IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. Y, MONTH 1999
100
Estimation and Detection of Myocardial Tags in MR image Without User-De ned Myocardial Contours Thomas S. Denney Jr.
Abstract | Magnetic resonance (MR) tagging has been shown to be a useful technique for non-invasively measuring the deformation of an in vivo heart. An important step in analyzing tagged images is the identi cation of tag lines in each image of a cine sequence. Most existing tag identi cation algorithms require user-de ned myocardial contours. Contour identi cation, however, is time consuming and requires a considerable amount of user intervention. In this paper, a new method for identifying tag lines, which we call the ML/MAP method, is presented that does not require user-de ned myocardial contours. The ML/MAP method is composed of three stages. First a set of candidate tag line centers are estimated across the entire region-ofinterest (ROI) with a snake algorithm based on a maximumlikelihood (ML) estimate of the tag center. Next a maximum a posteriori (MAP) hypothesis test is used to detect the candidate tag centers that are actually part of a tag line. Finally a pruning algorithm is used to remove any detected tag line centers that do not meet a spatio-temporal continuity criterion. The ML/MAP method is demonstrated on data from ten in vivo human hearts. Keywords | MR tagging, feature extraction, MR cardiography, image sequence analysis.
M
I. Introduction
AGNETIC resonance (MR) tagging [1,2] techniques have shown great potential for non-invasively measuring the local mechanical wall function (strain) in an in vivo left ventricle (LV). Tagged images appear with a spatially encoded pattern of dark lines called tag lines that move with the myocardium and can be analyzed to reconstruct LV deformation and strain. MR tagging techniques have largely remained in the research setting, however, because of the time and manual intervention required for a user to identify the left-ventricular contours in each image in the study. Typically 3-4 hours are required to process a 200 image study [3]. In a clinical setting, however, a strain map must be reconstructed quickly and with a minimal amount of user intervention. For MR tagging techniques to be viable in this setting, the requirement for user-de ned contours must be eliminated. The basic procedure for analyzing tagged MR images consists of three steps. First the user identi es the endocardial and epicardial contours of the LV in each image with a semi-automated algorithm [4,5]. Second, tag line positions are identi ed (tracked) in each image. Third, the tag line positions are used to t either a parameterized model of cardiac displacement [6,7,8,9,10,11]. or reconstruct a T.S. Denney is with the Department of Electrical and Computer Engineering, Auburn University, Auburn, Alabama. E-mail:
[email protected]. This work was supported by a Biomedical Engineering Research Grant from the Whitaker Foundation..
dense displacement eld [12,13]. Strain is then computed by di erentiating the deformation model or displacement eld with respect to spatial coordinates. User-de ned LV contours are used in both the tag tracking and displacement reconstruction steps. For MR tagging techniques to be clinically viable, the need for user-de ned contours must be eliminated in both of these steps. In this paper, we propose a method for tracking tag lines that eliminates the need for user-de ned contours. The problem of reconstructing LV displacement without user-de ned contours will be addressed in a future paper. Several methods have been proposed for identifying tag lines in tagged images, but most require user-de ned epicardial and endocardial contours. Guttman, et al. [4] used a template matching approach along with a least-squares error criterion. Contours are used in this approach to restrict the domain over which the template match is performed. Young, et al. [6], Amini, et al. [9], Radeva, et al. [14], and Kumar and Goldgof [15] used snakes to identify tag lines based on image intensity and spatial continuity constraints. Contours are used in these approaches to break the continuity constraints at the epicardial and endocardial boundaries. In [5], Guttman, et al. proposed a tag identi cation algorithm that does not require contours. In this approach, a snake based tag identi cation algorithm is applied to a a high-pass ltered version of the original image data. A heuristically chosen threshold applied to the high-pass ltered image is used to determine which snake points are part of a tag line, and spatial continuity constraints are only applied to these points. In this paper a new method for identifying tag lines without user-de ned contours, which we call the ML/MAP method, is presented. In this method, the tag identi cation problem is formulated as a combined parameter estimation and signal detection problem. The ML/MAP method is composed of three stages. First a set of candidate tag line centers are estimated across the entire region-of-interest (ROI) with a snake algorithm [16,17] based on a maximumlikelihood (ML) estimate of the tag center. A maximum a posteriori (MAP) hypothesis test is used to detect boundaries between myocardium and non-myocardium regions of the image. Spatial continuity of the snakes is not enforced across these boundaries. In the second stage, another MAP hypothesis test is used to detect the candidate tag centers that are actually part of a tag line. The third stage consists of a pruning algorithm that removes any detected tag centers that do not meet a spatio-temporal continuity cri-
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. Y, MONTH 1999
101
s(x)
FWHM
1
1-A/2
A
x
Fig. 2. Intensity pro le of a tag along a line perpendicular to the tag line.
A. Tag Line Model It is known from the imaging protocol [18] that along a line perpendicular to the tag line, tags have approximately a Gaussian pro le [4,9,19] as shown in Figure 2. We model this pro le s(x) as
s(x) = 1 ? A(t)e?(4 ln 2)x2 =FWHM2 ;
Fig. 1. Tagged images of the left ventricle of a normal human volunteer at early systole (47 ms) and end-systole (340 ms).
terion. This paper is organized as follows. The tag estimation, detection, and pruning algorithms are developed in Section II. In Section III these algorithms are demonstrated on data from ten in vivo normal human hearts. Conclusions and directions for future work are presented in Section IV. II. Algorithm Development
(1)
where 0 A(t) 1 is the tag amplitude, and FWHM is the full width at half maximum of the tag, which is determined by the imaging protocol [18]. The tag pattern amplitude decays with time due to T1 relaxation [1]. We model this decay as A(t) = e?t=T1nom ; where T1nom is the nominal T1 of the myocardium listed in Table I. The contraction and expansion of the myocardium during the cardiac cycle will cause the FWHM of the tag to change with time and also with position along the tag line. The tag pro le will also skew in some parts of the myocardium. For the sake of computational simplicity, however, we will assume a symmetric tag pro le and the same FWHM for each tag pro le in the image. Tagged MR image intensity is a function of the tag pattern, proton density, T1 , T2, the imaging protocol and noise (precise mathematical models are given in [20,21,22]. For the purposes of tag estimation and detection, however, we use a simpli ed model that approximates the intensity pro le in the neighborhood of a single tag line as the intensity of the untagged myocardium mmyo(x), which includes T1 , T2 , and other e ects, multiplied by the tag pro le s(x) plus noise [19]
Tags are at saturation planes which are applied to the heart usually at end-diastole [1,2]. As shown in Figure 1, images acquired on planes orthogonal to these tag planes show the tags as dark lines which are nearly straight in images taken shortly after end-diastole and are curved in later images. In this section we use a priori knowledge of the tagging process to derive a tag line model, which we w(x) = mmyo(x)s(x ? ) + n(x) ; (2) then use to develop algorithms for estimating and detecting tag lines. We then use a priori knowledge of the spatial and where is the position of the tag center, which changes temporal continuity of tag lines to develop an algorithm for from time frame to time frame in a cine sequence due to removing false tag point detections due to noise. the motion of the myocardium.
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. Y, MONTH 1999
B. Tag Estimation The problem of estimating tag lines consists of estimating a set of candidate tag line centers from the image data. In the case of parallel vertical tags as shown in Figure 1, for each tag we estimate a candidate tag center for each image row.
B.1 Maximum Likelihood Estimate of the Tag Center We rst consider the estimation of a single tag center from a set of Ns image pixels along a line perpendicular to the tag line. These pixels can be stacked into an Ns -vector w 1. These image pixels are samples of the intensity pro le in Equation (2), so we can express them as
w = S()mmyo + n ;
(3)
where mmyo and n are Ns -vectors of samples of the myocardium and noise signals respectively, and
2 s(?) 66 s( ? ) S() = 4
s((Ns ? 1) ? )
3 77 5;
where is the image pixel spacing. The vector mmyo contains the local myocardial signal intensity in the vicinity of the tag line. In order to use the statistical estimation and detection methods described in the sequel, we assume the myocardial signal has a Gaussian probability distribution [25,26,27]. The myocardium signal intensity can vary with spatial position in a given slice, so we model mmyo myo = m as a Gaussian random vector with mean m myo1, where 1 is an Ns -vector of ones, and covariance matrix 2 ji?j j . This model allows the local myCmyo(i; j ) = myo myo ocardial signal to vary smoothly within the vector w and to vary with each tag center in the image. The myocardial signal varies from subject to subject, from scan to scan, and even from time frame to time frame, so values 2 are estimated from the image data for for m myo and myo each time frame in a cine sequence (see Section III). The parameter myo describes the amount of spatial correlation between the myocardial signal samples in mmyo. A value of myo = 0 means that the samples are uncorrelated (white noise), while a value of myo close to one means that the signal varies smoothly with spatial position. The value of myo could in principle be estimated from the image data, but for the sake of computational simplicity, we use a single value of myo = 0:9. The noise vector n is modeled as a zero mean white Gaussian vector [32] with covariance n2 I. A value for n2 is estimated from the image data for each time frame in the sequence (see Section III). We formulate the estimation of the tag center from the data w as a maximum likelihood estimation problem. In 1 In general the pixels in w can be obtained by interpolating along
an estimated local tag line normal. For vertically oriented tags, the implementation of the ML/MAP algorithm presented in this paper uses a set of Ns pixels along an image row. When the tag lines deform, this choice of w results in an increase in the FWHM of the tag pro le, but this e ect has not been found to be signi cant.
102
this formulation, is considered to be an unknown parameter. An Ns -variate probability density function (PDF) is derived for w as a function of , and the optimal tag center estimate ^ is the tag center that maximizes the PDF. Given the above models for the myocardium signal, tag line, and noise, the observation vector w is a Gaussian random vector whose PDF is parameterized by . The PDF of the observation vector is given by [23] 1 p f (w; ) = (2)Ns =2 jCT ()j 1 ? 1 T exp ? 2 [w ? mT ()] CT ()[w ? mT ()] ; (4) wherej j denotes determinant, mT () is the mean of w given by mT () = S()m myo ; and CT () is the covariance matrix of w given by
CT () = S()CmyoS() + n2 I : After taking the natural logarithm of f (w; ) and deleting
constant terms, the maximum likelihood estimate of the tag center ^ is given by
^ = arg min L(; w) ;
(5)
where
T ()]T C?T 1 ()[w ? m T ()] L(; w) = 21 [w ? m + 21 ln jCT ()j (6)
where ln() denotes natural logarithm. A plot of L(; w) for a 5-pixel neighborhood of a tag line in an in vivo heart image is shown in Figure 3. Note than in actual images, it is possible that w will contain portions of pro les from neighboring tags. Minimizing L(; w) will still result in the true tag center provided the optimization algorithm is started suciently close to the true tag center. The issues of optimizing L(; w) and initialization are addressed in Section II-B.2. The log likelihood function in Equation (6) is similar in spirit to the tag energy functions on other model-based tag tracking methods [4,5,9,14] In the log likelihood function, however, the inverse covariance matrix C?T 1 () provides a set of weights for comparing the data and tag template based on the physics of the tagging process and image statistics. If the myocardium signal is assumed to be con2 = 0) and the stant with unit intensity (i.e. m myo = 1, myo noise is assumed to have unit variance, L(; w) reduces to a discrete template match cost function similar to the one proposed by Guttman, et al. [4,5] (7) L(; w) = 21 [w ? s()]T [w ? s()] ; where s() = S()1 is the tag template. If the inner product in (7) is expanded and terms 21 wT w and 21 sT ()s()
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. Y, MONTH 1999
103
energy function
100 Likelihood function Image Intensity
NX c ?1
w
[i j L(i j ; i j ) i=0 + i j Esep (i j+1 i j ) + i j Esep (i j
E ( j ) =
80
?
+Estretch (vj ) + Ebend (vj ) ;
60
(9)
where Nc is the number of centers in each tag line and vj is the displacement of the tag line j from its initial (undeformed) position3 pj such that j = pj + vj . L(i j ; wi j ) is the likelihood function de ned in (6) for the ith center on the j th tag line and i j is a normalizing factor to be de ned later. Esep is a tag separation constraint given by
40
20
0 108
? i j?1 )]
109
110 Image column
111
112
Fig. 3. A plot of the tag center log likelihood function L(; w) versus position (image column) for a 5-pixel neighborhood of a tag line in an in vivo heart image. The image pixel intensities are denoted by circles. The position where L(; w) is minimum is the maximum likelihood estimate of the tag center.
are removed2, L(; w) reduces to a correlation function similar to the one proposed by Amini, et al. [9,14]
L(; w) ?
k=X Ns ?1 k=0
w(k)s(k ? ) :
(8)
The template match cost function in [4,5] and the correlation function in [9,14] both have normalization terms that adjust for amplitude di erences between the tag template and the data. In the log likelihood function in Equation (6), this adjustment is accomplished by the mean myocardium signal intensity m myo and a spatially varying normalization term, ij , in the snake algorithm described in the next section. B.2 Snake Algorithm In principle, a tag line could be estimated by applying Equation (5) independently to a set of neighborhoods perpendicular to each tag line. Imaging artifacts and noise spikes, however, can cause large estimation errors, and since each tag line is estimated independently, there is nothing to prevent two tag lines from occupying the same physical position. For these reasons, we optimize Equation (6) for each tag center subject to spatial continuity constraints and a constraint on tag separation [5]. We denote the vector of candidate tag centers for the j th tag line as j = [0 j 1 j Nc?1 j ]T , where each i j is a real valued position on the ith image row. We de ne the optimal tag line as the tag line that minimizes the following
(
0 for x dsep 1 (dsep ? x)2 for x < dsep ; 2 where dsep is the minimum distance between tag points.
i j is a user-speci ed parameter that controls the degree to which the separation constraint is enforced and is set to zero if i j is outside the myocardium (see Section II-C.1). Estretch and Ebend are constraints on the stretch and bending of the tag displacements [16,17,6] given by
Esep (x) =
NX c ?1
i j (v ? v )2 h i j i?1 j i=1 NX c ?2 i j (v ? 2v + v )2(10) Ebend (vj ) = 21 ij i?1 j ; h2 i+1 j i=1
Estretch (vj ) = 21
where h is spacing between candidate tag centers, which for the case of a candidate center for each row is the MR image pixel spacing. The weights i j and i j vary with each tag center depending on whether or not the center and its neighbors are determined to be inside the myocardium. Speci cally, i j = if i j and i?1 j are inside the myocardium and set to zero otherwise. Similarly, i j = if i+1 j , i j , and i j?1 are inside the myocardium and set to zero otherwise. and are user-speci ed parameters. Setting the weights to zero in this manner ensures that tag centers outside the myocardium cannot in uence the position of tag centers inside the myocardium through the smoothing constraints. The energy function in Equation (9) is optimized using the method proposed by Kass, et al. [16]. First Equation (9) is di erentiated with respect to vj , which results in
Aj vj ? qj (j?1 ; j + j+1 ) = 0 ; (11) where Aj is an Nc Nc pentadiagonal matrix obtained
from di erentiating the the internal energy terms in Equation (12) with respect to vi j (see [17] for details) . The vector qj contains force terms, which will be de ned in the sequel. Following [16,17] the right-hand side of (11) is set
2 This assumes that the power in the tag template, sT ()s(), is 3 The initial position of each snake is speci ed by the user (see IIIconstant with respect to a shift. For the tag templates used in this paper, this is approximately true for small shifts ( FWHM=2). B.1). j
j
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. Y, MONTH 1999
equal to the negative time derivative of vj , which yields the following iteration for the j th optimal tag line displacement ?1 ) ; (12) (I + Akj )vjk = vjk?1 + qj ( jk??11 ; jk?1 + jk+1 where k is the iteration index, and is a step size. There is an Equation (12) for each tag line in the ROI, and these equations are solved simultaneously using a banded system solver [24]. After each iteration of Equation (12) each tag center is tested to see if it is inside or outside the myocardium. If the status of a tag center changes, then a new Akj is computed. The initial snake displacement vj0 is computed by linearly extrapolating its displacement from its displacement in the previous two time frames. In the rst time frame, the initial snake position is the initial tag position and vi0j = 0. In the second time frame, the initial snake displacement is the nal snake displacement from the rst time frame. The vector qj contains \force" terms derived from the log likelihood function and the tag separation constraint. The ith component of qj is qj i (j?1 ; j ; j+1 ) = d L( ; w ) ?i j d ij ij + i j [(dsep ? i j + i j?1 )u(dsep ? i j + i j?1 ) ?(dsep ? i j+1 + i j )u(dsep ? i j+1 + i j )] ; (13) where u(x) = 1 for x 0 and u(x) = 0 otherwise. i j is a normalizing factor de ned as 1 i j = d L(i j ; wi j )j maxi2[i?Ns =2;i+Ns =2] j d
if i j is in the myocardium and zero otherwise. This normalization stabilizes the snake algorithm by keeping the forces acting on the snake from being too large relative to the iteration step size [17]. Equation (12) leaves the position of candidate centers outside the myocardium unchanged. After the iteration in Equation (12) is complete, the position of candidate centers outside the myocardium is computed by using a smoothing spline to interpolate between the positions of candidate centers inside the myocardium as shown in Figure 7. This smoothing spline minimizes the stretching and bending energies described in Equation (10) with the exception that the smoothness constraints are applied to candidate centers outside the myocardium. Connecting centers outside the myocardium in this manner makes it easier for a candidate center to move inside the myocardium in the next time frame. The positions of candidate centers inside and outside the myocardium are computed separately to ensure that the position of centers inside the myocardium in uence the position of centers outside the myocardium, but not vice versa. C. Hypothesis Testing The tag center estimation algorithm described in the previous section applies di erent smoothing constraints and
104
image forces to a candidate tag center depending on if the center is in the myocardium or not. Ideally, once the estimation algorithm is complete, any candidate tag center considered to be inside the myocardium would be on a tag line. Because of estimation errors, however, it is possible for a candidate tag center to be in the myocardium but not on a tag line. So to ensure that only candidate tag centers on tag lines are considered valid, we also explicitly test each tag center to determine if it is part of a tag line. In this section, two maximum a posteriori (MAP) hypothesis tests [23] are developed based on the signal model derived in Section II-B. The rst test determines if a candidate center is inside the myocardium, the second test determines if a candidate center is part of a tag line. The region of interest (ROI) can contain several di erent tissue types in addition to the myocardium such as saturated blood, air, and tagged non-myocardium tissues such as the liver. It is possible for the signal vector w corresponding to a given candidate tag center to contain pixels from multiple tissues such as tagged myocardium and saturated blood and even other tag pro les. Ideally, there should be an hypothesis for each of these cases, but this would require a procedure for determining the intensity characteristics for these tissues in a given image, which would greatly increase the computational complexity of the algorithm. Instead, we consider dark tissues such as saturated blood and air to be \background", all bright tissues to be \myocardium," and all tagged, bright tissues to be \tagged myocardium." Methods for di erentiating between tag centers in myocardial versus non-myocardial tissue during the deformation reconstruction process are discussed in Section IV. The above assumptions yield the following three mutually exclusive hypotheses: Hypothesis T: Candidate tag center is part of a tag line and is in the myocardium. Hypothesis M: Candidate tag center is in the myocardium but not on a tag line. Hypothesis B: Candidate tag center is not in the myocardium. We assume that the probability of each hypothesis is known and denote PT , PM , PB (PT + PM + PB = 1) as the probabilities of Hypotheses T, M, and B respectively. For Hypothesis B we assume that the candidate center ^ is located in non-myocardium tissue, which we will refer to as background tissue. Under Hypothesis B, the observation vector w is modeled as
w = mback + n ;
(14)
where mback is an Ns -vector of background signal samples, and n is the same noise vector described in Section IIB.1. The background vector mback is modeled as a random back = m back 1 and covariance matrix vector with mean m i?j j 2 are 2 jback . Values for m back and back Cback(i; j ) = back estimated from the image data for each time frame in the sequence (see Section III). The PDF of the observation vector conditioned on Hy-
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. Y, MONTH 1999
105
pothesis B is given by [23] 1p f (wjHB ) = N = s (2) 2 jCB j back]T C?B1 [w ? m back] ; (15) exp ? 21 [w ? m where CB is the covariance matrix of w under Hypothesis B and is given by
CB = Cback + n2 I : Under Hypothesis M, the observation vector w is modeled as w = mmyo + n ; (16) where mmyo is an Ns -vector of myocardium signal samples, and n is the same noise vector described in Section IIB.1. The myocardium vector mmyo is modeled as a ran myo = m myo1 and covariance madom vector with mean m 2 ji?j j . Values for m 2 are trix Cmyo(i; j ) = myo myo and myo myo estimated from the image data for each time frame in the sequence (see Section III). The PDF of the observation vector conditioned on Hypothesis M is given by [23] 1p f (wjHM ) = N = (2) s 2 jCM j myo]T C?M1 [w ? m myo] ; (17) exp ? 12 [w ? m where CM is the covariance matrix of w under Hypothesis B and is given by Fig. 4. The results of the MAP myocardium detector applied to each pixel in the images in Figure 1. CM = Cmyo + n2 I : The signal model under Hypothesis T (center is part of a tag line) is described in Section II-B.1. The PDF of the observation vector conditioned on Hypothesis T is therefore given by [23] f (wjHT ) = f (w; ^) ; (18) where f (w; ) is de ned in Equation (4). C.1 Myocardium Detection A candidate tag center ^ is considered to be inside the myocardium if it is more likely to be either part of a tag line (HT ) or in untagged myocardium tissue (HM ) than in the background tissue (HB ). In terms of the PDF's de ned in Equations (15), (17), and (18) and the known probabilities PB , PM , and PT , the MAP decision rule is that ^ is inside the myocardium if [23] f (wjHM ) PB f (wjHT ) PB (19) f (wjHB ) > PT or f (wjHB ) > PM : Taking the natural logarithm of both sides of the inequalities in (19) and cancelling terms results in the rule that an estimated tag center ^ is inside the myocardium if max[L(^; w); M (w)] < B (w) ;
where M (w) and B (w) are given by myo]T C?M1 [w ? m myo] M (w) = 21 [w ? m + 12 ln jCM j ? ln(PM =PT )
back]T C?B1 [w ? m back] B (w) = 21 [w ? m + 12 ln jCB j ? ln(PB =PT ) :
Note that the prior probabilities PB , PM , and PT e ectively bias the decision rule in favor of one hypothesis or another. We use PB = PM = PT = 1=3, which means that all three hypotheses are equally likely. In this case, ln(PM =PT ) = ln(PB =PT ) = 0. Results of the myocardium detector in Equation (20) applied to the images in Figure 1 are shown in Figure 4.
C.2 Tag Line Detection In principle, once the tag center estimation algorithm has converged, any candidate tag center inside the my(20) ocardium should be the center of an actual tag line. The
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. Y, MONTH 1999
myocardium detector, however, is not perfect and sometimes non-myocardium tissue is classi ed as part of the myocardium. For this reason, it is necessary to explicitly test each candidate tag center to determine if it is part of an actual tag line. A candidate center ^ is considered to be part of a tag line if it is more likely to be part of a tag line (HT ) than either untagged myocardium (HM ) or background tissue (HB ). The MAP decision rule based on the PDF's de ned in Equations (15), (17), and (18) and the known probabilities PB , PM , and PT is to choose HT if [23] f (wjHT ) PM f (wjHT ) PB f (wjHM ) > PT and f (wjHB ) > PT : (21) Taking the natural logarithm of both sides of the inequalities in (21) and cancelling terms results in the rule that an estimated tag center ^ is part of a tag line if L(^; w) < min[M (w) ; B (w)] : (22) D. Removal of False Tag Points With real image data, the tag detector in Equation (22) sometimes detects a false tag. These false tag detections usually occur in small, isolated clusters that spontaneously appear and disappear in time. As a result after the tag lines have been estimated and detected, we use a pruning algorithm, which consists of two subroutines. The rst, called removeShortRuns() deletes all tag points that appear in contiguous groups of less than Nmin in size. We use Nmin = 3, which removes isolated tag points and isolated pairs of tag points. The second, called applyTemporalContinuity(), deletes all tag points who do not have a neighboring tag point in the same tag line within Nnbhd points in either the previous or next time frame. This allows the tag line to either grow or shrink by Nnbhd points due to the expansion or contraction of the myocardium between time frames. The value of Nnbhd should be roughly proportional to the temporal resolution. For a temporal resolution of 32.5ms, we use Nnbhd = 1. After the tags have been estimated and detected, these subroutines are called for each time frame in the following order: 1. removeShortRuns() 2. applyTemporalContinuity() 3. removeShortRuns() The rst call to removeShortRuns() deletes isolated tags and isolated tag pairs in each image, which are usually the result of imaging noise creating a dark pixel between two relatively bright pixels in untagged regions such as the blood pool. The call to applyTemporalContinuity() deletes tag points in regions of the image that are not persistently tagged. This procedure sometimes creates new groups of isolated tag points, which are removed by the second call to removeShortRuns().
106 Set Initial Conditions
Compute Image Statistics
Estimate Candidate Tag Line Centers
Next Time Frame
Detect Tag Line Centers
Prune Tag Lines
Fig. 5. Flow chart of the tag tracking and detection algorithm.
imaging protocol [18]. Four short-axis sequences and one long-axis image sequence were acquired. The short-axis image sequences consisted of seven parallel 8mm thick slices with no separation between slices. Each short-axis slice was imaged with tag plane orientations of 0, 90, +45, and -45 degrees. Di erent tag plane orientations were acquired by keeping the tag lines in a vertical orientation and rotating the slice prescription by the appropriate angle [18]. The long-axis image sequence consisted of six 8mm thick slices radially oriented around the long-axis of the LV with an angular separation of 30 degrees. Each long-axis slice was imaged once with tag planes oriented parallel to the shortaxis image planes. The resulting images are 256 256 with a pixel size of = 1:25mm. The tag planes were separated in the reference state (end-diastole) by 6mm. Ten cardiac phases were imaged spaced 32.5ms apart through systole for a total of 350 images acquired during the study. Nine additional normal human volunteers (NV2-NV10) were imaged with the same protocol but with 6 short-axis slices (imaged with 0 and 90 degree tag angles only), tags spaced 7mm apart, and a pixel size of 1.41mm. B. Algorithm Implementation B.1 Algorithm Overview The tag estimation, detection, and pruning algorithms described in Section II were implemented as shown in Figure 5. First the user speci ed a rectangular region of interest (ROI) and a rectangular region over which the image noise statistics were computed (see below). These regions are shown in Figure 6a. The same regions were used for each time frame in a given slice. The user also speci ed the III. In Vivo Heart Experiment initial position of the tag lines as shown in Figure 6b. The A. Imaging spacing between undeformed tag planes is known from the The LV of a normal human volunteer (NV1) was imaged imaging protocol, so only a single user-speci ed o set was using a cine, black blood, breath-hold parallel planar tag needed to set the initial position of all tags in the ROI. The
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. Y, MONTH 1999
107
TABLE I
Parameters used in tag tracking and detection experiment.
(a)
(b)
Fig. 6. (a) Region of interest (ROI) for both noise computation (upper box) and tag tracking (lower box). (b) Initial tag position.
same set of initial conditions (two rectangular regions and the initial position) were used for each slice in a stack of short-axis images with the same tag plane orientation. For the long-axis images, the initial conditions were set separately for each individual slice because of the di erences in orientation of the image planes. These initial conditions were the only user intervention used by the algorithm. At this point for each imaged phase (time frame) of the cardiac cycle, the algorithm rst computed the image statistics (see below), ran the tag estimation algorithm [Equation (12)]. The snake iteration in Equation (12). was performed for each tag line until either the maximum change in tag position was less than 0.01 or 150 iterations were completed. Next the tag detection routine [Equation (22)]. was applied to each time frame. Finally after all time frames were processed, the tag pruning algorithm [Section II-D] was run over all time frames.
Tag Line Model Parameters FWHM tag full width at half maximum myo correlation of myocardium signal T1nom nominal T1 of the myocardium Tag Tracking Parameters snake continuity weight snake bending weight dsep minimum tag separation snake algorithm time step Ns tag center neighborhood size Tag Detection Parameters back correlation of background signal PT prior probability of the tag hypothesis PM prior probability of the myocardium hypothesis PB prior probability of the background hypothesis Tag Line Pruning Parameters Nmin spatial neighborhood size Nnbhd temporal neighborhood size
2.0 pixels 0.9 500 ms 10 1 3 pixels 0.1 5 pixels 0.9 1/3 1/3 1/3 3 pixels 1 pixel
B.2 Choice of Algorithm Parameters The parameters used in the ML/MAP algorithm for all ten imaging studies are listed in Table I. The tag full width at half maximum (FWHM) was determined from the imaging protocol [18], and the tag center neighborhood size (Ns ) was set to the smallest odd integer larger than twice the FWHM. The minimum tag separation dsep was chosen based on a visual inspection of an end-systolic short-axis image. The prior probabilities PT , PM , and PB must sum to one as described in Section II-C, and were chosen such that all probabilities were equal. This choice re ects a assumption that a candidate center is as likely to be on a tag line as it is to be in untagged myocardium or background. The remaining parameters were chosen experimentally. B.3 Computation of Image Statistics In order to run the tag estimation and detection algorithms, an estimate of the noise variance n2 and the mean 2 ) and and variance of the myocardium pixels (m myo, myo 2 background pixels (m back , back ) must be computed for each image in the sequence. Accurately determining these parameters requires segmenting and classifying the tissues in the ROI, which is a dicult problem [25,26,27,28] Instead, we used a computationally simple approach that yielded a rough estimate of these parameters. The noise variance was computed using the pixels enclosed by the rectangular region speci ed by the user as a sample. We assumed that the enclosed pixel intensities represented the absolute value of a zero mean white noise process, and set the noise variance equal to the mean-square value of the enclosed
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. Y, MONTH 1999
pixel intensities. The method used for computing the myocardium and background statistics was motivated by the observation that the tagged tissue is among the brightest tissues in the ROI. The myocardium and background statistics were computed by rst performing a morphological closing operation over the ROI to remove the tag lines as described in [4]. The resulting pixel intensities were then 2 were set equal to the sample mean sorted. m myo and myo and variance of the brightest 50% of the pixel intensities. 2 were set equal to the sample mean and m back and back variance of the darkest 50% of the pixel intensities. C. Experiments C.1 Quantitative Accuracy The accuracy of the ML/MAP tag tracking algorithm for each imaging study was evaluated by comparing the tag center locations to a set of tag points estimated using previously identi ed myocardial contours and edited by an expert user [29] The location of tag centers in the expert edited data sets is approximately 0:1mm at early systole and 0:3mm at end-systole [19]. For each tag center identi ed by the ML/MAP algorithm, the minimum distance to the expert tag line was computed. The distances were averaged over all tag points inside the myocardial contours and over all slices for each tag plane angle and time frame.
C.2 Parameter Sensitivity The sensitivity of the ML/MAP algorithm to both the parameters in Table I were studied by varying the parameters and running the ML/MAP algorithm on all time frames in the mid-ventricular short-axis slice in Figure 8 (0 degree tag angle) and the long-axis slice in Figure 9 The resulting tag line positions from each parameter combination were compared to the expert edited tag lines described above. The number of tag points detected inside the myocardium was also computed for each parameter combination using a set of expert edited contours to segment the myocardium in the ROI. The true number of tag points in the myocardium was determined by visual inspection and was found to be 278 points in the short-axis slice and 289 points in the long-axis slice. To reduce the number of parameter combinations, related groups of parameters were varied while the remaining parameters were set to their nominal values listed in Table I. The following parameter groups were studied: the tag model parameters FWHM and T1nom , the snake stretching and bending weights and , the correlation coecients myo and back , the tag center neighborhood size Ns , the minimum tag separation dsep , the prior probabilities PT , PM , PB , and the tag pruning parameters Nmin, Nnbhd . The sensitivity of the ML/MAP algorithm to the ve im2 ,m 2 ) was studage statistics (n2 , m myo, myo back , and back ied by scaling the means and standard deviations by factors from 0.1 to 2.0 in steps of 0.1 and then running the ML/MAP algorithm on all time frames in the short-axis and long-axis slice described above. In each run, one parameter was varied while the others were xed at their
108
original values. The resulting tag line positions from each combination were compared to the expert edited contours described above. D. Results The ML/MAP algorithm summarized in Figure 5 was run on all image sequences in each imaging study (NV1NV10). The parameters listed in Table I were used for both short-axis and long-axis images in all studies. Figure 7a. shows a mid-ventricular short-axis image from the NV1 study overlaid with the results of the tag estimation algorithm. Note that the continuity of each snake is broken near the boundaries of the myocardium and that the tag separation constraint keeps the tags from crossing one another. The results of the tag detection algorithm are shown in Figure 7b. Most of the tag points inside the myocardium are preserved, and the false tag points occur in small, isolated clusters. The tag pruning algorithm results are shown in Figure 7c. Most of the the false tag centers are removed while preserving the tag centers inside the myocardium. Note that the third tag line from the left is attracted to a dark pixel in Figure 7a, but the erroneous tag centers are removed by the tag detection (Figure 7b) and pruning (Figure 7c) stages. The nal results of the ML/MAP tag tracking algorithm for a selection of short-axis and long-axis images from the NV1 study are shown in Figures 8 and 9. Only the portion of the image contained in the ROI is shown. The 90 degree tag plane angle images are acquired by keeping the tag lines in a vertical orientation and rotating the slice prescription +90 degrees [18], The 90 degree images in Figure 8 have been rotated -90 degrees so that the heart is in the same orientation in both the 0 and 90 degree tag orientation images. Note that the algorithm will preserve tag centers in any persistently tagged tissue as seen on the right hand column Figure 8 and 9. In the long-axis images, the contrast between the myocardium and the blood pool is much lower than in the short-axis images, which violates the assumption in Section III-B.3. that the myocardium pixels are the brightest 50% of the pixels in the ROI. In spite of this violation, the ML/MAP algorithm still does a good job of tag line identi cation. The root-mean-square (RMS) di erence between the ML/MAP tags and contour-based tags for the NV1 study is plotted versus time frame in Figure 10. Each data point represents an average over all slices for a given tag orientation (approximately 1000 tag points). Note that the RMS di erence is within the precision of the expert edited tags. The di erences were equally distributed around zero. In an apical slice of the +45 degree short-axis sequence, a single tag line was attracted to non-tag feature. This tag line was excluded from the di erence analysis. Table II shows the RMS di erences at end-systole for all ten normal human studies. Each entry represents approximately 700 points in these studies because of the larger pixel size and tag spacing. In all the studies, the RMS di erence is within the precision of the expert edited tags, and the di erences were equally distributed around zero.
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. Y, MONTH 1999
109
Early systole (47 ms)
Mid-systole (177 ms)
End-systole (340 ms)
Fig. 8. Results of the ML/MAP tag tracking algorithm for a mid-ventricular slice for two di erent tag plane angles (0 and 90 degrees).
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. Y, MONTH 1999
110
(a) Early systole (47 ms)
(b) Mid-systole (177 ms)
(c)
Fig. 7. Intermediate results in ML/MAP tag tracking algorithm: End-systole (340 ms) (a) Result of the snake tag estimation algorithm. Candidate tag centers considered to be inside the myocardium are shown with Fig. 9. Results of the ML/MAP tag tracking algorithm for a long-axis +'s, and centers outside the myocardium are shown by 's. (b) slice. Result of the MAP tag detector. (c) Final result after the spatiotemporal pruning algorithm.
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. Y, MONTH 1999
111
TABLE IV
RMS difference in mm
0.5
0.4
RMS difference in mm between ML/MAP tag lines and expert-edited tag lines for different values of the snake stretching and bending parameters and .
SA +00 SA +90 SA +45 SA -45 LA
Short-Axis Long-Axis 0.1 1.0 10.0 0.1 1.0 10.0 1.0 0.25 0.25 0.27 0.84 0.18 0.18 10.0 0.23 0.23 0.25 0.13 0.14 0.17 100.0 0.37 0.36 0.40 0.22 0.22 0.24
0.3
0.2
0.1
0.0 0.0
100.0
200.0 Time from QRS in msec
300.0
400.0
Fig. 10. Quantitative accuracy results for normal human volunteer NV1. TABLE II
RMS differences between ML/MAP tag lines and expert-edited tag lines at end-systole for 10 different normal human volunteers.
RMS Di erence (mm) Short-Axis 0o Short-Axis 90o Long-Axis NV1 0.19 0.23 0.16 NV2 0.35 0.27 0.25 NV3 0.22 0.29 0.23 NV4 0.24 0.21 0.16 NV5 0.19 0.23 0.15 NV6 0.14 0.15 0.34 NV7 0.15 0.23 0.19 NV8 0.18 0.19 0.19 NV9 0.25 0.24 0.20 NV10 0.25 0.20 0.14 In two of the studies (NV2, NV10), rapid contraction during early diastole caused a loss of tag line correspondence in some long-axis slices (1 slice in NV2, 3 slices in NV10). Also in the NV3 study, a single tag line in an apical 90 degree short-axis was attracted to non-tag feature in the ROI. These slices were excluded from the di erence analysis. Table III shows the e ect on RMS di erence at endsystole when the tag pro le model parameters FWHM and T1nom are varied. The ideal tag FWHM (determined from the imaging protocol) is 2.0 pixels. The RMS di erence is fairly insensitive to FWHM for values of 2.0 and higher, but degrades slightly when FWHM=1.5. The RMS di erence is does not change signi cantly with T1nom provided the FWHM is 2.0 or higher. A similar variation was observed in the number of tags detected in the myocardium (not shown). Table IV shows the e ect on RMS di erence at endsystole when the snake stretching and bending parameters and are varied. The RMS di erence is more sensitive to
TABLE VI
RMS difference in mm and percent of tag centers in the myocardium detected by the ML/MAP algorithm for different values of the tag center neighborhood size Ns .
RMS Di erence
% Detected
Ns Short-Axis Long-Axis Short-Axis Long-Axis 3 5 7
0.23 0.23 0.21
0.15 0.14 0.12
87% 86% 80%
79% 79% 76%
these parameters than the tag line model parameters, since and control the amount of smoothness enforced in the tag line optimization. The RMS di erence is still reasonable, however over approximately a two decade range of and . A similar variation was observed in the number of tags detected in the myocardium (not shown). Table V shows the e ect on RMS di erence at endsystole when the myocardium and background correlation coecients myo and back are varied. The RMS di erence tends to improve slightly when both coecients are 0.99, but as shown in Table V, the percentage of tags detected in the myocardium decreases. This decrease in tag detection is because = 0:99 re ects an assumption that the myocardium and background signals are almost constant in the neighborhood of a candidate tag center. This assumption is violated at the edges of the myocardium where the neighborhood may contain both myocardium and background pixels. Table VI shows the e ect on RMS di erence at endsystole when the tag center neighborhood size Ns is varied. The RMS di erence tends to improve slightly as the neighborhood size is increased, but as shown in Table VI the tag detection tends to degrade. This degradation is because larger neighborhoods are more likely to include both myocardium and background signals or multiple tag pro les. Table VII shows the e ect on RMS di erence at endsystole when the minimum tag separation dsep is varied. The minimum tag separation constraint is a nonlinear constraint that is not active unless two tag lines get within dsep pixels of each other. In both the short and long-axis
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. Y, MONTH 1999
112
TABLE III
RMS difference in mm between ML/MAP tag lines and expert-edited tag lines for different values of the tag profile parameters FWHM and T1nom .
FWHM (pixels) 1.5 2.0 2.5 3.0
250 0.23 0.20 0.21 0.24
Short-Axis 500 750 0.29 0.28 0.23 0.20 0.23 0.19 0.20 0.18
T1nom (msec) 1000 0.28 0.22 0.17 0.17
250 0.13 0.13 0.14 0.16
Long-Axis 500 750 0.17 0.20 0.14 0.11 0.14 0.14 0.15 0.15
1000 0.23 0.11 0.14 0.13
TABLE V
RMS difference in mm and percent of tag centers in the myocardium detected by the ML/MAP algorithm for different values of the correlation coefficients myo and back .
back (RMS Di erence) back (% Detected) Short-Axis Long-Axis Short-Axis Long-Axis myo 0.80 0.9 0.99 0.80 0.9 0.99 0.80 0.9 0.99 0.80 0.9 0.99 0.80 0.23 0.24 0.24 0.14 0.15 0.13 91% 90% 88% 78% 80% 82% 0.90 0.22 0.23 0.25 0.14 0.14 0.13 87% 86% 88% 78% 79% 80% 0.99 0.18 0.18 0.18 0.13 0.15 0.09 65% 83% 84% 18% 25% 67% TABLE VII
RMS difference in mm between ML/MAP tag lines and expert-edited tag lines for different values of the minimum tag separation distance dsep .
dsep Short-Axis Long-Axis 2.0 2.5 3.0 3.5 4.0
0.22 0.20 0.23 0.29 0.82
0.14 0.14 0.14 0.14 0.27
slices, the RMS di erence is fairly constant until dsep approaches the original tag separation of 4.8 pixels. There is more variation in the short-axis slice because of circumferential contraction of the myocardium. A similar behavior was observed in the number of tags detected in the myocardium (not shown). Table VIII shows the number of tag centers detected in the myocardium at end-systole as a percentage of the true number tag centers when the prior probabilities PT , PM , and PB are varied. The percentage detection increases slightly when tag line and myocardium hypotheses are assumed to be more probable. The RMS di erence (not shown) did not vary signi cantly with changes in the prior probabilities. Table IX shows the e ect of varying the pruning parameters Nmin and Nnbhd on the percentage of tag points detected in the myocardium at end-systole. There is not much variation with respect to these parameters although
TABLE VIII
Percent of tag centers detected in the myocardium by the ML/MAP algorithm for different values of the prior probabilities PT , PM , and PB .
PT
1/3 1/3 1/3 1/2 1/6 1/2 1/6
PM PB 1/3 1/2 1/6 1/3 1/3 1/6 1/2
1/3 1/6 1/2 1/6 1/2 1/3 1/3
Short-Axis Long-Axis 86% 79% 89% 84% 84% 74% 91% 85% 81% 73% 86% 79% 85% 80%
the percent detected does degrade slightly as the minimum run length is increased and the temporal neighborhood size is decreased. The RMS di erence (not shown) did not vary signi cantly with changes in the pruning parameters. Figure 11 shows the e ect of errors in estimating the image statistics n , mmyo, myo, mback , and back on the RMS di erence at end-systole. In the short-axis slice, the RMS di erence is relatively constant when the statistics are individually scaled by factors from 0.1 to 2.0, which represents a variation of -90% to +100% of the original values. This trend also occurs in the long-axis slice except for isolated cases when the scale factor is particularly large or small. The o -scale points in the mmyo and mback plots are 0.69mm and 0.85mm respectively. These points correspond to cases when mmyo and mback were scaled such that mmyo < mback , which cannot occur in the estimation procedure described
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. Y, MONTH 1999
113
TABLE IX
Percent of tag centers detected in the myocardium by the ML/MAP algorithm for different values of the pruning parameters Nmin and Nnbhd .
Nmin
(pixels) 2 3 4 5
1 87% 86% 84% 79%
7 87% 87% 87% 82%
σn
0.2
1 80% 79% 75% 74%
0.4 m myo
0 100
0.4
100
σmyo
0
50
0
0.4
100 m back
0.0
0.2
50
0.0
0
0.4
100
0.2
0.0 0.0
0.5
1.0 Scale Factor
1.5
2.0
Short-Axis Long-Axis
50
0.0
0.2
7 83% 83% 83% 77%
50
0.0
0.2
Long-Axis 3 5 83% 83% 83% 83% 82% 83% 77% 77%
100
Short-Axis Long-Axis
σback
m back
σmyo
m myo
σn
0.4
σback
Nnbhd (pixels)
Short-Axis 3 5 87% 87% 87% 87% 86% 87% 81% 82%
50
0 0.0
0.5
1.0 Scale Factor
1.5
2.0
Fig. 11. RMS di erence between ML/MAP tag lines and expert- Fig. 12. Percentage of tag centers detected in the myocardium by the ML/MAP algorithm when the image statistics were scaled. edited tag lines when the image statistics were scaled.
in Section III-B.3. The increased sensitivity in the long-axis slice is because the blood pool intensity is brighter in these images, which makes the assumptions underlying the statistical parameter estimation less valid. The e ect of errors estimating the image statistics is more signi cant in the tag detection results in Figure 11 because the ML/MAP algorithm depends on these statistics to distinguish between tagged myocardium and other tissues. The detection is particularly poor when the myocardium standard deviation (myo) is severely underestimated because only pixels in a narrow intensity range are considered to be in the myocardium. Detection also degrades as back is overestimated because a wider range of pixel intensities are considered to
be background. As with the RMS error, if mmyo is underestimated or mback is overestimated by a large amount, then mmyo < mback results and the detection degrades. The detection is fairly insensitive to the noise and myocardium standard deviations, however, for factors of 0.5 and higher. Finally, we note that scale factors of 0.8 and 0.9 result in increased detection with the same RMS di erence, which suggests that the procedure in Section III-B.3. tends to slightly overestimate the tissue statistics. IV. Discussion
For MR tagging techniques to be clinically viable, the need for user-de ned contours must be eliminated in
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. Y, MONTH 1999
both the tag identi cation and displacement reconstruction phases of tagged cardiac image analysis. In this paper, we addressed the problem of identifying tag lines without user-de ned contours. An algorithm, which we call the ML/MAP algorithm, was presented that identi es tag lines in the myocardium without user-de ned contours based on a priori knowledge of the tag shape and spatio-temporal continuity of the tag lines. The algorithm was demonstrated on in vivo tagged image data from ten normal human volunteers. The ML/MAP estimated tag positions inside the heart were compared to a set of tag positions estimated using prior contour knowledge and edited by an expert. The RMS di erences between the ML/MAP estimated tag positions and the expert veri ed tag positions were within the precision of the expert veri ed tag positions. From these results we conclude that the ML/MAP algorithm and the parameters in Table I are stable across a range of normal subjects. Subjects su ering from ischemic disease have di erent myocardial signal and motion characteristics. The behavior of the ML/MAP algorithm on these subjects is a topic of future research. The ML/MAP algorithm worked well over a number of di erent images and regions of interest. In particular, the ML/MAP algorithm worked well on the long-axis images where the contrast between the myocardium and the blood pool is relatively small. This is an important result because both the signal model and the method used to compute image statistics assume that there is a signi cant di erence between the myocardial signal and blood pool signal. The robustness of the ML/MAP algorithm to the choice of user de ned parameters and errors in estimating the image statistics were demonstrated by running the algorithm on both a short-axis slice and a long-axis slice with di erent parameter combinations. The resulting tag line positions were compared to a set of expert edited tag lines. The RMS di erence between the ML/MAP and expert edited tag lines was within the precision of the expert edited tag lines for reasonable variations in the parameters. The RMS di erence is also small in most cases when the image statistics are varied. During the detection phase, however, the algorithm is sensitive to large variations in the image statistics. This sensitivity is because there are no user-de ned contours and the algorithm must rely on di erences in image intensity described by these statistics to distinguish between tag centers inside and outside the myocardium. Because the ML/MAP does not use any prior knowledge of the myocardial contours, some tag points are detected outside the myocardium, particularly in tagged non-myocardium tissue. While these tag points could be removed with user supervised processing, this approach would not be practical in a clinical environment. Since non-myocardium tag points tend to move di erently than myocardium tag points, however, a more interesting alternative is use an algorithm that identi es the myocardial contours based on di erences in the spatio-temporal motion of both myocardium and non-myocardium tag points. A reconstruction algorithm based on this approach is currently under development. A preliminary version is de-
114
scribed in [30]. As with other tag identi cation algorithms, ML/MAP tags can sometimes be attracted to a non-tag feature in the ROI. When this situation occurs, the tag line must be manually edited using a graphical user interface such as the one described in [31]. Methods for reducing or eliminating this type of error by exploiting tag continuity between slices and between tag plane orientations are currently under investigation. Also, particularly in early time frames of long axis sequences, it is possible that tags can move more than one half tag spacing between time frames and cause a tag identi cation algorithm to skip a tag line in the image data. In the ML/MAP algorithm, tag line correspondences are not explicitly constrained, but a temporal prediction model is used to extrapolate the initial position of each tag line from the previous two time frames. In 8 of the 10 experiments presented in this paper, the temporal prediction model produced initial tag positions suciently close to the true tag line that tag line correspondences were maintained. Methods for explicitly constraining tag line correspondences are a topic of future research. The ML/MAP algorithm contains a tag center estimation algorithm that uses a Gaussian tag pro le to generate image forces for a snake algorithm. This approach is similar in spirit to snake-based tag estimation algorithms proposed by other researchers [5,6,15,9]. The ML/MAP algorithm, however, di ers from these other snake-based algorithms in the following ways. First, the maximum likelihood (ML) estimation framework provides a set of spatially varying weights for comparing the image data with the Gaussian tag template based on the physics of the tagging process and signal intensity and noise statistics computed from the image data. In previous tag tracking methods, the points used to compare the image data with the tag template are equally weighted. Second, the myocardium is automatically segmented in the ROI by using a maximum a posteriori (MAP) hypothesis test. Previous tag tracking algorithms require user-de ned contours to segment the myocardium. User-de ned contours greatly enhance the ability of an algorithm to identify tag lines in the myocardium. Since the ML/MAP algorithm does not use any prior contour knowledge, the tag line positions extracted using the ML/MAP algorithm will not, in general, be as accurate or as dense as those identi ed with user-de ned contours. The ML/MAP algorithm, however, is intended for use in a clinical environment where user supervised processing of each individual image would not be practical. In a research lab or other setting where user-de ned contours can be obtained, one of the contour-based algorithms listed above should be used. At present, the algorithm assumes a parallel tag pattern like the one shown in Figure 4a, but the methods presented in this paper could easily be extended for use in other radial [1] or grid [2] tag patterns. In tag patterns such radial tags [1], where the undeformed tag lines are at an oblique angle relative to the slice and read gradient directions, the tag pro le samples would still be taken perpendicular to
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. Y, MONTH 1999
the initial (undeformed) tag line position. The tag line normal would no longer be an image row but an oblique line, and tag center displacements would be computed in the direction of the tag line normal. In summary the ML/MAP algorithm presented in this paper is capable of identifying tag lines without userde ned myocardial contours and has the potential to remove some of the technological roadblocks keeping MR cardiac tagging techniques from routine clinical use. Acknowledgments
The author would like to thank Elliot R. McVeigh, PhD for making the in vivo human data available to him and for helpful discussions on MR tagging and tag identi cation. References [1] E.A. Zerhouni, D.M. Parish, W.J. Rogers, A. Yangand, and E.P. Shapiro, \Human heart: tagging with MR imaging | a method for noninvasive assessment of myocardial motion," Radiology, vol. 169, pp. 59{63, 1988. [2] L. Axel and L. Dougherty, \MR imaging of motion with spatial modulation of magnetization," Radiology, vol. 171, pp. 841{845, 1989. [3] A. Bazille, M.A. Guttman, E.R. McVeigh, and E.A. Zerhouni, \Impact of semi-automated versus manual image segmentation errors on myocardial strain calculation by MR tagging.," Investigative Radiology, vol. 29, no. 4, pp. 427{433, 1994. [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, vol. 13, no. 1, pp. 74{88, 1994. [5] 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, Nice, Aug. 1995, SMR, vol. 1, p. 41. [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, vol. 14, no. 3, pp. 413{421, Septmeber 1995. [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, vol. 270, pp. H281{H297, 1996. [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, vol. 195, pp. 829|835, 1995. [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 Arti cial Intelligence (AAAI) Spring Symposium Series. Applications of Computer Vision in Medical Image Processing. March 1994, pp. 126{ 129, The AAAI Press. [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, vol. 66, no. 2, pp. 163{178, May 1997. [11] A. Amini, R. Curwen, and J. Gore, \Snakes and splines for tracking non-rigid heart motion," in Lecture Notes in Computer Science, Cipolla and Buxton, Eds., vol. 1065. Springer-Verlag, Berlin, April 1996. [12] 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, vol. 14, no. 4, pp. 625{635, December 1995. [13] T.S. Denney Jr. and E.R. McVeigh, \Model-free reconstruction of three-dimensional myocardial strain from planar tagged MR images," Journal of Magnetic Resonance Imaging, vol. 7, no. 5, pp. 799{810, September/October 1997. [14] P. Radeva, A. Amini, J. Huang, and E. Marti, \Deformable Bsolids and implicit snakes for localization and tracking of MRISPAMM data," in IEEE Workshop on Mathematical Models in Biomedical Image Analysis, San Francisco, CA, June 1996.
115
[15] S. Kumar and D. Goldgof, \Automatic tracking of SPAMM grid and the estimation of deformation parameters from cardiac MR images," IEEE Transactions on Medical Imaging, vol. 13, no. 1, pp. 122{132, 1994. [16] M. Kass, A. Witkin, and D. Terzopoulos, \Snakes: active contour models," International Journal of Computer Vision, vol. 1, pp. 321{331, 1988. [17] L.D. Cohen, \Note on active contour models and balloons," CVGIP: Image Understanding, vol. 53, no. 2, pp. 211{218, 1991. [18] E.R. McVeigh and E. Atalar, \Cardiac tagging with breathhold cine MRI," Magnetic Resonance in Medicine, vol. 28, pp. 318{327, 1992. [19] E. Atalar and E.R. McVeigh, \Optimization of tag thickness for measuring position with magnetic resonance imaging," IEEE Transactions on Medical Imaging, vol. 13, no. 1, pp. 152{160, 1994. [20] S.E. Fischer, G.C. McKinnon, S.E. Maier, and P. Boesiger, \Improved myocardial tagging contrast," Magnetic Resonance in Medicine, vol. 30, no. 2, pp. 191{200, August 1993. [21] S.B. Reeder and E.R. McVeigh, \Tag contrast in breath-hold CINE cardiac MRI," Magnetic Resonance in Medicine, vol. 31, no. 5, pp. 521{5, May 1994. [22] W.R. Crum, E. Berry, J. Ridgeway, U.M. Sivananthan, L-B Tan, and M.A. Smith, \Simulation of two-dimensional tagged mri," Journal of Magnetic Resonance Imaging, vol. 7, no. 2, pp. 416{ 24, Mar-Apr 1997. [23] K.S. Shanmugan and A.M. Breipohl, Random signals: detection, estimation and data analysis, John Wiley and Sons, Inc., 1988. [24] G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins University Press, Baltimore, 2nd ed. edition, 1989. [25] W.M. Wells, W.E.L. Grimson, R. Kikinis, and F.A. Jolesz, \Adaptive segmentation of MRI data," IEEE Transactions on Medical Imaging, vol. 15, no. 4, pp. 429{442, August 1996. [26] R. Guillemaud and M. Brady, \Estimating the bias eld of MR images," IEEE Transactions on Medical Imaging, vol. 16, no. 3, pp. 238{251, June 1997. [27] D.H. Laidlaw, K.W. Fleischer, and A.H. Barr, \Partial-volume Bayesian classi cation of material mixtures in MR volume data using voxel histograms," IEEE Transactions on Medical Imaging, vol. 17, no. 1, pp. 74{86, February 1998. [28] H.R. Singleton and G.M. Pohost, \Automatic cardiac MR image segmentation using edge detection by tissue classi cation in pixel neighborhoods," Magnetic Resonance in Medicine, vol. 37, no. 3, pp. 418{24, March 1997. [29] C.C. Moore, C.H. Lugo-Olivieri, L.Lee, J.A.C. Lima, E.R. McVeigh, and E.A. Zerhouni, \3D deformation of the normal human left ventricle: assessment by MR tagging," in Proc. SMR/ESMRMB, Nice, Aug 1995, SMR, vol. 3, p. 1427. [30] L. Yan and T.S. Denney, \Joint reconstruction of 2-D left ventricular displacement and contours from tagged magnetic resonance images using Markov random eld edge prior," in Proceedings of the 1998 IEEE Workshop on Biomedical Image Analysis, San Diego, CA, June 26-27 1998. [31] M.A. Guttman, E.A. Zerhouni, and E.R. McVeigh, \Analysis and visualization of cardiac function from MR imgages," IEEE Computer Graphics and Applications, vol. 17, no. 1, pp. 30{38, January 1997. [32] R.M. Henkelman, \Measurement of signal intensities in the presence of noise in MR images," Medical Physics, vol. 12, no. 2, pp. 232{233, Mar/Apr 1985.
