Royal Stat. Soc. B, 48:259{302, [19] S. Z. Li. Markov Random Field Modeling in Computer Vision. Springer-Verlag, [20] E. Atalar and E.R. McVeigh. Optimization of tag thickness for measuring position with magnetic resonance imaging. IEEE Transactions on Medical Imaging, 13(1):152{160, [21] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling. Numerical Recipes in C. Cambridge University Press, [22] E. Waks, J.L. Prince, and A. Douglas. Cardiac motion simulator for tagged mri. In Proceedings of the IEEE Workshop on Mathematical Methods in Biomedical Image Analysis, San Francisco, CA, June