Bw

where, as a reminder, H = [h1, h2] defines the image orientation in 3D space. Thus, measuring $(y, t) provides information regarding both the reference map and the deformation gradient for every material point in the image.

To measure $(y, t), harmonic phase (HARP) images are obtained [36-39]. HARP images utilize the presence of distinct spectral peaks in the Fourier transform of tagged images, as illustrated in Fig. 8. The locations of the peaks are, by the modulation theorem, determined by the spatial frequency of the tag pattern. In particular, there will be a spectral peak located at w = HTg. If all spectral information except the single spectral peak centered at w is removed through filtering, and the inverse Fourier transform is applied, the resulting image is very nearly where f is the position in Fourier space and BW is the filter bandwidth. In practice, BW must be chosen carefully to minimize the influence of other spectral peaks, while maximizing information obtained from the desired peak. Transforming back to the spatial domain results in the complex HARP image.

Then, a phase measurement for each pixel in the HARP image is easily obtained from the real and imaginary components using the arctangent operator. This measurement is, however, subject to wrapping (see Fig. 8c) because the range of the arctangent is only [— n, n). We refer to the wrapped phase as ^^(y, t), which must be unwrapped to obtain $(y, t). To unwrap t), points of constant phase are tracked by identifying the nearest point in each subsequent image with the same wrapped phase. The absolute phase at time t is then determined by the known phase of the corresponding point at t = 0. In the time between consecutive images, LV motion is assumed to be small enough that phase ambiguity is not a problem.

Alternatively, tracking and phase unwrapping may be avoided by directly calculating the 2D apparent strain associated with the apparent motion (see Fig. 8d). For this, the 2D deformation gradient Fapp of apparent motion is needed, the inverse of which is given by

This 2D deformation gradient is related to the 3D deformation gradient F by

Comparison of this expression to that for Vy$(y, t) (11) shows them to be very similar. In fact, if two HARP images are obtained with associated tag directions g1 and g2 that are linearly independent combinations of h1 and h2, then we may calculate app

gjK gjhi gjh2 gJh2

where b(t) is related to tag fading. Thus, the HARP image takes on complex values with the phase of each pixel given by $(y, t). HARP has the added benefit of potentially rapid imaging because MR images are acquired in the Fourier domain and a single spectral peak can be acquired rapidly.

To filter one spectral peak and obtain a HARP image, a standard tagged image is transformed into the Fourier domain (or the raw image data are kept in the Fourier domain). Then, a filter is selected with central frequency w and smooth rolloff, such as where ^1(y, t) and $2(y, t) are the associated phase maps of the HARP images. The reason this calculation may be performed without phase unwrapping is that the spatial derivatives of $(y, t) and t) are the same except at points of discontinuity in the wrapped phase. Those points of discontinuity are easily addressed by a local unwrapping procedure. Then, determining phase gradients through finite difference approximations leads directly to apparent strain. The down side of this approach is that apparent strain may not be closely related to the true strain. For example, a strain-free rotation can produce a nonzero apparent strain.

FIGURE 8 In HARP processing, one spectral peak is extracted from the raw Fourier data of a tagged MR image (a). Taking the inverse Fourier transform yields a complex image with a magnitude (b) and wrapped phase value (c) for each pixel. The images show a close-up of the LV and in (c), black is a phase of — n and white is + n. For clarity, the phase has been masked using a thresholded version of the magnitude image. Combination of two HARP images with different tag directions allows apparent strain to be depicted (d). In this case circumferential shortening is shown for an LV undergoing abnormal contraction — evidenced by the bright regions at the upper left that indicate elongation. See also Plate 56.

In cardiac applications, however, there is generally good correlation between apparent strain and true strain.

HARP also has the potential to provide out-of-image-plane motion measurements because the phase of a HARP image is given by (10) regardless of the direction of g. If three independent tag directions are used that span 3D space, HARP can, in theory, estimate p(y, t) completely. In practice, however, this requires out-of-plane tags that are affected by the finite thickness of image slabs. This effect has yet to be resolved so that 3D measurements from HARP have not been demonstrated.

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