K

Repeating this process for additional lines of pixels results in a discrete approximation of the tag line.

The tag point found through template matching is determined with subpixel precision; in theory, the error in the measurement can have a standard deviation as low as 0.9/CNR pixels [19], where CNR is the contrast-to-noise ratio of the tag pattern. In typical tagged MR images, this translates to an error of 0.1 pixels, an accuracy level that has been confirmed experimentally [19]. To achieve such a low level of error, the template must accurately reflect the true profile, which depends on the applied tag pattern, fading, the line spread function of the scanner, local myocardial properties, and the motion of the tissue that we are trying to resolve. One approach for obtaining an accurate template is to simulate the physics of MR tagging and imaging [20]. On the other hand, a generally good approximation is the Gaussian template g(s) = Io(1 - de-s2), where I0 is the local tissue brightness, d is the tag depth, and w is a width parameter. These parameters may also be estimated in the process of identifying a point on a tag line [21].

Although finding the optimal position s0 of the template match accurately identifies a point on a tag line, it does not determine m, which relates the point to a specific tag plane. To determine m, a recursive tracking procedure is used such as that proposed by Guttman et aZ. [20]. Beginning with a usersupplied set of initial tag line locations, search regions are identified in each subsequent image based on the estimated tag position at the previous time. The best template match within the search region is considered to be a point on the tag line corresponding to the same tag plane. For better error performance and outlier rejection, Kerwin and Prince [21] added a spatiotemporal filter to refine the tag point estimates between recursions. Improved performance may also be attained by Fourier filtering of the tagged images [24].

One challenge for template matching procedures is that image features other than tags, such as boundaries, are often mistaken for tags in template matching. To avoid this problem, search regions are restricted to the myocardium, the borders of which must be identified by hand or with user-intensive semiautomatic procedures [20]. Alternatively, Denney [25] uses the spatial and temporal characteristics of myocardial tags to weed out misidentified points.

Active Geometry

In active geometry approaches, tag identification is accomplished using a curve segment called a snake that conforms itself to a tag line in the image. The snake is assigned an energy function and moves in an effort to minimize that energy. The energy function has two components: Eimage, determined by the image in such a way that a snake will have low energy along the feature of interest, and Einternal, determined by derivatives of the curve and forcing continuity and smoothness in the snake. For tag line tracking, Eimage is usually related to image brightness because minimizing this energy will force a snake toward brightness minima, i.e., tag lines.

In practice, a snake is approximated by a set of K points along a tag line, the kth point at the nth time beingy(k, tn). To determine the set of points y(1, tn),..., y(K, tn) that best identifies a tag line, we begin with the solution for time tn_ 1, or for the first time, with a user-specified line of points. We then find a set of 2D displacements d(k, tn) that move the snake at tn_ 1 to the optimal position at tn. Thus, the solution is given by

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