Open dynamic problem questions

The research on the dynamic problem provided a general estimation framework and showed two methods for dynamically determining input vectors that provide superior MR image estimates over the current state of the art. In sharp contrast to previous methods which focused on the acquisition of contrast change sequences, these new methods are particularly well suited to handle many types of image changes including bulk motion and jitter. Thus, it is expected that these methods are quite well suited to handle clinical imaging scenarios.

However, comparing the image quality of the realizable low-order methods with the theoret-ically optimal method, we see that there is significant room for improvement in the realizable methods. For example, the image estimate error in theoretically optimal estimates appears to be uncorrelated with the structure of the original image. This is not the case for the realizable methods. Specifically, errors often appear at the edges of regions showing motion change. Thus the open question discussed in this section is, How can one determine realizable inputs that are close to the theoretically optimal?

The crux of this problem is somewhat obscure. The basic concept of the prior SVD methods is to track the images and determine new inputs from the previous estimates. We showed in Section 5.2.1 that this strategy is somewhat limited due to the bias of future inputs to previous inputs. A second point of view is to track the dominant subspace of the underlying image system.

In general, this is very difficult to do. Consider an example given by Stewart and Sun in [40]. The right singular vectors of the matrix



In the limit that  −→ 0, the right singular vectors of the matrix become

V =

This example shows that the transition of the right singular vectors of a matrix can be discontinuous even for very small changes in the matrix itself. However, keep in mind that in the adaptive framework the theoretically optimal method uses differences between matrices to identify suitable input vectors. Generally speaking, there is little reason to suspect that the dominant singular vectors of difference matrices are at all similar over the course of a dynamic MRI sequence. Thus, we conclude that tracking singular vectors is of limited utility.

Of the methods examined thus far, the linear predictor methods have the greatest potential for approaching the estimate quality provided by the theoretically optimal method. Two main avenues of research are available. First, the linear predictor methods presented in this thesis used temporal prediction on a pixel-by-pixel basis. Given the fact that bulk motion changes so strongly affect the estimate quality, it is reasonable to expect that including spatial changes in the predictor will provide superior performance. Second, incorporating image change models will likely improve the image estimate quality. These models would likely need to be case specific such as a temporal model matched to the stimulus used in functional MRI studies or a periodic bulk motion model to predict changes introduced by breathing or cardiac activity.

There are other questions of secondary priority that remain open at this juncture. These relate to specific design criteria in the acquisition of clinical dynamic sequences. For example, currently there is no clear guideline for selecting the number of input vectors to use for each image acquisition/reconstruction. The examples given in Chapter 5 used a ratio of partial sums of singular values from the reference image. This appeared to give consistent quality for a variety of sequences, but better selection methodologies for block acquisition size may be needed in clinical settings. Furthermore, this number was assumed to be fixed in Chapter 5. It may in fact be advantageous to dynamically set the number of new input vectors for each new image acquisition.

Conceivably, this selection would depend on a measured quality of output data, growing larger if the output data is dramatically different than expected and growing smaller if the output data closely matches the expected data. This approach is bolstered by the fact that the theoretically

optimal method can reconstruct high quality images of the synthetic knee sequence using only one input vector.

Other secondary issues include how often data needs to be acquired and whether the acqui-sition and image estimation should be performed a single vector at a time or in a block fashion.

Preliminary analysis shows that block-vector acquisitions does not track the image changes quite as well as single-vector acquisitions, but that they are not significantly different. Implementation in a clinical setting may in fact be the driving force in this design decision. Likewise, while most likely image sequence dependent, the rate of data acquisition may depend significantly on hardware constraints. Firm guidelines as to acquisition rates and algorithm robustness for certain clinical sequences needs to be more fully addressed.

Finally, it would be preferable to have a “one size fits all” low-order acquisition method. With the adaptive estimate framework, this would most likely be accomplished with hybrid techniques.

The primary advantage of the linear system model is that inputs need not be constrained to a single basis. One could potentially use Fourier basis vectors in tandem with vectors identified through other means, e.g., via the SVD, CG-St, or linear predictor methods. This would provide the image estimation with the ability to track both contrast and motion changes while not restricted to one specific modality. Such hybrid techniques are worth further exploration.