Coil Sensitivity Estimation for Optimal SNR ... - Semantic Scholar

Coil Sensitivity Estimation for Optimal SNR Reconstruction and Intensity Inhomogeneity Correction in Phased Array MR Imaging Prashanthi Vemuri1, Eugene G. Kholmovski1, Dennis L. Parker1, Brian E. Chapman2 1 UCAIR, Department of Radiology, University of Utah, SLC, USA {pvemuri, ekhoumov, parker}@ucair.med.utah.edu 2 Department of Radiology, University of Pittsburgh, PA, USA {chapmanbe}@upmc.edu

Abstract. Magnetic resonance (MR) images can be acquired by multiple receiver coil systems to improve signal-to-noise ratio (SNR) and to decrease acquisition time. The optimal SNR images can be reconstructed from the coil data when the coil sensitivities are known. In typical MR imaging studies, the information about coil sensitivity profiles is not available. In such cases the sumof-squares (SoS) reconstruction algorithm is usually applied. The intensity of the SoS reconstructed image is modulated by a spatially variable function due to the non-uniformity of coil sensitivities. Additionally, the SoS images also have sub-optimal SNR and bias in image intensity. All these effects might introduce errors when quantitative analysis and/or tissue segmentation are performed on the SoS reconstructed images. In this paper, we present an iterative algorithm for coil sensitivity estimation and demonstrate its applicability for optimal SNR reconstruction and intensity inhomogeneity correction in phased array MR imaging.

1 Introduction Phased array coils (multiple receiver coil systems) have been extensively used for acquisition of MR images owing to their benefit of increased SNR, extended field-ofview (FOV), and reduced acquisition time. The optimal way to merge individual coil information into the composite image in terms of the maximum SNR has been proposed by Roemer et al. [1], where each voxel value is obtained by the combination of voxel by voxel coil data with each coil’s contribution weighted by the corresponding coil sensitivity. This algorithm can only be applied when the coil sensitivities are known. To find coil sensitivity profiles additional reference scans are required. Increase in imaging time and possible discrepancies between estimated and the true coil sensitivity profiles due to patient motion between reference and imaging scans make this approach difficult to use in clinical practice. Thus, the SoS algorithm [1] is typically used for image reconstruction from multi-coil data where the resulting (composite) image is obtained by the square root of the sum of the squares of the individual

coil images. In comparison with the images reconstructed by the SNR optimal approach [1], the SoS reconstructed images are modulated by a spatially variable function due to the non-uniformity of coil sensitivities and have systematic error (bias) in signal intensity which necessitate intensity inhomogeneity correction and bias compensation before using the images for quantitative analysis. A multitude of post processing algorithms [2-6] have been proposed to correct the intensity inhomogeneity in the SoS images. In all these techniques, it is assumed that the intensity inhomogeneity can be represented by a slowly varying function of position. The intensity inhomogeneity in the SoS image is estimated by fitting a low degree polynomial to the SoS image or from the low resolution image obtained by filtering out the higher frequency contributions in the frequency domain representation of the SoS image. Finally, the intensity inhomogeneity is corrected by dividing the SoS image by the estimate. Reliability of the given approach directly depends on the validity of the approximation of the intensity inhomogeneity in the SoS reconstructed images by a slowly varying function. The sensitivity of the individual coil element (which is typically a circular loop) can be described by a unimodal, smoothly varying function of position [7]. Such a function can be accurately represented by a low degree polynomial. However, the intensity inhomogeneity in the SoS images must be characterized by the square root of the sum of squares of the individual coil sensitivities. Thus, the sensitivity modulation in the SoS images is multimodal and has a large number of higher order terms. Therefore, it cannot be reliably approximated by a low order polynomial. In this paper, we have proposed to estimate the intensity inhomogeneity in the individual coil images instead of the intensity inhomogeneity in the corresponding SoS image. Using this approach, consistent estimates of coil sensitivities can be found by fitting low order polynomial function to the image regions occupied by a dominant tissue type. Furthermore, the resulting estimates can be used not only for inhomogeneity correction in the SoS image but also for optimal SNR reconstruction.

2 Theory The complex image acquired by the i-th coil can be described as a product of the true image I (r) and the complex coil sensitivity Si (r) with additive Gaussian noise: R i ( r ) = I ( r ) S i ( r ) + N i ( r ) , i = 1,2 ,.., L

(1)

where r denotes the position in the image space, L is the number of coils in the coil array and Ni (r) is the complex Gaussian noise. When the coil sensitivities are known, the optimal SNR image IOPT(r) is given by [1]: I OPT ( r ) =

R ( r ) ψ −1 S H ( r ) −1

(2)

H

S(r )ψ S (r )

where R(r) is the row vector of coil images: R(r) = [R1 (r), R2 (r),…, RL (r)]; S(r) is the row vector of coil sensitivities: S(r) = [S1 (r), S2 (r),…, SL (r)]; ψ is a Hermitian L

by L matrix which describes the coupling and noise correlations between the coil elements and H denotes a Hermitian transpose. In the cases where the coil sensitivities are unknown, the SoS algorithm is applied: I SoS ( r ) = R ( r ) ψ −1R H ( r )

(3)

Even though SoS images are within 10% of the maximum SNR limits of the optimal SNR reconstruction [1], the intensity of the SoS reconstructed image is modulated by a spatially variable function due to the non-uniformity of coil sensitivities, which limits the usage of the images for quantitative analysis and segmentation. To resolve this problem, the algorithm proposed in this paper estimates the sensitivity profiles of the individual coil elements by iteratively identifying the image region occupied by a dominant tissue type and fitting low order polynomial function to image intensity in the region. Finally, Eq. [2] is used to combine the individual coil images to obtain the optimal SNR image without any intensity inhomogeneity. 2.1 Preliminary Steps: Un-biasing, Estimation of Individual Coil Phase Maps and Noise Correlation Matrix The developed iterative algorithm has achieved more reliable results when the preliminary steps described in detail below have been applied before the algorithm initialization. The acquired coil images are corrupted by sensitivity inhomogeneity as well as complex Gaussian noise. Hence, it is important to remove the noise related bias in the magnitude coil images before they are used to estimate the corresponding sensitivity maps. The noise bias in each coil image is calculated by evaluating the standard deviation of noise from the histogram of the image. Then, the bias is removed by using the technique proposed in [8]. The resulting unbiased coil images Ric (r ) are used instead of the original coil images Ri(r) for the sensitivity estimation. For SNR optimal reconstruction it is essential to know both magnitude and phase of coil sensitivities. The phase variation in the coil images is mainly linear in nature [9] and can be accurately estimated from the low resolution complex coil images. It is to be noted that an apodization function (e.g. Hamming window) needs to be used to reduce the effect of Gibbs artifact when the phase map of each individual coil θˆi (r ) is estimated. The noise correlation matrix ψ can be found using either a pre-scan noise calibration [1] or from a set of noise samples in the image field-of-view (FOV) [10]. In the case when sufficient numbers of noise samples are not present in the FOV, the noise correlation matrix can be assumed to be an identity matrix. 2.2 Iterative Technique for Coil Sensitivity Estimation The developed technique has three main steps: First, dominant tissue spatial distribution (the region of support) is identified; second, for each coil element, the sensitivity maps Si(r) is estimated by fitting low degree polynomial function to the image intensity in the region of support and finally, the sensitivity maps are used in Eq. [2] to reconstruct the new image estimate. Since the estimation of the coil sensitivity

profiles crucially depends on the identification of the region of support, an iterative algorithm has been developed where the first iteration of the algorithm is initialized (0) by the SoS image: I OPT (r ) = I SoS (r ) . On subsequent iterations the current image estimate with substantially suppressed intensity inhomogeneity is used to refine the region of support. The flowchart for the algorithm is shown in Fig. 1 and the main steps of this algorithm are explained in detail below.

Fig. 1. Flow chart of the algorithm.

2.2.1 Identification of the Region of Support In high SNR MR images, the intensity distribution can be described as a linear combination of Gaussian distributions. The given assumption has been widely used in a number of statistics-based techniques for segmentation of brain tissues in the MR images [11-13]. A similar approach is utilized in our method to identify a spatial distribution of the dominant tissue type. The detailed description is illustrated in the context of brain MRI. The histogram h of the brain MRI image can be modeled by a linear combination of Gaussian distributions: 4

h = ∑ α k G(µ k , σ k )

(4)

k =1

where G(µk, σk) is the Gaussian distribution of the k-th tissue type (k=1 corresponds to air/bone, k=2 to CSF, k=3 to gray matter and k=4 to white matter) with mean µk

and variance σk . αk is the number of voxels corresponding to each tissue type within the image. It can be assumed that the intensity value corresponding to the global maximum of the image histogram (excluding the histogram peak related to tissue free image areas) corresponds to the mean of the dominant tissue type. Assuming for simplicity that the dominant tissue is white matter (k=4). Then, the region of support ( n −1 ) in the n-th iteration, M(n)(r), is identified using the image estimate I OPT (r ) obtained at the (n-1)-th iteration as follows: 1 M ( n ) (r ) =  0

( n −1 ) ( r ) < µ 4 + 2σ 4 µ 4 − 2σ 4 < I OPT

(5)

otherwise

where µ4 is equal to the image intensity value corresponding to the peak of the image ( n −1 ) intensity distribution (histogram) of I OPT ( r ) . Identification of the dominant tissue region support is preferable rather than identifying other tissue spatial distributions as dominant tissue region would present a larger number of voxels spread throughout the image. This way more precise estimates of coil sensitivities can be found. 2.2.2 Estimation of Coil Sensitivities Each coil image is modulated by complex coil sensitivity. The estimation of the phase maps θˆi (r ) is done as a preliminary step of the algorithm (Sec 2.1) and the magnitude of the coil sensitivity is estimated in the current step of the algorithm. In the absence of noise and image intensity modulation due to non-uniform coil sensitivities, the image intensity distribution should consist of a few peaks corresponding to different tissue types. Hence, when the image region defined by M(n)(r) is considered, the image intensity in the region should be equal to µ4. But in the images acquired by phased array coils the intensity distribution is mainly spread due to coil sensitivity non-uniformity. To restore the original intensity distribution, the coil sensitivities should be identified and their influence on the composite image intensity should be compensated. For each individual coil image, an estimate of the image intensity modulation due to the coil sensitivity magnitude | Sˆ in (r ) | is found by fitting a polynomial function to image intensity values in the region of support M(n)(r). A least-squares algorithm is used for polynomial fit because it is optimal for Gaussian noise found in MR images [14]. The estimate of the complex coil sensitivity is given by: ˆ Sˆ i( n ) (r ) =| Sˆ i( n ) (r ) | e jθ i ( r )

(6)

2.2.3 Reconstruction of Composite Image Once the sensitivity maps are found, new image estimate is calculated using the following equation:

(n) I OPT (r ) =

R ( r ) ψ −1 (Sˆ ( n ) ( r )) H Sˆ ( n ) ( r ) ψ −1 (Sˆ ( n ) ( r )) H

(7)

where Sˆ ( n ) ( r ) = [ Sˆ 1( n ) ( r ), Sˆ 2( n ) ( r ),..., Sˆ L( n ) ( r ) ] . The images obtained in this step of the algorithm have substantially suppressed intensity inhomogeneity in comparison with the original SoS images. 2.2.4 Termination of the Algorithm In each iteration, coil sensitivity maps and image estimate are updated based on the current region of support. Then the image estimate is used to re-identify the region of support for the next iteration. This process is repeated till the region of support does not change anymore; i.e. there are no more points added or removed from the identified dominant tissue region of support. This would cause the estimated sensitivity map to remain constant yielding the same image estimate. At this point when the region of support converges, the algorithm is terminated.

2.3 Application to Multi-contrast MR images The proposed technique can also be applied for inhomogeneity correction and reconstruction of multi-contrast images when they are acquired using the same coils over the same imaging volume. The only difference in the algorithm would be the identification of the dominant tissue spatial distribution. Since all of the multi-contrast images are modulated by the same coil sensitivities, the ratios between them are practically free from coil sensitivity intensity modulation and can be used to identify the dominant tissue spatial distribution. Given C1i (r), C2i (r),...,CKi (r), i=1,…L are individual coil images from a multi-contrast study with K ( n −1 ) ( n −1 ) ( n −1 ) different contrasts and C1OPT ( r ), C 2 OPT ( r ),..., CK OPT ( r ) are the corresponding composite images obtained by the (n-1)-th algorithm iteration and arranged in the order of decreasing SNR. The region of support of the dominant tissue in the n-th iteration, M(n)(r), is identified using the ratio of the highest SNR images as follows:  1 (n) M (r ) =  0 

γ − 0.05