Misalignment correction for T1 maps using a maximum likelihood estimator approach
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1
University of Antwerp, Belgium
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2
University of Antwerp, Department of Physics, Belgium
Magnetic Resonance Imaging (MRI) is a versatile and powerful technique where images with excellent characteristics can be obtained. However, MR often suffers from misalignment owing to motion or errors in the parameters defining the k-space Fourier integral. For example, considering the properties of the Fourier Transform (shift theorem), offset errors in the magnetic gradients lead to (at least at first order Taylor approximation) displacements in the reconstructed magnetic image which results in a misalignment between different images of the same patient. This fact affects negatively quantitative parameter estimation, for example, T1 maps in T1 weighted images. T1 maps represent, for each pixel, the longitudinal relaxation time (T1), which is tissue dependent [1]. Commonly, this misalignment problem is addressed before estimating T1 parameters. The most widespread approach consists of a two step procedure: misalignment correction of the intensity image and then T1 estimation. However, misalignment correction of the intensity image should be designed carefully because, due to noise, the assumption of gradient constraint, which states that the intensity of the shifted image is conserved, does not usually hold [2]. Consequently, the well-known misalignment methods such as phase correlation or optical flow methods are not suitable for intensity image and ad-hoc methods must be designed [2].
In this work, we propose a maximum-likelihood approach for misalignment correction of T1-weighted images that does not require a two step procedure. We assume that the probabilistic model of the intensity image is the well-known Rician Model [3] where the envelope parameter is the T1 magnetization curve which includes the standard deviation of the noise. In order to derive our method, we follow this assumption: "If there is a misalignment between the two true magnetic images, i.e. envelope parameters, the maximum likelihood estimator (MLE) of the envelope suffers from the same misalignment". This is a realistic assumption because it is well known that maximum likelihood estimator (MSE) is consistent (converges with probability one to the true parameter) [4].
Consequently, we can pose the problem of misalignment correction of intensity images, as the misalignment correction of MLE images (T1 parameters are considered to be a function of (x,y) ). The MLE of T1 parameter is obtained by maximizing the log-likelihood function of the envelope parameter with the Nelder-Mead algorithm [3]. Noise variance is estimated from the background area, where Rician model turns out to be Rayleigh, and the analytical MLE for variance exits [4].
Once MLE of T1 parameters for each two images have been obtained, we estimate the displacement error considering several strategies. For example, when T1 parameters are estimated, we can use the log-likelihood function as a similarity measure to detect most probable displacement. Furthermore, due to the robustness of MLE, the gradient constraint is more probably to hold in MLE images than in pure intensity images. Therefore, phase correlation or optical flow methods could be used and its suitability is discussed in this work [5].
Experiments have been carried out using both real and synthetic images. Synthetic images have been obtained from Kwan et al. brain simulator [6]. Several simulation with different values of the noise variance are performed in order to validate the accuracy of the proposed method. Real experiment is based on T1-weighted dataset where an horizontal displacement have been clearly observed probably due to a shift in b_0 magnetic field. In both cases, the proposed method based on MLE of T1 parameters provides satisfactory results and misalignment can be corrected. Moreover, after misalignment correction, T1 maps are ready to be used for clinical purposes.
Acknowledgements
The research which led to this work was supported by the Fund for Scientific Research-Flanders (FWO)
References
[1] Xue, Hui, et al. "Motion correction for myocardial T1 mapping using image registration with synthetic image estimation." Magnetic Resonance in Medicine 67.6 (2012): 1644-1655.
[2] Nestares, Oscar, and David J. Heeger. "Robust multiresolution alignment of MRI brain volumes." Magnetic Resonance in Medicine 43.5 (2000): 705-715.
[3] Sijbers, Jan, et al. "Parameter estimation from magnitude MR images."International Journal of Imaging Systems and Technology 10.2 (1999): 109-114.
[4] Sijbers, Jan, and A. J. Den Dekker. "Maximum likelihood estimation of signal amplitude and noise variance from MR data." Magnetic Resonance in Medicine51.3 (2004): 586-594.
[5] Foroosh, Hassan, Josiane B. Zerubia, and Marc Berthod. "Extension of phase correlation to subpixel registration." IEEE Transactions on Image Processing, 11.3 (2002): 188-200.
[6] Kwan, RK-S., Alan C. Evans, and G. Bruce Pike. "MRI simulation-based evaluation of image-processing and classification methods." IEEE Transactions on Medical Imaging, 18.11 (1999): 1085-1097.
Keywords:
MRI,
misalignment,
T1 map,
maximum likelihood estimation,
optical flow,
Rician
Conference:
Imaging the brain at different scales: How to integrate multi-scale structural information?, Antwerp, Belgium, 2 Sep - 6 Sep, 2013.
Presentation Type:
Poster presentation
Topic:
Poster session
Citation:
Ramos-Llordén
G and
Sijbers
J
(2013). Misalignment correction for T1 maps using a maximum likelihood estimator approach.
Front. Neuroinform.
Conference Abstract:
Imaging the brain at different scales: How to integrate multi-scale structural information?.
doi: 10.3389/conf.fninf.2013.10.00040
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Received:
31 Aug 2013;
Published Online:
31 Aug 2013.
*
Correspondence:
Mr. Gabriel Ramos-Llordén, University of Antwerp, Antwerp, Belgium, gabrll@gmail.com