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This work addresses the problem of recovering multi-echo T1 or T2 weighted images from their partial K-space scans. Recent studies have shown that the best results are obtained when all the multi-echo images are reconstructed by simultaneously exploiting their intra-image spatial redundancy and inter-echo correlation. The aforesaid studies either stack the vectorised images (formed by row or columns concatenation) as columns of a Multiple Measurement Vector (MMV) matrix or concatenate them as a long vector. Owing to the inter-image correlation, the thus formed MMV matrix or the long concatenated vector is row-sparse or group-sparse respectively in a transform domain (wavelets). Consequently the reconstruction problem was formulated as a row-sparse MMV recovery or a group-sparse vector recovery. In this work we show that when the multi-echo images are arranged in the MMV form, the thus formed matrix is low-rank. We show that better reconstruction accuracy can be obtained when the information about rank-deficiency is incorporated into the row/group sparse recovery problem. Mathematically, this leads to a constrained optimization problem where the objective function promotes the signal's groups-sparsity as well as its rank-deficiency; the objective function is minimized subject to data fidelity constraints. The experiments were carried out on

In multi-echo imaging, different images of the same cross-section are acquired by changing certain scan parameters, e.g., the echo times for T2 weighted images or the repetition times for T1 weighted images. The objective is to obtain images (of the same cross-section) with varying tissue contrasts. The details about the physics and techniques for acquiring these multi-echo MR images are found in [

Traditionally the K-space was obtained using full sampling on a uniform Cartesian grid. Each image was then reconstructed by applying the inverse Fast Fourier Transform (FFT). Full sampling of the K-space is however time consuming. Recent advances in Compressed Sensing (CS) allowed MRI researchers to reconstruct the MR images, almost perfectly, using partial,

Compressive Sampling (CS)-based MRI reconstruction has used the prior information that the images are spatially redundant, specifically that they have a sparse representation in a transform domain such as wavelets [

This work is based on the concepts introduced in [

Our work differs from its predecessors in the optimization problem used for reconstruction. As mentioned above, when the transform coefficients of the echo images are stacked in MMV form, the resulting matrix is row-sparse. Such a row-sparse matrix is low-rank as well (the rank is less than or equal to the number of non-zero rows). The key difference between this work and [

The problem of rank-deficient row-sparse MMV recovery has been studied before [

The next section describes the theory behind the proposed method. Section 3 describes the experimental results. The conclusions of the work are discussed in Section 4. The derivation of the algorithm is relegated to the

The previous work that is directly relevant to us is found in [^{th} group is formed by collecting the j^{th} coefficient of the transform coefficients of each one of the N echoes. There will be n such groups of size N and the concatenated wavelet coefficient vector will be group-sparse [

The aforesaid studies use two prior pieces of information—that the images are spatially redundant and are correlated with each other. The wavelet transform decorrelates the spatial redundancies in the individual images leading to a sparse representation; the fact that concatenated wavelet transform coefficients form a row-sparse matrix [

In this work, we follow the same assumptions as [^{th} echo image is as follows:
^{i}) of the image space into the K-space, x is the vector representing the image to be reconstructed, η is the system noise and N is the total number of multi-echo images. Since the K-space is partially sampled, solving

For ease of expression, we will drop the matrix and vector dimensions in the rest of the text since they do not pose any ambiguity. There are two ways to arrange the multi-echo images which will result in slightly different reconstruction problems. In [^{i} = F for all i's), therefore the acquisition model could be expressed in the MMV form.

^{i}'s, X is the matrix formed by vertically stacking x^{i}'s and N is formed by vertically stacking η^{i}'s.

Applying the wavelet transform (W) on X leads to a row-sparse MMV matrix A(=WX). This is because the resulting wavelet transform coefficients will only have high values along rows that correspond to edges in the multi-echo images. The MMV matrix is shown in _{i}'s are the transform coefficient vectors.

The row-structure is highlighted in ^{th} row corresponds to an edge it will have non-zero values throughout the row. To recover the row-sparse MMV matrix A, the following optimization problem was proposed in [_{2,1} is a mixed norm defined as the sum of the _{2}_{F} is the Frobenius norm (Euclidean norm of a matrix) and ε = mNσ^{2}, where σ is the standard deviation of noise. The _{2}_{2}

The problem with this approach [

In [^{i}'s, F is the block diagonal matrix with F^{i}'s in the diagonal blocks, x is the vector formed by concatenating x^{i}'s, η is the vector formed by concatenating η^{i}'s. Such a representation allows for different sampling masks for different echoes.

When the wavelet transform coefficients of all images are concatenated as vector α, then if all the coefficients corresponding to the same index in each image are grouped, only those groups that correspond to edges in the original image will have high valued coefficients, the rest will be zeroes or near about zeroes. This is shown in

We see how the transform coefficients are grouped according to original indices in ^{th} group (α^{(j)}) will have high value. Thus solving the wavelet transform coefficients of the images turns out to be a group-sparse optimization problem. In [

Here α is the vector formed by concatenating the wavelet transform coefficients of all the images, Φ is the block diagonal matrix consisting of F^{i}W^{H} as diagonal elements and ε is as defined before in _{2,1}_{2}_{2,1}_{2}_{2}

The aim of this work is to reconstruct multi-echo MR images. But this is the intermediate step. In quantitative MRI, the final objective is to compute the T2 or T1 maps. Generally, these maps are computed by non-linear curve fitting of the multi-echo images. This work aims at computing these maps from partially sampled K-space measurements. In this work, we show that if we can faithfully recover the images by our proposed method, the computed maps can be computed fairly accurately as well. There is one work which skips the intermediate step of reconstructing the images and directly computes the maps from the under-sampled K-space measurements [

However the aforementioned techniques are not standard ones in quantitative MRI. In this work, we follow the more conventional approach. The multi-echo images are reconstructed from which the maps are computed via non-linear least squares fitting.

Let us re look at the MMV model for multi-echo MR imaging [

Incorporating the wavelet transform into the data acquisition model we get:

In [

Since, the MMV matrix to be recovered is rank deficient; one can solve it via Nuclear-norm minimization [

Here ‖.‖_{NN} represents the nuclear norm of the Matrix which is defined as the sum of singular values of the matrix. Solving the reconstruction of A by utilizing only its rank-deficient property (solving Nuclear-norm minimization) will not yield good results. This is because the fact that A is rank-deficient is an effect of A being primarily row-sparse; it is low rank since only a few rows are non-zeroes. Thus row-sparsity is a more fundamental property and without it the reconstruction will not be good. The best solution can be achieved when the information about both its row-sparsity and rank-deficiency is incorporated into the reconstruction formulation simultaneously. The straightforward way to fuse the two pieces of information (rank-deficiency and row-sparsity) is to solve the following optimization problem:

Here γ is the term controlling the relative importance of the row-sparsity and rank-deficiency. Intuitively

The problem with the MMV formulation (as discussed above) is that it does not allow for separate sampling masks for each echo. To accommodate separate masks, the data acquisition equation should be expressed as

When the Wavelet transform is incorporated into the data acquisition model, we have the following form:

In _{i} = Wx^{i}'s are concatenated to form α. A group-sparse optimization problem was proposed in [

Group-sparsity and row-sparsity are two different ways of representing the same phenomenon. In this work, we use the extra information that the matrix formed by stacking the wavelet transform coefficients as its columns is a low-rank matrix. For row-sparse MMV recovery, the low-rank property is exploited as done in

Here the parameter γ also controls the relative importance of group-sparsity and rank-deficiency. The said optimization problem

Experts in medical physics pointed out that, for rigid tissues (like cartilages) portions of the tissues can have low relaxation times which will lead to change in the position of its edges across echoes. In such a situation, the transform coefficients will not have a common (sparse) support across echoes. As a result, the MMV matrix A

In such a case, the problem may be rectified by introducing non-convex row-sparsity or group-sparsity promoting norms:

Here, the non-convex norm ‖.‖_{1}

All animal experimental procedures were carried out in compliance with the guidelines of the Canadian Council for Animal Care and were approved by the institutional Animal Care Committee. One female Sprague-Dawley rat was obtained from a breeding facility at the University of British Columbia and acclimatized for seven days prior to the beginning of the study. Animal was deeply anaesthetized and perfused intracardially with phosphate buffered saline for 3 minutes followed by freshly hydrolysed paraformaldehyde (4%) in 0.1 M sodium phosphate buffer at pH 7.4. The 20 mm spinal cord centred at C5 level was then harvested and post-fixed in the same fixative. MRI experiments were carried out on a 7 T/30 cm bore animal MRI scanner (Bruker, Germany). Single slice multi-echo CPMG sequence [

A rectangular coil (22 × 19 mm) was surgically implanted over the lumbar spine (T13/L1) of a female Sprague-Dawley rat as described previously [

In this work we follow the same experimental methodology as in [

The algorithm used in this work requires certain parameters to be specified. In

We tested several families of wavelet transforms—Daubechies, fractional spline and complex dualtree. We found that the best results were obtained with the complex dualtree wavelets. Therefore all the results reported here use the complex dualtree wavelets as the sparsifying transform. For rank-deficient row-sparse MMV recovery we employed the optimization problem proposed in [

We use the Signal-to-Noise ratio (SNR) as the metric for comparing the quantitative reconstruction results. Our first set of experiments constrains all the echoes to have the same sampling mask. This corresponds to the situation studied in [

The results experimentally validate the intuition behind this work—the more information we use for reconstruction, the better is the result. In [

In the next set of experiments, we use a different sampling mask for each echo. Here instead of using the MMV formulation, we use the proposed optimization problem

Comparing

From

Numerical results do not always provide information about the qualitative nature of reconstruction. Thus, we provide reconstructed and difference images for visual inspection. Owing to limitations in space, we only show the results for 64 lines (this corresponds to a sampling ratio of 25%). Echo numbers 1, 5, 9 and 13 are shown in the following figures. In

Looking closely at these figures, especially at the edges, one can see that the reconstruction quality progressively improves as one move from top to bottom. The worst results are obtained when the same Fourier map is used for all the echoes and simple row-sparse MMV recovery is used. The results improve slightly when rank-deficiency is introduced, but the improvement is only slight because the condition of incoherence is not satisfied when the same Fourier map is used for all the echoes. When different Fourier maps are used for the different echoes, the results from both group-sparse recovery and rank-deficient group-sparse recovery show improvement in the reconstruction quality; this is because of the better incoherence in the measurement operator. However, the best results are obtained when rank-deficient group-sparse recovery is used with different Fourier maps for different echoes.

The differences in the reconstruction quality are best evaluated from the difference images. The difference is formed by taking the absolute difference between the reconstructed and the groundtruth images. The contrast of the difference images is enhanced five times for visual clarity.

The difference in the reconstruction quality is easily discernible in the difference images; see

This work assumes that if the multi-echo MR images reconstructed from partial K-space scans are similar to those from full K-space scans, the corresponding T2 maps will be similar as well. This assumption is validated experimentally. In the following figure, we show the T2 maps for the

From

In this work we address the problem of multi-echo MRI reconstruction form under-sampled K-space data. The experiments are carried out on T2 weighted images but the methodology developed here is applicable to other modalities (such as T1 weighted imaging) as well. This builds on our earlier work [

In this work, we propose the reconstruction problem as a synthesis prior problem. In synthesis prior formulation, the sparse transform coefficients are solved. Previously we have found that better reconstruction can be obtained if the analysis prior problem is solved instead. In the analysis prior problem, the images are directly reconstructed (and not their transform coefficients) [

The major limitation of this work (and the prior methods [

This work was supported by NSERC and by Qatar National Research Fund (QNRF) No. NPRP 09-310-1-058.

Here we derive the algorithm for solving:

Solving the constrained optimization problem directly is difficult. Therefore we propose to solve its unconstrained Lagrangian form:

There exists a relationship between the constrained form

First, we replace the Nuclear Norm in _{NN}^{H}X^{1/2}, Thus, we will solve the following equivalent problem:

We follow the Majorization Minimization (MM) approach to solve this problem [

Let

Set iteration count _{0}

Repeat Steps 2–4 until a suitable stopping criterion is met.

Choose _{k}(x)

_{k}

_{k}

Set _{k}_{+}_{1}_{k}(x)

Set k = k + 1, go to Step 2.

For our problem the function to be minimized is: _{2} + λ ‖_{2,1} + ^{H}X^{1/2}. There is no closed form solution to _{k}(x)^{H}A

Now _{k}(x)

Minimizing

The problem now is to minimize _{2}); “.” denotes element wise division.

Setting the gradient to zero, one gets:

The problem

Based on this derivation, we propose the following algorithm to solve (12 or 13):

Intitialize: _{0} = 0

Repeat until:

Step 1.

Step 2.

Step 3. Compute

End

Our discussion so far had been on solving the unconstrained problem

Intitialize: _{0} = 0, ^{T} y

Choose a decrease factor for cooling λ.

Outer Loop: While

Inner Loop: While ^{2}

Step 1.

Step 2.

Step 3. Compute

End While^{2}

End While^{1}

The stopping criterion here is based on the data consistency term. The iterations stop when
^{−3}. The maximum number of iterations that are allowed for the inner and the outer loops are 50 and 10 respectively.

Row structure in MMV matrix.

Grouping according to positions.

Images from

Images from

Difference Images from

Difference Images from

T2 maps for

T2 difference maps for

Row-sparse MMV [ |
11.8 | 14.3 | 15.3 |

Proposed row-sparse rank-deficient MMV | 13.7 | 15.6 | 16.4 |

Row-sparse MMV [ |
8.8 | 14.2 | 17.9 |

Proposed row-sparse rank-deficient MMV | 10.7 | 15.7 | 19.1 |

Group-sparse [ |
12.7 | 15.2 | 16.7 |

Proposed group-sparse rank-deficient | 14.6 | 16.7 | 18.1 |

Group-sparse [ |
10.7 | 16.3 | 20.3 |

Proposed group-sparse rank-deficient | 12.6 | 17.7 | 21.5 |