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Coupled B-Snake Grids and Constrained Thin-Plate Splines1 for Analysis of 2D Tissue Deformations from Tagged MRI Amir A. Amini?, Yasheng Chen? , Rupert W. Curwen??, Vaidy Maniz, and Jean Sun? CVIA Lab, Washinton University Medical Center? , St. Louis, MO CMA Associates?? , Schenectady, NY Iterated Systems, Inc.z , Atlanta, GA

Address for Correspondence: Amir Amini Cardiovascular Image Analysis Lab Campus Box 8086 660 S. Euclid Ave. Washington University Medical Center St. Louis, MO 63110-1093 Tel: (314)454-7408 FAX: (314)454-5350 E-mail: [email protected] www: http://www-cv.wustl.edu

Supported in part by a grant from Whitaker Biomedical Engineering Foundation, and grant IRI-9796207 from the National Science Foundation 1

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Abstract MRI is unique in its ability to non-invasively and selectively alter tissue magnetization, and create tagged patterns within a deforming body such as the heart muscle. The resulting patterns de ne a time-varying curvilinear coordinate system on the tissue, which we track with coupled B-snake grids. B-spline bases provide local control of shape, compact representation, and parametric continuity. Ecient spline warps are proposed which warp an area in the plane such that two embedded snake grids obtained from two tagged frames are brought into registration, interpolating a dense displacement vector eld. The reconstructed vector eld adheres to the known displacement information at the intersections, forces corresponding snakes to be warped into one another, and for all other points in the plane, where no information is available, a C 1 continuous vector eld is interpolated. The implementation proposed in this paper improves on our previous variational-based implementation and generalizes warp methods to include biologically relevant contiguous open curves, in addition to standard landmark points. The methods are validated with a cardiac motion simulator, in addition to in-vivo tagging data sets.

Keywords: Tagged MRI, Cardiac Motion, Deformable Models, B-Splines, Image Warps.

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1 Introduction 1.1 Tagged MR: Imaging Non-invasive techniques for assessing the dynamic behavior of the human heart are invaluable in the diagnosis of heart disease, as abnormalities in the myocardial motion sensitively re ect de cits in blood perfusion [16, 33]. MRI is a non-invasive imaging technique that provides superb anatomic information with excellent spatial resolution and soft tissue contrast. Conventional MR studies of the heart provide accurate measures of global myocardial function, chamber volumes and ejection fractions, and regional wall motions and thickening. In MR tagging, the magnetization property of selective material points in the myocardium are altered in order to create tagged patterns within a deforming body such as the heart muscle. The resulting pattern de nes a time-varying curvilinear coordinate system on the tissue. During tissue contractions, the grid patterns move, allowing for visual tracking of the grid intersections over time. The intrinsic high spatial and temporal resolutions of such myocardial analysis schemes provide unsurpassed information about local contraction and deformation in the heart wall which can be used to derive local strain and deformation indices from dierent myocardial regions. Several tagged imaging techniques have been proposed in the MR literature. The basic premise in all tagging schemes involve using a set of rf pulses to place tags in thin slices of the myocardium, perpendicular to the imaging plane which can then be tracked over the heart cycle. The method originally proposed by Elias Zerhouni [33] (star-burst tagging) uses a series of slice selective rf pulses (selective in a plane perpendicular to the imaging plane) with a range of gradients in the x and y direction. This produces a number of radial saturation stripes within the image. These saturation bands provide a means of measuring myocardial twist and, if striped tags are used, they can also yield information on motion in the radial direction. More recent developments is based on segmented kspace imaging with breath-hold Cine MR. The limited extent of the Fourier components of a parallel 3

line tagging pattern is used to reduce the data acquisition time for heart wall motion measurement. McVeigh reported the possibility of acquiring movie sequences of the heart with 24ms time resolution in 4-16 heart beats using the Dante/SPAMM pulse sequence [22]. SPAMM [7] is a technique for producing a regular grid pattern in the imaging plane, introduced by Axel and Dougherty. This method uses a binomial pulse to produce spatial modulation of spins in the tissue. A 1-3-3-1 pulse, applied in the presence of a magnetic eld gradient Gx, will produce parallel lines in the direction perpendicular to the applied gradient. If the 1-3-3-1 pulse is repeated a second time, this time using a gradient Gy , then a grid pattern is imposed within the tissue being imaged. The Dante sequence, which is used to create a grid pattern of signal voids, involves application of a series of short rf pulses in the presence of gradients rst in the x direction followed by gradients in the y direction yielding a rectangular grid pattern in the tissue. The total time taken for tagging is of the order of 2msec. It is important that the tagging be performed in as short a time as possible to avoid blurring of the tagged grid due to motion during its formation.

1.2 Tagged MR: Image Analysis Among the various approaches which have been proposed in the literature for analysis of tagged cardiac images, the work in [32] is probably most closely related to ours in that an analysis system based on snakes has been adopted. First, tag data in dierent slices are tracked with snakes in order to recover tag intersection motion within the image slices. This snake approach minimizes an external energy which is sum of intensities for each slice (i.e., assuming the tag points are the darkest image points), together with an internal energy which provides smooth displacements between snakes in successive time points. User interaction is provided through attachment of springs to various points on the snakes in order to guide the curves to fall into the correct local minima. Once the tag positions on the myocardium are 4

found, coordinates of these points in each deformed image are determined within a volumetric nite element model tted to endocardial and epicardial contours. To t a 3D Finite Element Model (FEM) to the stripe displacements, an objective function is formed which minimizes the perpendicular distance of the model point to all of the corresponding undeformed tag planes, thereby determining the FEM nodal parameters and deforming the FEM mesh [32]. Although the FEM model provides good local strain analysis, it results in a large number of model parameters. The work described in [25] considers geometric primitives which are generalization of volumetric ellipsoids. Use of parameter functions in this context allows for spatial variations of aspect ratios of the model to t the LV. The models are also further generalized to parameterize twisting motion about the long axis of the LV as a function of distance along the long axis as well as the radial direction. Forces are computed from SPAMM data points. Contour points apply forces to nodes of closest triangles on the surface in the initial time frame. Each SPAMM intersection point applies a force proportional to distance to the intersecting line of its two original, and perpendicular undeformed tag planes. The forces are applied to model points by distributing forces to FEM nodes with in a Lagrangian dynamics, physicsbased formulation, and the model comes to rest when all the applied forces come to an equilibrium. As the output, the recovered information includes parameter functions describing the aspect ratios of the model as well as parameters which quantify twisting about the long axis of the LV. Though the model provides useful information, it only utilizes information available at tag crossings. Methods based on optical ow have also been applied to the analysis of tagged MR images. An approach called Variable Brightness Optical Flow (VBOF) accounts for temporal variation of signal intensities, thus generalizing the Horn and Schunck's optical ow constraint equation. The primary assumption in deriving Horn and Schunck's optical ow constraint is that the intensity of a material point does not change with time. Hence, this approach is not directly applicable to tagged MR images. 5

The drawback of VBOF as stated in [18] is that the algorithm requires knowledge of the parameters

D0 (proton density), T1 , and T2 in the region of interest. A new algorithm described in [18] relaxes the intensity constancy constraint and allows for intensity variations to be modeled by a more accurate local linear transformation. Physics of MR is used to derive approximations for the intensity variations by assuming a local linear transformation of the intensity, no motion, and adequate temporal resolution which subsequently is utilized in a new optical ow algorithm requiring only approximate knowledge of

T1 in the image. The approach in [19] for analysis of radial tagged images determines the ventricular boundaries on a morphologically processed image. The morphological operators ll-in the tag points on the image. A graph-search technique then determines the optimal inner and outer boundaries of the myocardium. In order to determine the tag locations, the tag pro les are simulated as a function of time, assuming linear gradients and a sinc RF pulse for generating the ip angle of rotation for samples as a function of distance along the gradient. The simulated pro les are subsequently used for nding tag lines by nding points one after the other in a sequence, using initial search starting points on the determined LV boundaries. The outer boundary serves as a stopping criterion for the tag localization procedure in short axis images. Parallel tags in long axis images of the ventricle are processed similar to the short axis images. More recent work uses images from 3 sequences of parallel tags from segmented k-space imaging obtained at dierent times [24]. Following processing of the images as described in the previous paragraph for determining tag locations, the authors perform least squares tting of a truncated power series in the prolate spheroidal coordinate system on the whole of the myocardium in order to measure dense displacements. The steps in the tting procedure include rst, determining
x,
y, and
z in reference to tag planes in the initial undeformed state. The gross ane, linear t is then performed in Cartesian 6

coordinates, and is subtracted from the initial displacement data to remove large bulk motions and linear stretches and shears. A t is then performed in prolate-spheroidal coordinates with power series up to linear term in the radial direction, and up to 5th order in the azimuthal and angular directions. This leads to a 50 parameter t for each of the prolate spheroidal coordinate directions. An alternate approach to motion reconstruction developed in [13] utilizes a multidimensional stochastic model (each component of displacement eld is modeled by a Brownian surface) for the true displacement eld and the Fisher estimation framework to estimate displacement vectors in points on the lattice. The advantage of this framework is an error covariance which determines the number of tag lines needed to achieve a given estimation accuracy. In [1, 2, 3], tag lines are tracked with dynamic programming B-snakes and B-snake grids which will be described fully and validated in this paper. In [28], a volumetric B-solid model was proposed to concurrently track tag lines in dierent image slices by implicitly de ned B-surfaces which align themselves with tagged points. The solid is a 3D tensor product B-spline whose isoparametric curves deform under image forces from tag lines in dierent image slices. For the motion reconstruction problem, an original solution was proposed in [1, 4]. However, in this formulation only information at tag crossings were utilized as part of the reconstruction algorithm. In [2], the method was further extended to that of constrained thin-plate reconstruction of the displacement eld from points and lines based on a variational solution. In the present paper, we improve on the reconstruction technique in [2] and present validation of the methodologies. One advantage of the approach proposed in this paper is that it allows for reconstruction of dense deformations between 2 arbitrary frames in a sequence of tagged images, as motion reconstruction methods generally produce displacement vector elds relative to undeformed tags in the initial frame.

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1.3 Tagged MR: Myocardial Mechanics Many investigators have already utilized MR tagging for better understanding kinematics of the myocardium. It is now established that transmural gradient of shortening across the myocardium exists, where epicardially, there is less segment shortening. As one moves towards the endocardium, the segment shortening of the muscle bers at end-systole approaches twice that found at epicardium. In addition, as reported in [10, 14], a gradient from base to apex of this measure exists, endocardially, at mid-wall, and at the epicardium with more circumferential shortening at apex than base. Using radial tagging, when viewed from the apex, ventricular torsion has been reported to be counter-clockwise at apex, with the base twisting in the opposite direction [11]. It was shown that torsion increases with distance from base, and it varies with distance from the center of the LV cavity. Investigators have reported that torsion reverses rapidly during isovolumic relaxation before diastolic lling. Furthermore, during same period no signi cant circumferential segment length changes occur. However, following mitral valve opening, diastolic lling was found to result in signi cant circumferential segment length changes [27]. Investigators have also been able to demonstrate that the position of apex is essentially xed during systole and that the atrioventricular valve planes move toward the apex during myocardial shortening and ventricular emptying [20]. These reports establish that both the physics of MR tagging as well as its application to the understanding of regional LV function are maturing and that tagging is quite likely to play a major role in patient care in coming years. In order to handle the large magnitude of generated data, it is imperative to develop automated methods for analysis of tagged cardiac images. The organization of the paper is as follows. Section 2 presents a framework for automated tracking of SPAMM tags, and section 3 presents machinery for reconstruction of dense displacements with constrained thin-plate splines. Tracking of SPAMM tags is accomplished with coupled B-snake grids 8

requiring adjustment of few control points in order to specify the location of complete tag grids on an image frame. Once the location of tag lines on the myocardium are speci ed in two frames, the methods of section 3 are applied for reconstructing dense deformations. The reconstruction algorithm utilizes conjugate gradient optimization which converges in few minutes. Section 4 deals with validations of both tracking as well as of reconstruction methods.

2 Coupled B-Snake Grids B-spline curves are suitable for representing a variety of industrial and anatomical shapes [2, 8, 17, 21, 23, 28]. The advantages of B-spline representations are: (1) They are smooth, continuous parametric curves which can represent open or closed curves. For our application, due to parametric continuity, Bsplines will allow for sub-pixel localization of tags, (2) B-splines are completely speci ed by few control points, and (3) Individual movement of control points will only aect their shape locally. In medical imaging, local tissue deformations can easily be captured by movement of individual control points without aecting static portions of the curve. A B-spline curve is expressed as

(u) =

X p B (u)

N ?1 i=0

(1)

i i

where Bi (u) are the B-spline basis functions having polynomial form, and local support, and pi are the sequence of control points of the B-spline curve. Two remarks should be made regarding the sequence of control points: a) The number of control points is much fewer in number than a sampling of the curve (u) on a pixel grid, and b) Except for the case of a completely straight spline, pi 's rarely reside on the actual curve. Coupled snake grids are a sequence of spatially ordered snakes, represented by B-spline curves, which respond to image forces, and track non-rigid tissue deformations from SPAMM data. The spline grids 9

are constructed by having the horizontal and vertical grid lines share control points. By moving a spline control point, the corresponding vertical and horizontal snakes deform. This representation is reasonable since the point of intersection of two tag lines is physically the same material point, the tissues are connected, and furthermore through shared control points, a more ecient representation is achieved. We de ne a MN spline grid by f(M  N ) ? 4g control points which we represent by the set

ffp12 ; p13 ;


; p1;N ?1 g ; fp21 ; p22 ;


; p2;N g ;


; fpM;2; pM;3;


; pM;N ?1gg

(2)

where pij is the spline control point at row i and column j ( gure 1). To detect and localize SPAMM tag lines, we optimize grid locations by nding the minimum intensity points in the image, as tag lines are darker than surrounding tissues. However, there is an additional energy term present in our formulation which takes account of the local 2D structure of image intensity values at tag intersections. Although we can not specify an exact correspondence for points along a tag line, we do know the exact correspondence for points at tag intersections{ assuming 2D motion for the LV. 2 This is the familiar statement of aperture problem in image sequence analysis. The way to incorporate this familiar knowledge into our algorithm and therefore distinguish between 1D and 2D tagged points is to utilize the SSD (sum-of-squared-dierences) function [5] in the minimization

E (p12;


; pM;N ?1) = 1

X Z I ( (u))du + X SSD(v k

k

2

ij

ij )

(3)

where I is the image intensity function, k is a horizontal or vertical spline, vij denotes the intersection point on the pixel grid of snake curves, and 1 and 2 are pre-set constants. The SSD function determines the sum-of-squared-dierences of pixels in a window around point vij in the current frame (with intensity function I ) with a window around the corresponding B-snake grid intersection in the previous frame 2

We note that in general the motion of LV is three-dimensional. However, there is little through-plane motion in the

apical end of the LV especially during systole [29]

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(with intensity function J ). That is, when the location of the grid in I is sought

SSD(vij ) =

K X (I (q ) ? J (q0 ))2 i=1

i

(4)

i

for corresponding locations qi and qi0 in an image window with K total pixels. In order to minimize the discretized version of E , in each iterative step, we compute the gradient of E with respect to pij , perform a line search in the 5E direction, move the control points to the minimum location (which will result in real values for pij ), and continue the procedure until the change in energy is less than a small number, de ning convergence. To compute the gradient of E , it is helpful to note that with pij = (xij ; yij ),

@ E ' E (p12 ;


; (xij +
x; yij );


; pM;N ?1 ) ? E (p12 ;


; (xij ; yij );


; pM;N ?1 ) @xij
x @ E ' E (p12 ;


; (xij ; yij +
y);


; pM;N ?1 ) ? E (p12 ;


; (xij ; yij );


; pM;N ?1 ) @yij
y

(5)

In practice, we have an additional constraint in computing the energy function: we only use the intersections and points on the snake grid which lie on the heart tissue. To track the grid over multiple frames, the localized grid for the current temporal frame becomes the initial grid for the next frame, which is then optimized on the new data.

3 Constrained Thin-Plate Splines 3.1 Continuous Formulation Tracking tissue deformations with SPAMM using snake grids provides 2D displacement information at tag intersections and 1D displacement information along other 1D snake points [2]. The displacement measurement from tag lines however are sparse; interpolation is required to reconstruct a dense displacement eld from which strain, torsion, and other mechanical indices of function can be computed at all myocardial points. In this section, we describe an ecient solution to the formulation in [2] (this improves on methods in computation time and order of convergence) for reconstructing a dense 11

displacement vector eld using localized coordinates of tag positions. In this development, we assume only 2D motion (as is roughly the case towards the apical end of the heart [29]). Although thin-plate warps have been investigated by Bookstein [9], they have been used to interpolate a warp given speci ed landmarks. To proceed more formally, the vector eld continuity constraint is the bending energy of a thin-plate which is applied to the x and y component of the displacement eld (u(x; y); v(x; y)): 1 =

ZZ

u2xx + 2u2xy + u2yy dxdy +

ZZ

2 + 2v2 + v2 dxdy vxx xy yy

(6)

This serves as the smoothness constraint on the reconstructed vector eld, characterizing approximating thin-plate splines. With intersection \springs" in place, the intersections of two grids are \pulled" towards one another by minimizing 2 =

X(u ? u

int )2 + (v ? vint )2

(7)

In (7), uint and vint are the x and y components of displacement at tag intersections as well as intersections of myocardial contours with tag lines. The form of the intersection spring constraints is similar to depth constraints in surface reconstruction from stereo, and has also been used in a similar spirit in [32]. Assuming 2D tissue motion, a further physical constraint is necessary: any point on a snake in one frame must be displaced to lie on its corresponding snake in all subsequent frames. This constraint is enforced by introducing a sliding spring. One endpoint of the spring is xed on a grid line in the rst frame, and its other endpoint is allowed to slide along the corresponding snake in the second frame, as a function of iterations. We minimize 3 =

X n(x + u ? x)2 + (y + v ? y)2o

(8) 12

along 1D snake points. In the above equation, (x; y) are the coordinates of a point on the snake in the current frame, and (x; y) is the closest point to (x + u; y + v) on the corresponding snake in the second frame. An optimization function can be obtained by a linear combination of the three terms in (6), (7), and (8). There are two fundamentally dierent approaches to the minimization of this function, namely, nite elements and nite dierences. The method of nite elements involves guessing the form of the solution (a continuous function or a combination of continuous functions) and then calculating the parameters of this function. The implementation of a nite elements method is usually very fast since there are few parameters to be calculated. However, in many cases, the presumption about the form of the solution may be too restrictive. The nite dierences approach on the other hand needs no such initial guess. But this method yields the values of the solution only at selected grid points - values at points in between need to be interpolated. Often, the values have to be calculated at a large number of grid points and in these cases, the nite dierence algorithms are slower. In this paper, we consider only nite dierence techniques. Finite element methods will be investigated elsewhere. Again, there are a number of ways to proceed. We nd a quadratic approximation to the optimization function and then use a conjugate gradient or quasi-Newton algorithm to minimize this quadratic approximation. The reason of course is that quasi-Newton algorithms have quadratic convergence properties for functions which are almost quadratic.

3.2 Conjugate gradient and quasi-Newton algorithms Let (u(x; y); v(x; y)) be the displacement eld as before. The objective function (u; v) which needs to be minimized is the linear combination  = 1 1 + 2 2 + 3 3 :

(9) 13

Note that x and y are dependent on u and v respectively which makes the function 3 (u; v) nonquadratic. We can derive the Euler-Lagrange equations for the variational problem in (9) and solve the resulting system of equations [2]. In this paper, we develop a more ecient approach. We follow [15] and straightaway discretize the function  in (9). Assuming the distance between two adjacent grid points to be

ui+1;j ? uij = ui;j+1 ? uij = h;

(10)

the second order partial derivatives (uxx )ij , (uxy )ij and (uyy )ij at the point (i; j ) can be approximated by (uxx )ij = ui+1;j ? 2hu2i;j + ui?1;j

(uxy )ij = ui+1;j +1 ? ui+1h;j2 ? ui;j +1 + ui;j (u ) = ui;j +1 ? 2ui;j + ui;j ?1 yy ij

(11)

h2

The discrete form of the function 1 can be obtained by substituting the discrete derivatives into the rst equation in (6). The partial derivatives of 1 can be calculated using the computational molecule approach discussed in [30] though special attention should be paid in computing the molecules near the endocardial and epicardial boundaries where the smoothness constraint should break in order not to smooth over the motion discontinuity. The discretization of the function 2 and calculation of its partial derivatives is almost trivial. Let us consider the function 3 which is non-quadratic. The partial derivatives of 3 are (3 )u = (u + x ? x)(1 ? xu ) + (v + y ? y)(?yu ) (3 )v = (v + y ? y)(1 ? yv ) + (u + x ? x)(?xv )

(12)

For simpli cation, we now make two approximations. For vertical grid lines, the x-coordinates of curves only vary slightly, and as the vertical lines are spatially continuous, xu is expected to be small. 14

Furthermore, for vertical lines y changes minutely as a function of u, so that yu  0. For horizontal grid lines, the y coordinates of curves also vary slightly along the length of the lines, and since these are spatially continuous curves, yu is expected to be small. Note that these approximations will hold under smooth local deformations, as is expected in the myocardial tissue. Only xu for horizontal grid lines, and yv for vertical grid lines is expected to vary more signi cantly. The approximate derivatives are now given by: (3 )u ' (u + x ? x)(1 ? Thor xu ) (3 )v ' (v + y ? y)(1 ? Tver yv )

(13)

The variables Thor and Tver are predicates equal to one if the snake point of interest lies on a horizontal, or a vertical grid line. Needless to say, the above functions can be discretized by replacing the continuous values by the corresponding values at the grid points. After discretization, a typical quadratic optimization problem takes the following form:

f (x) = c ? bT x + 21 xT Ax

(14)

where x is the vector of variables, A is the constant Hessian matrix of second order partial derivatives and b and c are constant vectors. In the present problem, the terms 1 and 2 can be cast in the above form. Unfortunately, in the term 3 , the values x and y are dependent on x and y respectively which makes 3 non-quadratic. The discrete optimization function form of 3 is given by: (x) = c ? bT x + 12 xT Ax + 3 3 (x)

(15)

where A, b and c are constants and include the contributions from 1 1 and 2 2 . we are now ready to look at speci c minimization algorithms. The rst algorithm we investigate is the conjugate gradient (CG) algorithm. For an order N quadratic problem, the CG algorithm is guaranteed to converge in N iterations. Moreover, it does 15

not store the Hessian matrix and requires o(N ) storage for an order N optimization problem. Note that the CG algorithm does not explicitly calculate or store the Hessian matrix A and can be adapted to the function  in (15). We do not know the Hessian matrix for . However, we do know how to calculate the derivative r(p) using the derivatives for the functions 1 , 2 and 3 . We use this knowledge of the gradient of  in the implementation of the CG algorithm. As a nal point, since  is non-quadratic, the algorithm may not converge in N iterations. For a description of the CG algorithm please refer to the appendix. Quasi-Newton algorithm is a dierent optimization method that we have investigated. It diers from CG in that it has higher memory requirements but better convergence properties for non-quadratic functions. By quasi-Newton methods we mean techniques which use an approximation to the inverse Hessian matrix in each iteration as opposed to Newton methods which use the exact inverse. A generic quasi-Newton algorithm calculates and stores an approximation to the inverse Hessian matrix in each iteration. Hence for an order N optimization problem, this method needs o(N 2 ) storage. The advantage of a quasi-Newton algorithm lies in that it has quadratic convergence properties for general smooth functions (not necessarily quadratic). A speci c quasi-Newton algorithm is characterized by the approximation it uses for the Hessian matrix. The quasi-Newton method used in this paper is called the Davidon-Fletcher-Powell (DFP) algorithm, an overview of which is given in the appendix.

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4 Validations 4.1 Methods 4.1.1 Interactive B-Spline Environment We have built an environment where a user can interactively apply various deformations to a grid, and view the reconstructed displacements. The environment, implemented on an SGI INDY with Open Inventor represents a grid with coupled B-splines which can be molded interactively by moving the spline control points in the plane. The spline bases for this synthetic environment were non-periodic and of order 4 (i.e.; cubic polynomials). An example of an undeformed grid is displayed in gure 2. The user can create various deformations by interactively pushing and pulling on the control points.

4.1.2 Cardiac Simulator An environment based on a 13 parameter kinematic model of Arts et al. [6] has been implemented as was described in [31] for simulating a time sequence of tagged MR images at arbitrary orientation. Based on user-selected discretization of the space between 2 concentric shells and by varying the canonical parameters of the model, both a sequence of tagged MR images as well as a \ground truth" vector eld of actual material point deformations are available. A pair of prolate spheroids represents the endocardial and epicardial LV surfaces, and provides a geometric model of the LV myocardium. The motion model involves application of a cascade of incompressible linear transformations describing rigid as well as non-rigid motions. Once a discretization step is assumed and a mesh for tessellating the 3D space is generated, the linear matrix transformations are applied in a sequence to all the mesh points so as to deform the reference model. The parameters of 17

the motion model, referred to as k-parameters, and the transformations to which they correspond are stated in table 1. In order to simulate MR images, an imaging plane intersecting the geometric model is selected, and tagged spin-echo imaging equations are applied for simulating the in-vivo imaging process. Figure 3 displays an undeformed 3D model (with all the k-parameters identical to zero) and a simulated tagged image corresponding roughly to the middle of the LV. The imaging and geometric parameters of the model for this gure as well as the validation experiments are stated in table 2. In order to assess tracking performance with coupled B-snakes, starting with a user-initialized grid in frame 1, the algorithm described in section 2 was used to automatically track tag lines in a time sequence of simulated images. In a separate setting, an expert was also asked to outline the tags by manually adjusting the control point locations of a coupled B-snake grid in each frame of the same image sequence. The gure of merit proposed in [12] was then modi ed in order to compare each manually outlined B-snake grid, G1 , with the corresponding grid automatically determined by the algorithm, G2 :

d(G1 ; G2 ) = M + 1N ? 4

X 1 nmax min D( k

2

u

w

k;1 (u); k;2 (w)) + max w min u D(k;2 (w); k;1 (u))

o

(16)

where k;1 and k;2 are corresponding B-snake curves (horizontal or vertical), u and w are their corresponding spline parameters (a 100 points per span sampling of u and w was employed), M + N ? 4 is the total number of splines in the deformable grid, and D is the Euclidean distance metric. For the purposes of validating 2D displacement eld reconstructions, we have used the parameters k2 , k4 , k5 , and k10 for generating 2D deformations of the geometric model, based on which images and in addition 2D displacement vector elds of actual material points are produced. The error norms used in comparing the ground truth vector eld (Vg ) with the vector eld measured by our warp algorithm (Vm ) are:

"L = N1

X jjV j ? jV jj m g

(17) 18

and

" = P 1jV j g

X jV j: arccos Vg
Vm g

(18)

jVg jjVm j

where "L measures the average dierence in length between Vg and Vm , and " measures the deviation in angle between Vg and Vm . The latter measure requires further explanation. As can be seen from (18), we weigh individual angle deviations by the magnitude of the material point displacement vector; normalized by the sum of magnitude of all ground truth vectors. The reason for this is to emphasize angle deviation of points which have large displacements, and similarly to de-emphasize the angle deviation of points which have a smaller displacement.

4.1.3 In-Vivo Imaging A SPAMM pulse sequence was used to collect images of normal healthy volunteers. Multiple images in both short-axis (SA) and long axis (LA) views of the heart were collected to cover the entire volume without gaps. Immediately after the ECG trigger, rf tagging pulses were applied in two orthogonal directions. The repetition time (TR) of the imaging sequence was approximately 7.1 msec, the echo time (TE) was 2.9 msec, the rf pulse ip angle was 15 degrees, and the time extent of rf tag pulses was 22 msec. Echo sharing was used in collecting each time-varying image sequence for given slice position. Five data lines were collected for any time frame during each heart cycle, but two data lines were overlapped between two consecutive cardiac frames, resulting in an eective temporal resolution of approximately 22 msec. Other imaging parameters were: eld of view = 330mm, data acquisition matrix size = 160  256 (phase encoding by readout), in-plane resolution = 2:1  1:3mm2 , slice thickness = 7mm, and tag spacing = 7mm. The total imaging time was therefore 32 heart beats for each Cine sequence, and the subject was instructed to breath only following each Cine acquisition. Since there were 19 volumetric temporal frames and 17 spatial slices in each image volume, all of the images were 19

acquired in 544 heartbeats. As \out-of-plane" movement of the LV primarily occurs in slices close to the valve-plane; i.e., at the top of the LV, for the purposes of extraction of 2D deformations, here we only consider image slices near the ventricular apex.

4.2 Results 4.2.1 Coupled-Snake Tracking Cardiac Simulator Figures 7,8, and 9 show error plots (equation (16)) for our spline tracker (over entire image sequences) for a series of 2D deformations obtained by varying k2 , k4 , k5 , k10 , k11 , and k12 with a temporal resolution of 20 msec. In all experiments, 1 = 75, and 2 = 1. The details of the spline tracker were as follows: the splines were of order 4, and a B-spline span roughly corresponded to a tag spacing. To illustrate, gures 4 and 5 show the deformations of the undeformed model when varying k2 and k4 , and gure 6 shows the automatically localized grids overlaid on top of 4 frames of the simulated image sequence corresponding to k2 .

In-Vivo Tracking An in-vivo result from localization and tracking of spline grids with gradient descent are shown in gure 10. Note that the portions of the snake grids lying within the ventricular blood and in the liver do not contribute to the energy of the snake grid. The endocardial and epicardial contours were each manually segmented through-out the sequence using a 6 control point B-spline representation. 20

The contours delimit the part of snake grids on the myocardium which are used by the algorithm.3 Initialization of B-snake grids may be performed automatically based on the imaging protocol of section 4.1.3. which includes SA as well as LA acquisitions. Since location of one set of SA tag planes are related to LA image slices and furthermore, the tag interspacings are known, the B-snake grid is automatically initialized at the outset from image headers without any user intervention.

4.2.2 Constrained Thin-Plate Splines Interactive B-Spline Environment In gure 11, a realistic grid is displayed which closely follows actual myocardial deformations; i.e., subendocardial segment shortening more pronounced than subepicardial segment shortening (hence the bowing-in eect). Next, new location of actual sample points of the undeformed grid of gure 2 are high-lighted as small dark squares. As can be seen, the sample points on the undeformed grid coincide with points on the deformed grid. Finally, the reconstructed vector eld is displayed on the deformed grid. To elucidate, table 3 reports the values for 1 , 2 , and 3 in the beginning of iterations and at convergence for gure 11. Visually, results from DFP are identical to those of CG. As was concluded, on the average DFP performs slightly better than CG but at the cost of more storage. In the remainder of the paper we will adopt CG in performing displacement reconstructions.

Cardiac Simulator 3

We have found automatic determination of endocardial and epicardial contours in SPAMM image sequences to be a

formidable task due to presence of tag lines.

21

Figures 12 and 13 show the angle and length errors by comparing Vm and Vg as a function of a range of values of k2 , k4 , k5 , and k10 , keeping the rest of the k parameters constant. Additionally, as part of the validations and in order to test the sensitivity of the algorithms to dierent values of algorithm coecients 1 , 2 , and 3 , we varied each of these coecients individually in the range f0; 1;


; 10g, and kept the other 2 coecients at the constant value of 1 (excluding 1 = 0). The error bars in these plots show the 3 range on either side of the error mean for particular values of each k parameter. As can be seen from the gures, to a large degree the displacement reconstruction algorithm is insensitive to the exact values of i 's. An additional remarkable point regarding the error plots is the fact that for smaller motions, the value of " is larger than that for bigger motions. The reason for this unintuitive result can only be attributed to the larger percent inaccuracies in reconstruction of smaller displacements by the warping algorithm. Also, it should be noted that error plots in gures 12 and 13 subsume the errors incurred in localization of tags and myocardial contours (for these validations, tag and contour localization is performed through manual placement of control points of B-spline grids.) Although the magnitude of errors are bound to be smaller if accurate location of contour and tag lines in the simulated images were to be used, our complete system for tracking and reconstruction of tag lines would not be tested, and furthermore since the exact location of tags and contours are not known in real images, phantom validation results would not be a good model of realistic situations. Finally, gure 14 displays true and reconstructed vector elds corresponding to rotation and torsion of the computational phantom. In-Vivo Validations

The third component of our warp validation studies includes reconstruction of myocardial displace22

ments in normal human volunteers. Results of application of techniques to in vivo data with i = 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 dierent 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 dierent 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 C 1 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 defor23

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 constrained 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, ConjugateGradient 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 24

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 vector x0 . Let g0 = h0 = ?rf (x0) where h0 is the initial descent direction.

 Step 2: Minimize along the current descent direction hi and calculate the next solution vector

xi+1.  Step 3: Calculate the vector gi+1 = ?rf (xi+1).  Step 4: Calculate the next descent direction hi+1 using the following formulae:

i = (gi+1 g? :ggi ):gi+1 i i

hi+1 = gi+1 + ihi :

(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 vector x0 , the inverse Hessian H0 = I, I being the identity matrix, and the initial descent direction d0 = ?rf (x0 ).

 Step 2: Minimize the function along the current direction di and calculate xi+1 . 25

 Step 3: Update the inverse Hessian matrix Hi+1 at the point xi+1 according to:

Hi+1 = Hi + correction

(20)

 Step 4: Calculate the next descent direction di+1 using

di+1 = ?Hi+1:(rf (xi+1)):

(21)

 Step 5: Check termination criterion. Go back to step 2 if necessary.

References [1] A. A. Amini. Automated techniques for measurement of cardiac motion from MR tagging. Proposal funded by The Whitaker Foundation, 1992. [2] A. A. Amini, R. W. Curwen, and John C. Gore. Snakes and splines for tracking non-rigid heart motion. In European Conference on Computer Vision, pages 251{261, University of Cambridge, UK, April 1996. [3] A. A. Amini and et al. Energy-minimizing deformable grids for tracking tagged MR cardiac images. In Computers in Cardiology, Durham, North Carolina, pages 651{654, October 1992. [4] A. A. Amini and et al. MR physics-based snake tracking and dense deformations from tagged MR cardiac images (oral presentation). In AAAI Symposium on Applications of Computer Vision to Medical Image Processing, Stanford University, Stanford, California, March 1994.

[5] P. Anandan. A computational framework and an algorithm for the measurement of visual motion. International Journal of Computer Vision, 2:283{310, 1989.

[6] T. Arts, W. Hunter, A. Douglas, A. Muijtjens, and R. Reneman. Description of the deformation of the left ventricle by a kinematic model. J. Biomechanics, 25(10):1119{1127, 1992. 26

[7] L. Axel and L. Dougherty. MR imaging of motion with spatial modulation of magnetization. Radiology, 171(3):841{845, 1989.

[8] A. Blake, R. Curwen, and A. Zisserman. A framework for spatio-temporal control in the tracking of visual contours. International Journal of Computer Vision, 11(2):127{145, 1993. [9] F. Bookstein. Principal warps: Thin-plate splines and the decomposition of deformations. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-11:567{585, 1989.

[10] M. Buchalter, C. Sims, A. Dixon, and et al. Measurement of regional LV function using labelled magnetic resonance imaging. The British Journal of Radiology, 64:953{958, 1991. [11] M. Buchalter, J. Weiss, W. Rogers, and et al. Noninvasive quanti cation of left ventricular rotational deformation in normal humans using magnetic resonance imaging myocardial tagging. Circulation, 81:1236{1244, 1990.

[12] V. Chalana and Y. Kim. A methodology for evaluation of boundary detection algorithms on medical images. IEEE Transactions on Medical Imaging, 16(5):642{652, 1997. [13] T. Denney and J. Prince. Reconstruction of 3-d left ventricular motion from planar tagged cardiac MR images: An estimation-theoretic approach. IEEE Transactions on Medical Imaging, 14(4):625{ 635, December 1995. [14] N. R. Clark et al. Circumferential myocardial shortening in the normal human left ventricle: Assessment by magnetic resonance imaging using spatial modulation of magnetization. Circulation, 84:67{74, 1991. [15] W. Grimson. An implementation of a computational theory of visual surface interpolation. Computer Vision, Graphics, and Image Processing, 22:39{69, 1983.

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[16] W. Grossman. Assessment of regional myocardial function. J. of Amer. Coll. of Cardiology, 7(2):327 { 328, 1986. [17] A. Gueziec. Surface representation with deformable splines: Using decoupled variables. IEEE Computational Science and Engineering Magazine, pages 69{80, Spring 1995.

[18] S. Gupta and J. Prince. On variable brightness optical ow for tagged MRI. In Information Processing in Medical Imaging (IPMI), pages 323{334, 1995.

[19] M. Guttman, J. Prince, and E. McVeigh. Tag and contour detection in tagged MR images of the left ventricle. IEEE Transactions on Medical Imaging, 13(1):74{88, 1994. [20] E. Homan, J. Rumberger, L. Dougherty, and et al. A geometric view of cardiac eciency. J. Amer. Coll. of Cardiology, 13(2):86A, 1989.

[21] A. Klein, F. Lee, and A. Amini. Quantitative coronary angiography with deformable spline models. IEEE Transactions on Medical Imaging, 16(5):468{482, 1997.

[22] E. McVeigh and E. Atalar. Cardiac tagging with breath-hold cine MRI. Magnetic Resonance in Medicine, 28:318{327, 1992.

[23] S. Menet, P. Saint-Marc, and G. Medioni. B-snakes: Implementation and application to stereo. In Proceedings of the DARPA Image Understanding Workshop, Pittsburgh, PA, pages 720{726, Sept.

1990. [24] W. O'Dell, C. Moore, W. Hunter, E. Zerhouni, and E. McVeigh. Three-dimensional myocardial deformations: Calculation with displacement eld tting to tagged MR images. Radiology, 195(3):829{835, 1995.

28

[25] J. Park, D. Metaxas, and L. Axel. Volumetric deformable models with parameter functions: A new approach to the 3d motion analysis of the LV from MRI-SPAMM. In International Conference on Computer Vision, pages 700{705, 1995.

[26] W. Press, B. Flannery, S. Teukolsky, and W. Vetterling. Numerical recipes in C. Cambridge University Press, Cambridge, 1988. [27] F. Rademakers, M. Buchalter, and et al. Dissociation between left ventricular untwisting and lling: Accentuation by catecholamines. Circulation, 85:1572{1581, 1992. [28] 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 1997. [29] N. Reichek. Magnetic resonance imaging for assessment of myocardial function. Magnetic Resonance Quarterly, 7(4):255{274, 1991.

[30] D. Terzopoulos. Multiresolution Computation of Visible Representation. PhD thesis, MIT, 1984. [31] E. Waks, J. Prince, and A. Douglas. Cardiac motion simulator for tagged MRI. In Proc. of Mathematical Methods in Biomedical Image Analysis, pages 182{191, 1996.

[32] A. Young, D. 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, September 1995. [33] E. Zerhouni, D. Parish, W. Rogers, A. Yang, and E. Shapiro. Human heart: Tagging with MR imaging { a method for noninvasive assessment of myocardial motion. Radiology, 169:59{63, 1988.

29

P12

P13

P14

P15

P22

P23

P24

P25

P21

P26 P32

P33

P34

P35

P31

P36

P42

P43

P44

P45

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.

30

Figure 2: An undeformed coupled B-spline grid. The squares at grid intersections high-light location of control points.

31

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.

32

Figure 4: Deformed models of the LV resulting from change of k2 from 0:2 to 0:8 in increments of 0:2

33

Figure 5: Deformed models of the LV resulting from change of k4 from ?0:02 to ?0:08 in increments of ?0:02

34

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.

35

0.25

0.2

0.2

cm

cm

0.25

0.15

0.15

0.1

0.1

0.05

0.05

0 −1

−0.8

−0.6

−0.4

−0.2

0 k2

0.2

0.4

0.6

0.8

0 −0.1

1

−0.08

−0.06

−0.04

−0.02

0 k4

0.02

0.04

0.06

0.08

0.1

0.25

0.25

0.2

0.2

cm

cm

Figure 7: The gure of merit (equation (16)) for coupled B-snake tracking as a function of k2 and k4 .

0.15

0.15

0.1

0.1

0.05

0.05

0 −0.06

−0.04

−0.02

0 k5

0.02

0.04

0 −0.25

0.06

−0.2

−0.15

−0.1

−0.05

0 k10

0.05

0.1

0.15

0.2

0.25

0.25

0.25

0.2

0.2

cm

cm

Figure 8: The gure of merit (equation (16)) for coupled B-snake tracking as a function of k5 and k10 .

0.15

0.15

0.1

0.1

0.05

0.05

0 −0.2

−0.18

−0.16

−0.14

−0.12

−0.1 k11

−0.08

−0.06

−0.04

−0.02

0 −0.2

0

−0.18

−0.16

−0.14

−0.12

−0.1 k12

−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 . 36

Figure 10: Results of tracking SPAMM images with deformable spline grids ( 1 = 100, 2 = 1).

37

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.

38

angle error

angle error

0.4

0.4

0.3

0.3

PI

0.5

PI

0.5

0.2

0.2

0.1

0.1

0

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

0

1

−0.1

−0.08

−0.06

−0.04

−0.02

length error

0

0.02

0.04

0.06

0.08

0.1

0.02

0.04

0.06

0.08

0.1

length error

0.08

0.08

0.06

0.06

CM

0.1

CM

0.1

0.04

0.04

0.02

0.02

0

−1

−0.8

−0.6

−0.4

−0.2

0 K2

0.2

0.4

0.6

0.8

0

1

−0.1

−0.08

−0.06

−0.04

−0.02

0 K4

Figure 12: The error plots for angle and length error for k2 and k4 . Please see text for details.

angle error

angle error

0.4

0.4

0.3

0.3

PI

0.5

PI

0.5

0.2

0.2

0.1

0.1

0

−0.06

−0.04

−0.02

0

0.02

0.04

0

0.06

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−0.2

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−0.1

−0.05

length error

0

0.05

0.1

0.15

0.2

0.25

0.05

0.1

0.15

0.2

0.25

length error

0.08

0.08

0.06

0.06

CM

0.1

CM

0.1

0.04

0.04

0.02

0.02

0

−0.06

−0.04

−0.02

0 K5

0.02

0.04

0

0.06

−0.25

−0.2

−0.15

−0.1

−0.05

0 K10

Figure 13: The error plots for angle and length error for k5 and k10 . Please see text for details.

39

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).

40

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).

41

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. 42

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.

43

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 k k  R R  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 x

y

i

o

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.

44

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, and 3 = 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 (in mm) as a function of 1 , 2 , and 3 . The \ground-truth" in this case was chosen to be the reconstructed displacement eld with i = 1.

45