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Field inhomogeneities in Magnetic Resonance Imaging (MRI) can cause blur or image distortion as they produce off-resonance frequency at each voxel. These effects can be corrected if an accurate field map is available. Field maps can be estimated starting from the phase of multiple complex MRI data sets. In this paper we present a technique based on statistical estimation in order to reconstruct a field map exploiting two or more scans. The proposed approach implements a Bayesian estimator in conjunction with the Graph Cuts optimization method. The effectiveness of the method has been proven on simulated and real data.

Magnetic Resonance Imaging (MRI) is a coherent imaging technique consisting of detecting signals induced by nuclei of the object being imaged in complex domain. To allow nuclei to produce signals, the object has to be placed in a uniform magnetic field and sequentially excited with suitable RF impulses.

Some imaging techniques show high sensitivity to the non uniformities of the applied magnetic field, particularly when exploiting long readout times, for example echo-planar imaging and spiral scans. The most primary effects of field inhomogeneities in MR images are blur and distortion. Such errors cannot be removed unless an accurate field map is available and used to compensate the complex data [

Field map can be estimated from different scans (at least two) acquired at different echo times. The phase difference between the acquired images is due to the different precession frequencies, which are related to the field map via a linear relation.

Besides the trivial estimation consisting of dividing the phase difference by the delay time between acquisitions _{TE}

Non statistical approaches are mainly based on retrieving the field map using linear regression techniques [

Statistical approaches are based on the exploitation of the noise statistics in order to obtain a more efficient estimation from the information theory point of view. A Penalized Maximum Likelihood Estimator exploiting two or more images (

In this paper we propose a novel multi-acquisition Maximum _{TE}

In Section 2 the field map estimation problem is briefly addressed. In Section 3 the proposed method will be presented. The optimization algorithm will be explained in Section 4 and the results will be shown and discussed in Section 5. Finally, we draw some conclusions about the presented technique.

In MRI the signal comes from nuclei with spin which rotate at a certain frequency, called the Larmor angular frequency. The precession depends on the kind of nuclei and the energy of the state in which the nuclei are in a magnetic field _{0} [_{1} taken at an echo time _{E,1}; the analytical expression of the complex MR image _{1} is given by:
_{1} represents the amplitude of the signal and _{1} the phase at the time _{E,1}; we can represent the phase _{1} as the sum of an initial phase _{0}

If we consider a second complex image taken at an echo time _{E,2}, the complex image will be:
_{2} represents the amplitude of the signal and _{2}:

Note that

Using _{1} and _{2} we can estimate the Larmor angular frequency _{1} and _{2} and dividing by _{TE}_{E,2} – _{E,1}). Once obtained

This seemingly simple approach would generate properly reconstructed field maps if there are no 2_{2} – _{1}) phase data, otherwise phase unwrapping operation will be needed before computing _{1} and _{2}.

Let us consider _{1}, _{2}…_{N}^{T} obtained at echo times _{E}_{E,1}, _{E,2},…_{E,N}^{T}. By generalizing _{0}, the relation between the

Our idea is to perform the field map estimation exploiting jointly the phases of images

In order to obtain the likelihood function, we investigate the pdf of the involved noise. In the

Expression (7) can be approximated well with the following probability density function [^{2}/^{2}). The relation between them is empirically found fixing an SNR and looking for the

The likelihood function can be obtained starting from the pdf (8) (or from _{p}_{p}

Considering _{E,n}_{p}_{p,}_{1}, _{p,}_{2}, ⋯ _{p,N}^{T}

Let us now consider the _{1} _{2} _{3…}_{P}^{T} is the collection of the Larmor angular frequencies related to the _{p}

Given the likelihood Function (10) and given the

Once this maximization has been performed, the field map for the whole image can be computed by simply inverting

If the likelihood Function (10) shows more than one maximum, in order to obtain the uniqueness of the solution of the multi-acquisition Phase Unwrapping problem, the single acquisition likelihood functions need to have different periods, which is achieved considering a not rational value for the ratio between _{E}

In order to obtain the field map estimation,

To overcome this problem, the optimization algorithm that we use in this paper is based on the Graph Cut theory [

Graph Cut theory has already been applied in the MRI field by Hernando

The Ishikawa algorithm is based on computing a minimum cut in a particular graph. The graph

A simplified representation of the Ishikawa graph for the 1 dimensional case is shown in

Each of the edges has a certain _{p}

Note that the proposed

A minimum cut on the graph consists of separating the special vertexes

In this section, some case studies are presented in order to show the performances achievable by the presented method. Results are obtained applying the method to both simulated and real data sets. For all the presented cases, a constant magnetic field of _{0} = 1.5 T, corresponding to a central Larmor angular frequency of ^{8}Hz, and different echo time and SNR configurations are considered. The size of the images is set to be 128x128 pixels and we use

In the first case study a discontinuity free field map is estimated considering four images with SNR = [6 5 4 3] dB and echo times _{E}_{E}_{E}

As expected, the phase of the first image presents fewer fringes than the other one. We apply to the four acquisition data set our proposed approach based on the maximization of

In order to evaluate the advantage of the multi-acquisition configuration, we perform the reconstruction using only two and three acquisitions, instead of four. The results of the reconstruction in terms of normalized mean square errors are shown in

For the second case study a simulated scenario with air/tissue discontinuities is considered. This simulation, although not completely realistic in electromagnetic terms of the magnetic field local strength, is shown to remark the robustness of the proposed algorithm in tackling the phase unwrapping problem when discontinuities are present. Configuration parameters are the same used in the previous case, in a higher noise case. As before, we use four different acquisitions (four phase images). In

Also in this case, we perform the reconstruction using only two and three acquisitions, instead of four. The results of the reconstruction in terms of normalized mean square errors are shown in

It is interesting to compare

As a last study case, we consider the same study case of the second one, but with more noisy data (SNR lowered of 2.5 dB). This time the SNR is set to be [3.5 2.5 1.5 0.5] dB. The results of the estimation are show in

As expected, the error increases compared to the second study case, but, even in presence of noisy data, the algorithm is able to provide a good solution. We underline that for all the three case studies the proposed algorithm is able to provide the solution in less than one minute using a SUN Ultra 40 Workstation.

Finally, we test the method on a real data set. The data set consists of two head images acquired in axial position with echo times equal to _{E}_{E}

Note that in all reconstructions only signal relative to water component of the tissue is considered. The presented method can be applied also in case of superposed fat component signal. Under the hypothesis of selecting echo times in order to obtain in phase superposition of the two components, fat signal becomes undetectable and does not influence the field map estimation. In our approach this is possible since we are not limited in the range and the spacing of _{E} we have to use.

In this paper a novel approach for the field map estimation problem in Magnetic Resonance Imaging has been presented. The main characteristics of proposed method are the statistical approach and the fast optimization algorithm based on Graph Cuts. The algorithm has shown to correctly retrieve the field map in a wide range of scenarios, both on simulated and real data. It is able to solve the phase unwrapping problem and to work properly both with low and high SNRs as with different echo times. We have shown that the approach is able to correctly manage the sharp discontinuities that arise at air/tissue boundary. Moreover, due to the piecewise smooth nature of field maps, the proposed

The authors would like to thank John Pauly from Stanford University, for providing real MRI complex data.

Probability density function of MRI phase signal noise, real pdf (blue), approximated pdf (red), Gaussian pdf (black): (a) low

One dimensional Ishikawa graph construction: circles represent the vertexes

First case study: (a) first phase image, (b) fourth phase image, (c) estimated field map using proposed technique, (d) estimated field map using conventional ML technique.

Second case study: (a) first phase image, (b) fourth phase image, (c) estimated field map using proposed method, (d) estimated field map using the approach of paper [

Real case study: (a) first phase image, (b) second phase image, (c) estimated field map.

Normalized mean square error for different number of available acquisitions—discontinuities free field map case.

2 acquisitions (_{E} |
0.0117 |

3 acquisitions (_{E} |
0.0098 |

4 acquisitions (_{E} |
0.0091 |

Normalized mean square error for different number of available acquisitions—air/tissue discontinuities field map case.

2 acquisitions (_{E} |
0.1010 |

3 acquisitions (_{E} |
0.0685 |

4 acquisitions (_{E} |
0.0392 |

Normalized mean square error for different number of available acquisitions—low SNR case.

2 acquisitions (_{E} |
0.1634 |

3 acquisitions (_{E} |
0.1028 |

4 acquisitions (_{E} |
0.0880 |