## Optical Flow

Although tag line tracking provides highly accurate measurements of the reference map, those measurements are limited to the relatively sparse regions along tag lines. To produce denser measurements from tagged MR images, optical flow [26] techniques have been used [27-29] in which the velocity is estimated for every point in an image. In the case of 2D tagged MR images, optical flow processing yields an estimate of the 2D apparent velocity field v(y, t).

The basis for measuring velocity using gradient-based optical flow techniques is an identity known as the brightness constraint equation. It is derived by taking the material time derivative of the brightness function — that is, the partial derivative of I(y, t) holding q constant. This derivative is complicated by the fact that y itself is an implicit function of q and t given by the forward map of apparent motion y(q, t). Application of the chain rule to the material time derivative therefore yields

where Vy is the gradient with respect to y and It (y, t) is the partial derivative of I with respect to t (holding y constant). Equation (8) is the brightness constraint equation.

In order to use (8) for estimating the velocity field v(y, t) it is necessary to calculate spatiotemporal derivatives of the available image data. The spatial gradient VyI(y, t) at each image pixel is approximated using finite differences involving neighboring pixels. The temporal derivative It (y, t) is approximated using the change in pixel intensity from one image to the next in the sequence. Finally, a value for I(y, t) is needed. It is most common in the computer vision literature to assume that a particle maintains a constant brightness [30], so that I(y, t) = 0, an assumption that has been applied to tagged MRI [28]. In tagged MRI, however, the brightness of a particle does change — the so-called tag fading problem.1 To account for this effect, Prince and McVeigh propose variable brightness optical flow (VBOF) [27], which estimates I(y, t) using a model based on the imaging protocol and physical tissue properties. Alternatively, the framework of Gennert and Negahdirapour's optical flow (GNOF) [31] may be used, in which brightness variation is assumed to satisfy the linear model

Then, tag fading is estimated as part of the algorithm itself [29].

through-plane motion also causes brightness variations, but this effect is typically small compared to tag fading.

Regardless of the assumption made about brightness variation, the brightness constraint equation only provides one equation, where as the unknown v(y, t) has two components. To solve this ill-conditioned problem, optical flow is cast as a variational formulation involving the minimization of the functional

BCE "

a 2Ec

'smooth

where ebce ^

a term that encourages agreement with the brightness constraint equation, and Esmooth is a differential smoothness penalty on v(y, t), typically given by ||Vyv(y, t)||2. The regularization parameter a determines the trade-off between velocity smoothness and agreement with the brightness constraint equation.

Minimizing (9) can be done analytically to yield a set of coupled partial differential equations. These, in turn, can be solved numerically using the calculated spatiotemporal image derivatives to yield a sequence of estimated velocity fields (see [27]). It should be noted that in GNOF, the multiplier and offset fields, mt (y, t) and ct (y, t), must be estimated along with v(y, t). Also, in VBOF I(y, t) is estimated recursively using past estimates of v(y, t). Details can be found in the respective papers.

Gradient-based optical flow techniques are well suited for processing tagged MR images because tagging produces the spatial brightness gradients on which the brightness constraint equation depends. Best results are obtained when two SPAMM patterns are applied using only two RF pulses per pattern and orthogonal gradient pulse directions g: and g2. The result is a tag pattern that varies sinusoidally in both gradient directions, as shown in Fig. 7. Also shown in Fig. 7 is the velocity field obtained by applying optical flow to the two images. This motion pattern reveals the early rotation of the left ventricle in this midventricular cross-section. Within a few frames (not shown), this rotation gives way to the strong radial contraction typical of normal motion. One characteristic of these estimates is the smoothness of the velocity field across object boundaries— endocardium and epicardium, in particular — which is caused by the regularizing term in (9). The use of bandpass optical flow, which uses Hilbert transform techniques to shift the phase of the sinusoidal tag patterns, has shown some success in reducing the effects of regularization [32].

In addition to visualization of left-ventricular motion, there are other potential uses for computed velocity field sequences. These applications are identical to the uses described for phase contrast MRI [33], which also produces a sequence of velocity fields using a totally different approach. For one, it is possible

FIGURE 7 Results of optical flow techniques: The images on the left are two in a sequence of images depicting LV contraction after tagging with bidirectional sinusoidal tags. Application of optical flow to these two images produces the velocity field on the right, showing the region around the LV.

FIGURE 7 Results of optical flow techniques: The images on the left are two in a sequence of images depicting LV contraction after tagging with bidirectional sinusoidal tags. Application of optical flow to these two images produces the velocity field on the right, showing the region around the LV.

to compute spatial derivatives of the velocity fields and compute the strain rate [34]. This quantity has shown promise in the characterization of hypertrophic cardiomyopathy [35]. It is also possible to numerically integrate the velocity fields through time, obtaining an estimate of the motion map y(q, t) [27, 33]. Lagrangian strain can be computed from this map.

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