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Recently, there has been a significant interest in applying reconstruction techniques, like constrained reconstruction or compressed sampling methods, to undersampled k-space data in MRI. Here, we propose a novel reordering technique to improve these types of reconstruction methods. In this technique, the intensities of the signal estimate are reordered according to a preprocessing step when applying the constraints on the estimated solution within the iterative reconstruction. The ordering of the intensities is such that it makes the original artifact-free signal monotonic and thus minimizes the finite differences norm if the correct image is estimated; this ordering can be estimated based on the undersampled measured data. Theory and example applications of the method for accelerating myocardial perfusion imaging with respiratory motion and brain diffusion tensor imaging are presented.

There
has been large interest in speeding the acquisition of MRI data by acquiring
fewer samples in _{1}

In this paper, we propose a technique to improve the reconstruction of general signals that may not fit the TV constraint well. The technique uses preprocessing of the measured undersampled data to determine an improved ordering of the pixel intensities of the image estimate. If an ordering that improves the match of the estimated images and the constraint being used within the reconstruction can be found, an improved reconstruction can result. The image estimates are reordered solely to be used with the constraint or regularization term in the iterative reconstruction. The reordering approach is general in its applicability and can be used in contexts which are based on regularization techniques and in which ordering of the image intensities can be determined a priori. In the next sections, we give a brief overview of the compressed sampling or constrained reconstruction method for MRI from a regularization point of view and then present the theory and applications of the reordering method.

The
compressed sampling method is described rigorously from a mathematical
standpoint recently in a series of papers [_{1}_{1}

Although
the _{1}_{1}_{0}_{1}_{0}

The
compressed sampling method when applied to MR image reconstruction can be
thought as a constrained reconstruction method in an inverse problem framework
[^{-1}

Regularization
techniques can be used to solve this ill-posed problem. The existence of the
solution is imposed by considering least-square solutions which minimize the functional _{2}_{1}

Reconstruction
is performed by minimizing a convex cost function (

The
method can be extended to 2D and multi-image dimensions and it works very well
when the

For clarity, the reordering method is first described for the 1D case and then the method is extended to 2D and multidimension cases. Applications of the reconstruction method with reordering for dynamic myocardial perfusion imaging with respiratory motion and for brain DTI data are presented.

When
the signal of interest is varying rapidly and is not smooth or the data are not
piecewise constant, the total variation of the signal is already high and hence
reconstruction from undersampled Fourier domain samples can be inaccurate.
Consider, for example, a smoothly varying 1D signal and a rapidly varying
signal that are labeled “original full data” as shown in Figures

(a) A fully sampled smoothly varying 1D signal and the corresponding
signal reconstructed using IFT from its incomplete Fourier data undersampled by
a factor of two (

Reconstruction
using (

Better
reconstruction from the undersampled Fourier samples is obtained when the
reordered curve is used in the constraint term, as the a priori assumption that the curve has lower variation is better
satisfied. So the new reconstruction from undersampled data is performed
according to (

Note
that the reordering in (

The
reconstruction obtained with reordering is shown in Figure

From
the compressed sampling point of view, reordering the data can lead to sparser
representations of the data and hence higher acceleration factors.
Alternatively, better reconstructions for a given acceleration factor can be
obtained. Figure _{1}

Comparison of sparsity of the
fully sampled original nonsmooth signal in Figure

Choosing
the optimal regularization parameter

Comparison of optimal
regularization weights without and with reordering.

From the above, it is apparent that correct reordering can help in better reconstruction from undersampled Fourier data when the signals are not smoothly varying. In the above experiment, we used full data to determine the optimal ordering. To be able to use the reordering method, we need to have an ordering that makes the original signal best match the constraint. In practice, it is likely not possible to get the exact ordering of the signal curves or images as that obtained using fully sampled Fourier data due to various factors like blurring of the prior signal, noise in the prior signal, and so on. To simulate this case, we randomly perturbed the exact sorting order to see the effect of having inexact ordering on the performance of the algorithm.

In Figure

Comparison of errors in the
reconstruction for the nonsmooth signal in Figure

The
reordering method described above for the 1D case can be extended to 2D and
applied in the context of images. As in the 1D case, reordering in 2D for
images helps in better reconstruction when the images of interest are nonsmooth
or are not piecewise constant. For
example, Figure _{Rx}_{Ix}_{Ry}_{Iy}

Reordering method for 2D
images. (a) Simulated piecewise
constant heart image. (b) Image
reconstructed using IFT from ~15% of the full Fourier data, undersampled in a
variable density random fashion. (c) Image
reconstructed from undersampled data using a TV spatial constraint. (d) Actual MR magnitude image of the
short-axis slice of a heart at a single point in a perfusion sequence
reconstructed from fully sampled

Figure

Comparison of errors in the reconstruction for the actual MR heart
image in Figure

Perturbation for
a given row or a column is done independently for the entire length of the
sorting-order vector as described for the 1D case in Section

The
reordering method described above can be extended to multi-image MR
acquisitions like dynamic myocardial perfusion imaging and brain DTI. In
perfusion imaging, a series of images of the heart are acquired to track the
uptake and washout patterns of the contrast agent in the myocardium. DTI
requires the acquisition of multiple images with diffusion weightings in different
directions. Reordering can be done in the multi-image dimension—in the time
dimension for the case of myocardial perfusion imaging and in the diffusion
encoding dimension for the case of DTI. As in the 1D case, reordering in the multi-image dimension for the
images can give a better reconstruction when the signal changes in the
dimension are not smoothly varying which is the case for perfusion imaging with
respiratory motion and for DTI. The
constraint for the reordering in the multi-image dimension is represented as

Reconstruction
can then be performed by using TV constraints in both space and multi-image
dimensions and with reordering in the corresponding dimensions
as follows:

The reordering method for
multi-image acquisitions (

The reordering method was applied
on dynamic myocardial perfusion imaging with respiratory motion and on brain
DTI data. Full Cartesian raw ^{°}, Gd dose = 0.025 mmol/kg, slice thickness = 6 mm, and
acquisition matrix = 192 × 96. FOV = 380 × 285 mm^{2}. The data were acquired with
informed consent in accordance with the University of Utah Institutional Review Board. Brain DTI image data were
acquired on a GE 3T Scanner and full

The
results of the reordering method on the multi-image phantom data are shown in
Figure

Results of multi-image
reordering method on dynamic phantom data. Image at a time point reconstructed (a) from full

Figure

The
results of the reordering method for dynamic myocardial perfusion imaging are
shown in Figure

Result of multi-image
reordering method on dynamic myocardial perfusion data. (a) Images at two different time points in the sequence
reconstructed from full

The
reordering helps in reconstruction when there is a significant respiratory motion in the
data. We previously reported higher acceleration
factors (

The
results obtained by applying the reordering method on brain DTI data are shown
in Figure

Result of the reordering
method on multi-image brain DTI data. (a) Image of a single diffusion encoding direction reconstructed from full Fourier
data. A line for comparison of pixel intensity profiles for different
reconstructions is also shown. (b) Corresponding encoding direction reconstructed from

Figure

This
paper introduces a modified constraint term for compressed sampling and
constrained image reconstruction approaches. In general, it is possible to
choose a regularization or constraint term which is a good model for the image
being reconstructed. The basic idea of the reordering method is that it is
possible to tailor these regularization or constraint operators to improve the
reconstruction by reordering the signal. From a compressed sampling point of
view, various transforms have been proposed to enforce sparsity in the data.
Reordering can be thought as a new set of data-specific “transforms” that
further improve the sparsity. Recently, a new method using a prior image
constraint [_{1}

Reordering
can be done in multiple dimensions to improve the sparsity when the signals are
not smoothly varying. Here, we used images from the central low-resolution data
to determine orderings initially in the multi-image dimension and then used the
resulting images to obtain the spatial reordering for each image separately.
This is because the central low-resolution data are more faithful in the multi-image dimension
than they are in
the spatial dimension. Reordering in the multi-image dimension offered more
significant improvements as compared to reordering in the spatial dimensions.
This is because the temporal constraint generally plays a more important role
in resolving the artifacts as compared to the spatial constraint for dynamic
imaging [

The
reordering method incorporates the ordering information of the signal to better
match the total variation constraint assumption and thus improves the
reconstruction from undersampled data. Methods like adaptive regularization
[

The
reordering method may not be appropriate when the ordering is incorrect in such
a way that the total variation of the reordered full image sets is increased as
compared to that of the original full data. In practice, it might not be
possible to know this information beforehand. In such cases,

(a)

The
reconstruction time with image reordering was higher than the standard _{1}

A method involving reordering in time and space dimensions of the image estimates to better match the chosen constraints of an inverse problem-type reconstruction was presented. The method uses non-reordered reconstructions to obtain information about the signal to be reconstructed to determine the orderings of the pixel intensities. The orderings can be estimated from the low-resolution images when a variable density undersampling scheme is used, and from non-reordered constrained reconstructions. The method can be forgiving to errors in the images used to choose the orderings as the method does not use the data directly but uses only its ordering information. The method was shown to have promise for cardiac perfusion imaging and offered some small improvements for DTI data. Future improvements in finding more optimal reorderings, perhaps as part of the estimation process, may make the approach useful in a wide array of applications.

The authors would like to thank Dr. Eun-Kee Jeong for providing the phantom data;
Dr. Andy Alexander for providing brain DTI data; Chris McGann, Henry Buswell,
Melody Johnson, and Nathan Pack for their help with data acquisition of
myocardial perfusion data. This publication was made possible by Grant Number R01EB006155 from the National Institute of Biomedical Imaging and Bioengineering. Its contents are solely the responsibility of the authors and do not necessarily represent the official views of the NIH. The authors also thank the Ben B. and Iris M. Margolis Foundation for support. Part of this work was presented at the 2008
ISMRM [