The third component of our warp validation studies includes reconstruction of myocardial
displace-ments in normal human volunteers.
Results of application of techniques to in vivo data withi= 1 are shown in gures 15, 16, 17, and 18. In gure 17, dense motion is computed from two deformed grids. In order to assess the sensitivity of the vector eld to di erent i coecient values, the following study was undertaken. The displacement vector eld between 0 and 180 msec image for the uniform weight factors 1 = 1,2 = 1, and 3 = 1 ( gure 18) was chosen to be the ground-truth. Vector elds corresponding to di erent coecient values were subsequently compared with this vector eld using equations (17) and (18). Results are illustrated in table 4.
5 Conclusions
In conclusion, we have described new computational algorithms suitable for analysis of SPAMM tagged data. We have argued that in comparison to other forms of parameterization, use of B-splines for representing tag curves has several advantages, including parametric continuity, as well as the need to only optimize the location of few control points in order to determine the location of a complete tag line.
We have described new methods for ecient reconstruction of dense displacement vector elds from SPAMM grids. The constrained thin-plate methods warp an area in the plane such that two embedded grids of curves are non-rigidly registered, thereby interpolating a dense displacement vector eld. The new warp method treats intersection points of SPAMM grids as standard landmark points and forces these to come together. Furthermore, it corresponds complete tag curves and brings these into alignment.
Finally, where no information is available, it interpolates a C1 continuous vector eld.
In addition to the developed machinery in this paper, evaluation of the methods were undertaken based on a) A synthetic environment where coupled B-spline curves model tag lines, and realistic
defor-mation can be obtained by pushing an pulling on spline control points. b) A cardiac tagging simulator making use of Arts's kinematic model of myocardial deformations as described in [31]. For the purposes of this paper, only parameters of the model which rendered 2D deformations were utilized. c) In-vivo data from normal human volunteers. The coupled-snake tracking algorithm was tested using simulated and in-vivo data sets and the warping algorithm was tested for accuracy in length as well as angle of the reconstructed displacement vectors from the known ground-truth, and the results indicate that con-strained thin-plate reconstructions of myocardial deformations is suciently accurate for measurement of in-plane tissue deformations within a reasonable time and between any two frames in a sequence of tagged images. The extension of the techniques to 3D is topic of current research. In order to extract 3D displacements, tag planes in SA and LA acquisitions will be modeled by continuous B-spline surfaces.
The deformation vector elds are then interpolated given any 2 pairs of SA and LA images acquired synchronized to the ECG.
Acknowledgements
We would like to thank John C. Gore and Todd Constable of Yale School of Medicine for helpful discussions and data which were used in developing previous generation of methods in [2, 3].
Appendix
In this appendix we give a brief description of the two non-linear optimization methods, Conjugate-Gradient Descent and Davidon-Fletcher-Powell, which we have utilized in this paper. The reader is referred to Numerical Recipes for further details [26]. The conjugate gradient is basically a form of
steepest descent algorithm, except for the fact that the descent directions are chosen very eciently.
The following sequence of operations are performed for the objective function.
Conjugate Gradient Algorithm
Step 1
: Initialize the solution vectorx
0. Letg
0 =h
0=?rf(x
0) whereh
0 is the initial descent direction.
Step 2
: Minimize along the current descent directionh
i and calculate the next solution vectorx
i+1.
Step 3
: Calculate the vectorg
i+1=?rf(x
i+1).
Step 4
: Calculate the next descent directionh
i+1 using the following formulae:i= (
g
i+1?g
i):g
i+1g
i:g
ih
i+1 =g
i+1+ ih
i: (19)
Step 5
Check termination criterion. Go back to step 2 if necessary.A speci c quasi-Newton algorithm is characterized by the inverse Hessian updating method used in step 3. The Davidon-Fletcher-Powell is described below.
Davidon-Fletcher-Powell
Step 1
: Initialize the vectorx
0, the inverse HessianH
0 =I
,I
being the identity matrix, and the initial descent directiond
0 =?rf(x
0).Step 2
: Minimize the function along the current directiond
i and calculatex
i .
Step 3
: Update the inverse Hessian matrixH
i+1 at the pointx
i+1 according to:H
i+1 =H
i+correction (20)
Step 4
: Calculate the next descent directiond
i+1 usingd
i+1 =?H
i+1:(rf(x
i+1)): (21)
Step 5
: Check termination criterion. Go back to step 2 if necessary.References
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P P P P
P
P P P P
P
P
P P P P
P
P P P P
12 13 14 15
21
22 23 24 25
26
32 33 34 35
36
42 43 44 45
31
Figure 1: The spatial organization of control points for a coupled B-snake grid. Dependence of horizontal and vertical splines of deformable grids is captured by the shared control points.
Figure 2: An undeformed coupled B-spline grid. The squares at grid intersections high-light location of control points.
Figure 3: The undeformed prolate spheroidal model of the LV in the reference state. A tagged image corresponding to a selected imaging plane is shown on the right. The width of the \donut" is about 1.5 cm, and the discretization step is 0.05 cm.
Figure 4: Deformed models of the LV resulting from change ofk2 from 0:2 to 0:8 in increments of 0:2
Figure 5: Deformed models of the LV resulting from change of k4 from ?0:02 to?0:08 in increments of
?0:02
Figure 6: Results of coupled B-snake tracker on a simulated image sequence ( 1 = 75, 2 = 1). From top-left (k2 = 0:2) to bottom-right (k2 = 0:8) in increments of 0:2. Temporal resolution is 20 msec.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
Figure 7: The gure of merit (equation (16)) for coupled B-snake tracking as a function ofk2 and k4.
−0.060 −0.04 −0.02 0 0.02 0.04 0.06
Figure 8: The gure of merit (equation (16)) for coupled B-snake tracking as a function ofk5 and k10.
−0.20 −0.18 −0.16 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0
Figure 9: The gure of merit (equation (16)) for coupled B-snake tracking as a function of k11 and k12.
Figure 10: Results of tracking SPAMM images with deformable spline grids ( 1 = 100, 2 = 1).
Figure 11: The reconstructed displacement vector eld computed by CG, super-imposed on the deformed grid. New location of sample points of initial undeformed grid as determined by CG optimization are high-lighted as small dark squares. The discretization which was employed, divides each horizontal (vertical) spline segment between two vertical (horizontal) splines in to 4 equidistant intervals on the undeformed grid.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
Figure 12: The error plots for angle and length error fork2 and k4. Please see text for details.
−0.06 −0.04 −0.02 0 0.02 0.04 0.06
Figure 13: The error plots for angle and length error for k5 and k10. Please see text for details.
Figure 14: Comparison of computed (left) and true (right) displacement vector elds corresponding to rotation (k10=?0:25, top) and torsion (k2 =?1:0, bottom).
Figure 15: In-vivo tagged slice towards the apical end of the LV in end-diastole (0 msec after ECG trigger) (a), 90 msec (b), and 180 msec (c).
Figure 16: The reconstructed vector eld by CG computed between deformable grids in (a) and (b) of gure 15 (1 = 2 = 3 = 1). The vector eld is displayed on the myocardial region in the 0 msec image.
Figure 17: The reconstructed vector eld by CG computed between deformable grids in (b) and (c) of gure 15 (1 = 2 = 3 = 1). The vector eld is displayed on the myocardial region in the 90 msec image.
Figure 18: The reconstructed vector eld by CG computed between deformable grids in (a) and (c) of gure 15 (1 = 2 = 3 = 1). The vector eld is displayed on the myocardial region in the 0 msec image.
k1 Radially dependent compression k2 Left ventricular torsion
k3 Ellipticalization in long-axis (LA) planes k4 Ellipticalization in short-axis (SA) planes k5 Shear in x direction
k6 Shear in y direction k7 Shear in z direction k8 Rotation about x-axis k9 Rotation about y-axis k10 Rotation about z-axis k11 Translation in x direction k12 Translation in y direction k13 Translation in z direction
Table 1: The thirteen k-parameters of the kinematic model.
TS IP D0 TE TR T1 T2 kx ky Ri Ro ss
0.9 cm 0.5 cm 300 0.03 sec 10 sec 0.6 sec 0.1 sec 7 rad/cm 7 rad/cm 45 deg. 0.25 0.6 4 cm 0.05 cm/pixel Table 2: Imaging parameters and dimensions of geometric model. Please note that TS is tag separation,
IP is the image plane position, Ri and Ro are the inner and outer radii of the 2 prolate spheroids, and ss is the sample size.
1 2 3 Calculation Time Iter 1 11,902.02 290.18 3.113
CG, Iter 55 0.691266 0.001271 0.006792 116 sec.
DFP, Iter 115 0.695445 0.001299 0.006330 491 sec.
Table 3: The values for 1, 2, and 3 in the beginning and after minimization for CG and DFP. In both cases, 1 = 1, 2 = 10, and3 = 20. Reported calculation times are \terminal times" on an SGI INDY.
1 2 3 " "L
1 1 1 0 0
5 1 1 0.006 0.019 10 1 1 0.008 0.029 1 5 1 0.005 0.012 1 10 1 0.006 0.015 1 1 5 0.005 0.015 1 1 10 0.008 0.021 1 1 0 0.007 0.018 1 0 1 0.011 0.043
Table 4: The values for angle (fraction of ) and length errors (inmm) as a function of1,2, and 3. The \ground-truth" in this case was chosen to be the reconstructed displacement eld withi= 1.