## Displacement Estimates

The second task in tagged MR image analysis is to compile the information obtained from 2D images into a complete 3D description of LV motion. This is most conveniently done by estimating the function w(x, t)=p(x, t)—x, (12)

which describes the displacement necessary to move a material point from its deformed position x to its reference position p. To estimate this function, we assume we have obtained images from many spatial positions such as those depicted in Fig. 3 and that those images contain various combinations of three orthogonal tags orientations g 1, g2, and g3.

From the images, the same fundamental information for estimating w(x,t) is available, regardless of whether tag line tracking, optical flow, or HARP analysis is used for processing the images. To explain what that information is, suppose a measurement is obtained for a position y in an image with tag direction g{. Then that measurement gives

Ui(x, t) = eTw(x(y), t), where ei is the unit vector in the tag direction defined by

The measurement ui(x, t) is a 1D component of the 3D displacement vector w(x,t) evaluated at x(y). This concept is readily apparent for tag line tracking by referring to Fig. 4. For optical flow methods the displacement component is obtained by backward integration of the velocity field, and in HARP the displacement component is obtained by relating a phase measurement to a plane of constant phase at the reference time. We divide the measurements for all images into three sets r2, and r3 where r, contains all points x at which the component of displacement in the direction e, is known.

Often, two components of displacement are measured for the same point in the image. In tag line tracking this occurs at the tag line intersections of Fig. 6; in optical flow and HARP methods, two components are normally available for every pixel in the image. In general, though, it is not possible to simultaneously measure three components of motion for a given point. Additionally, there is typically 5 to 8 mm of separation between image planes such as those depicted in Fig. 3, accounting for large regions of space for which no measurements are available. The estimate of w(x, t) must therefore be the map that best reconciles the known mechanical properties of the LV with the measured components of displacement u¡(x, í), xe r¿, and interpolates between measurements.

Which image processing method to use for obtaining the sets of measurements is open to debate. Existing methods for determining 3D motion [4,7,40,41] are tailored around tag line tracking, although they will readily incorporate data from optical flow or HARP processing techniques as well. The primary advantage of tag line tracking is that it has the best demonstrated accuracy. In comparison, optical flow suffers from accumulating errors that result from numerical approximations of derivatives and integrals. HARP may prove to have accuracy comparable to tag line tracking methods, but HARP techniques have not had the rigorous testing necessary to prove their accuracy.

The disadvantage of tag line tracking is the sparsity of data, relative to optical flow and HARP methods, which produce data for every pixel. To illustrate this sparsity, suppose it were possible to view the deformation of an entire tag plane, not just the tag line formed where it intersects with an image plane. Then, use of three orientations of tag planes would induce a 3D grid that deforms with LV motion as depicted in Fig. 9. The motion of this grid would, however, be insufficient for determining the exact 3D motion of points except at the grid intersections where all three components of motion are known. This concept is, in fact, the basis of the MR markers method , where the intersections are tracked using splines to reconstruct the deforming grid. For typical tagged MR images, however, these intersections are spaced apart by several millimeters and only a few hundred exist within the LV myocardium.