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Adaptive Volterra Filter for Parallel MRI Reconstruction
EURASIP Journal on Advances in Signal Processing volume 2019, Article number: 34 (2019)
Abstract
Parallel magnetic resonance imaging (MRI) technique is able to accelerate MRI speed for reducing costs and enhancing patient’s comfortability. Parallel MRI can be categorized into two types: imagebased and kspacebased methods. For kspacebased parallel MRI, missing kspace data is reconstructed by interpolating existing acquired kspace data with appropriate coefficients, which is generally considered as a linear process. However, noise cannot be suppressed or removed during the linear reconstruction process and therefore reconstructed image often suffers serious noise, especially when the acceleration factor is high. Nonlinear filters are known to remove nonlinear noise better. Based on the Volterra series that discovers and removes the secondorder nonlinear noise, we proposed a nonlinear reconstruction strategy called adaptive Volterra generalized autocalibrating partial parallel acquisition (AVGRAPPA) to reconstruct the unacquired kspace signals. For the proposed AVGRAPPA, optimal selection of the secondorder Volterra series terms is adjusted and determined for optimizing reconstruction quality. Experimental results show that the proposed method is able to better remove the reconstruction noise and suppress aliasing artifacts.
1 Introduction
Magnetic resonance imaging (MRI) [1,2,3] is a noninvasive imaging technique. Different from computed tomography and other imaging technologies, MRI has the advantages of nonionizing radiation, multiple parameter imaging, high contrast, etc. It has become an important diagnostic tool in clinical imaging. However, MRI has its own shortcomings, such as long scan time, which limit its application in many clinical situations. Therefore, investigators have been exploring rapid magnetic resonance imaging methods since the invention of MRI technology. This has led to the development of methods for spiral acquisition, radial acquisition, parallel imaging, and so forth. Those methods have greatly improved the imaging speed to some extent. In MRI, the speed of imaging is very important. Early MRI scan often took several hours, and then the speed of imaging has increased dramatically due to improvements in field strength, gradient, and pulse sequence. However, rapid field gradients and highdensity continuous radio frequency (RF) pulses will result in higher specific absorption rate (SAR) that is unsustainable by the physiological limits of organ tissues in the human body. Therefore, the imaging speed cannot increase further.
With the application of complex computer image reconstruction algorithms with a phased array coils, imaging speed of MRI can be greatly improved. This technique is often referred to as parallel imaging technology. Parallel imaging includes simultaneous acquisition of spatial harmonics (SMASH) [4], sensitivity encoding (SENSE) [5] parallel acquisition technique, and generalized autocalibrating partially parallel acquisitions (GRAPPA) [6]. Parallel MRI (pMRI) reconstruction is an image reconstruction technique for rapid acquisition. It utilizes the spatial sensitivity difference of phased array coils for spatial encoding and simultaneous acquisition with phased array coils to achieve fast imaging speed. Currently, the acceleration in clinical applications can be about 2–6 times faster or even higher imaging speed. To use parallel imaging technology, new requirements are needed for MRI systems, such as multiple receiver channels, multiple array coils, and coil sensitivity calibration.
SMASH is a parallel acquisition and reconstruction method that uses coil sensitivity to fit spatial harmonic functions and fills undersampled data. The characteristics of the algorithm include the summation of the data of all channels and fitting procedure. There is a large error in the fitting calculation of the traditional algorithm, resulting in a serious artifact and a low signaltonoise ratio (SNR) of the SMASH image. Subsequently, GRAPPA enhances SMASH technique. GRAPPA uses the sampling data of all channels for fitting and recovers in the undersampled data of each channel. Then fully reconstructed image of each channel after the fitting is combined. Final image is calculated with sumofsquare on all channel images. The GRAPPA algorithm reduces the calculation error of fitting procedure and improves image quality.
When the number of acquired lines and the convolution kernel size change, GRAPPA reconstruction results also vary (corresponding to the estimation error that varies). For example, autocalibration signal (ACS) lines are usually sampled in the central region of kspace (lowfrequency region) rather than outer kspace (highfrequency region), based on which the fitting weights are derived. Therefore, interpolation would be inaccurate when they are used for reconstructing missing points at outer kspace region. Although variable density sampling strategy [7] has been proposed to solve this problem to some extent, estimation error still exists since it is impossible to fully sample kspace for reconstruction (otherwise, it is meaningless for pMRI). For both errors, Nana et al. proposed the crossvalidation model for estimating coefficients [8]. Since kspace signals cannot be fully acquired for GRAPPA reconstruction, model error is difficult to be reduced. During the matrix inversion process [9], measured noise will be propagated and amplified in the fitting and interpolation procedures. In this paper, we focus on how to reduce the second kind of error: noiserelated error. Furthermore, some works [10,11,12,13] on combining parallel imaging with compressed sensing and lowrank algorithms have been proposed. Reconstruction quality and imaging speed have been greatly enhanced.
Although some methods, including regularizationbased method [14, 15] and iterative reweighted leastsquares method [16], reduce the noise level in matrix inversion process, they fail to consider the noise generation routine in GRAPPA reconstruction. We analyze it and discover that nonlinear noise is generated in the propagation process. Finite impulse response (FIR) model is currently used in GRAPPA reconstruction [17, 18]. It models GRAPPA as a linear filter. The linear filter cannot remove nonlinear noise. This article presents a new method—adaptive Volterra GRAPPA (AVGRAPPA)—to address the poor SNR problem in GRAPPA. Based on the theory of adaptive Volterra series, adaptive Volterra filter can be used for suppressing nonlinear noise. The first part of the paper introduces the existing problems. The second section presents the background. The third part gives the proposed method. Experimental results are provided in the fourth part. Conclusion is given in the fifth part.
2 Theory
GRAPPA reconstruction can be generalized as a fitting and interpolation processes in the following equation [6]:
where S is kspace signal, w denotes weight coefficient set, R represents reduction factor, j is the target coil, l counts all coils, b and h are transverse neighbor acquired points. Indices k_{x} and k_{y} count through frequency encoding and phase encoding directions, respectively.
The weight coefficients are calculated by leastsquares:
Generally, the errors in the inverse process can be reduced in regularization SENSE and GRAPPA reconstruction [16, 17]:
where λ is regularization parameter. If noise can be suppressed with keeping a low level of aliasing artifacts, regularization strategy is actually effective for reconstruction. Otherwise, aliasing artifacts will deteriorate image quality seriously even noise can be reduced to some extent.
Because the linear filter has a wellestablished theoretical basis, simple mathematical analysis, and easy design and implementation, it has been widely used. However, it is found that in the presence of nonlinearity such as satellite links, highspeed communication channels, and echo cancelation, the linear filter’s performance is not ideal due to the inherent disadvantages of the linear adaptive filter. In order to overcome the shortcomings of linear filters and improve system performance, nonlinear filter theory research has gradually become popular. In recent years, a variety of nonlinear adaptive filtering methods have been proposed, such as morphological filter, homomorphic filter, order statistics filter, Volterra filter, and other polynomial filters. Volterra filter [19,20,21] studies both of the linear structure and nonlinear structure of the system. It is suitable for constructing nonlinear models of various systems and has broad application prospects. In actual application, the system features are unknown or timevarying, and parameters of the filter cannot be known in advance. Nonlinear active noise control (ANC) plays an important role in the field of nonlinear signal processing. The nonlinear characteristics contained in the Volterra filter are suitable for creating a nonlinear system model and it can effectively realize nonlinear active noise control. Therefore, the nonlinear active noise control based on Volterra filter may become feasible for adaptive signal processing in parallel MRI reconstruction.
The flowchart of the proposed method is presented in Fig. 1. Given the noisy input signal samples A and noisy desired output signal samples b, we are trying to estimate the true model parameters x^{′}, which is driven by na oiseless input signal A^{′} and produced noiseless output signal b^{′}. Volterra series has been widely used in signal processing, which was developed for modeling nonlinear behavior [20]. Conventional GRAPPA can be considered as a linear and timeinvariant system via convolution formula for the output v(t) in terms of the input z(t):
where z(t) and v(t) represent the input and output, respectively, and s(t) is the impulse response of the system. A nonlinear system is characterized by the Volterra series as
where s_{n}(τ_{1}, τ_{2}, …, τ_{n}) are called Volterra series coefficients. In mathematics, Volterra series represent an expansion of dynamics, nonlinear, and timeinvariant function. Volterra series can describe the input and output transfer characteristics of a nonlinear system, which is also referred to as Volterra filter. The major drawback of Volterra filter is that a large number of weights are needed to be computed which deteriorates computation cost. Furthermore, if n = 1 as shown in Eq. (5), it is a linear filter.
GRAPPA is essential a fitting and interpolation problem, which needs data consistency to obtain optimal fitting weights. If weights are more accurate, reconstructed missing kspace signals are more accurate. Otherwise, serious noise and artifacts will be appearing in the reconstructed image. The proposed adaptive Volterra filterbased GRAPPA reconstruction is presented as the following equation
where w denotes weight set, which has constant, firstorder, and secondorder parts, respectively. The firstorder part of the equation is equivalent to the conventional GRAPPA that linearly combine the data acquired from both phase (k_{y}) and frequency (k_{x}) directions from all coils. The constant and secondorder parts of Eq. (6) suppress the secondorder nonlinear noise. Noting that this relation is nonlinear in terms of input of acquired kspace data, but linear in terms of the weights w, it is possible to calculate the weights in matrix form similar to Eq. (2), which can simplify the process of solving weights. After the missing kspace, data are reconstructed, existing ACS lines and acquired lines are used to replace the corresponding reconstructed kspace data for the final sum of squares reconstruction.
A large computation cost is needed to solve the weights presented in Eq. (6), which is unbearable if all secondorder terms are calculated. Truncated Volterra series representation via random and pseudorandom inputs has been proposed for identifying the nonlinear system [21]. In order to enhance the reconstruction speed, only a subset of the secondorder terms that are randomly selected takes part in computing the weights. So, AVGRAPPA can be reformulated as follows:
where S_{pt} and S_{qt} are randomly selected from
The reconstruction quality also depends on the choice of the total number of the secondorder terms. When the number of terms U is too small, reconstructed image has low SNR, and if the number of terms were too large, aliasing artifacts would appear with high SNR. Similar to the choice of convolution kernel size for optimal reconstruction result, the number of the secondorder terms can also be chosen appropriately to achieve good reconstruction quality.
3 Methods
The proposed method is evaluated on one phantom dataset and two in vivo brain datasets. The Gaussian noise is added in this phantom dataset for testing performance. Informed consent was obtained from the volunteer in accordance with the institutional review board policy. Sensitivity information of each coil was obtained by prescanning. Based on empirical observation of multiple reconstructions, we choose 4 × 7 convolution kernel size. Furthermore, we choose 64, 48, and 38 ACS lines for the phantom, fourchannel brain, and eightchannel brain data sets, respectively. For the proposed AVGRAPPA, the number of the secondorder terms is set as 672, 784, and 896 for the phantom, fourchannel brain, and eightchannel brain data sets, respectively.
The proposed AVGRAPPA is also compared to reconstructed images by Tikhonov regularization [14] and iterative reweighted leastsquares (IRLS) [16] for performance evaluation. For visual evaluation, apart from directly comparing reconstructed images, difference maps and local patches are also used for comparison. Furthermore, with sum of squares (SoS) as the gold standard, reconstructions are also compared quantitatively in terms of the normalized mean squared error (NMSE) [22], which is defined as the normalized difference square between the reconstructed image (I_{estimated}) and the SoS as the gold standard (I_{standard}):
When the value of NMSE is larger, the quality of the reconstructed image is deteriorated more seriously, which suggests both increased image artifacts and noise.
For two comparison methods that reduce the noise level, Tikhonov regularization method for GRAPPA has an automatic mechanism to tune the parameter λ. Following the increasing of λ, the noise of reconstructed image will gradually disappear and aliasing artifacts will emerge. If both noise and aliasing artifacts exist in the content of the image simultaneously, the regularization does not have the power to reconstruct highquality image, since regardless of decreasing or increasing λ, reconstructed image is always deteriorated by noise or aliasing artifacts. We manually tune the parameter λ for Tikhonov regularization method to observe the gradual changing of reconstruction results. For IRLS, two implementations: slow robust GRAPPA and fast robust GRAPPA were proposed. Authors stated that both robust GRAPPAs have the similar performance, so we choose slow robust GRAPPA for the comparison without having to choose “outlier ratio” for different data sets, which should be done in the fast robust GRAPPA implementation.
4 Results and discussion
Figure 2 shows results of the reconstruction on phantom data set. The difference is noticeable among these methods. This experiment demonstrates that noised phantom can be recovered to highquality image with less noise by the proposed method. Figures 3 and 4 show reconstruction results of both in vivo data sets. For fourchannel brain imaging, reconstructions by the proposed method achieve a good performance compared to other methods. For eightchannel brain imaging, the difference maps also show promising results as the previous data sets. The proposed method focuses on removing noise rather than suppressing aliasing artifacts, so the result of the proposed AVGRAPPA still has tiny aliasing artifacts. It generally has the overall superior performance compared to other methods.
Table 1 shows the NMSEs for the reconstruction methods, in which columns represent different reduction factors, and rows denote different data sets. We fixed the kernel size as 4 × 7, and changed reduction factor for each data set. AVGRAPPA generally generates less reconstruction error compared to the conventional GRAPPA and existing methods. We still set the number of the secondorder terms as 672, 784, and 896 for the phantom, fourchannel brain, and eightchannel brain data sets, respectively. The proposed AVGRAPPA generally outperforms the other methods in terms of the NMSE measure. However, NMSE cannot describe the reconstructed image quality completely and accurately, because NMSE and other mean squared error measures do not correlate well with a subjective assessment of image quality. For eightchannel brain data reconstruction at reduction factor 6, reconstructed image by these methods has been seriously deteriorated by noise or artifacts, so NMSE measure cannot completely reflect the degree of deterioration. Although Tikhonov regularization and IRLS methods have lower NMSE value than that of the AVGRAPPA, they reconstruct the image with highlevel noise and aliasing artifacts. Another reason why the number of the secondorder terms is so high for eightchannel brain data reconstruction at reduction factor 6 is that insufficient ACS lines result in deterioration in reconstruction, which is presented in the discussion section.
Since GRAPPA constitutes both fitting and interpolation processes, nonlinear noise is generated in the final reconstructed kspace. Secondorder Volterra filter is able to remove it to achieve high SNR, while the previously existing methods do not count the influence of nonlinear noise. Tikhonov regularization presents good reconstruction results at low reduction factors. Although IRLS method can improve image quality to some extent, it still presents some noise. In the fitting process, small or zero weights are assigned to “outliers” in kspace, but these small or zero weights may be not accurate when they are used for estimating unacquired kspace signals in the interpolation.
This is one restriction of the AVGRAPPA. The main reason is that secondorder terms in adaptive Volterra filter will distort lowfrequency estimation to some extent. As we know, magnitudes of signals at lowfrequency are much larger than that of highfrequency. Outliers in lowfrequency regions will be enlarged by the secondorder terms to distort the estimation of missing signals at lowfrequency estimation. On the other hand, it is capable of estimating weights more accurately for the highfrequency region in the kspace. For this reason, the proposed AVGRAPPA is good at suppressing noise and has little ability to reduce aliasing artifacts. Furthermore, higherorder Volterra filter (e.g., the thirdorder, the fourthorder Volterra filters) will deteriorate the reconstruction more seriously in our experiments. The reason contains two folds: only the secondorder noise exists in the GRAPPA reconstruction so that the secondorder Volterra filter should be used, and higherorder terms will distort the estimation of missing signals of low frequency aggravatingly.
Because the secondorder part needs to be calculated by the adaptive Volterra filter compared to the conventional GRAPPA, in our experiments, reconstruction time is generally about 2–5 times of the conventional GRAPPA using the same convolution kernel size. Alternatively, the graph in Fig. 5 can help the user to tune the parameter U to achieve the optimal reconstruction results if the computation time is not the crucial factor when using the AVGRAPPA.
5 Conclusion
A nonlinear noise generation is analyzed in GRAPPA reconstruction. Based on the theory of Adaptive Volterra series, AVGRAPPA is proposed in this paper. It contains secondorder terms to suppress noise with the nonlinear structure of the reconstruction system. The future research will focus on how to solve the problem of reconstruction quality through using a smaller number of ACS lines and attempt to combine the proposed method with the previous nonlinear gradient applications [23, 24]. Since the secondorder terms can enlarge errors generated by outliers in the lowfrequency region, how to find a way to avoid the estimation error to reduce aliasing artifacts is also a major goal.
Availability of data and materials
Data sharing is not applicable.
Abbreviations
 ACS:

Autocalibration signal
 ANC:

Active noise control
 AVGRAPPA:

Adaptive Volterra generalized autocalibrating partial parallel acquisition
 FIR:

Finite impulse response
 GRAPPA:

Generalized autocalibrating partial parallel acquisition
 IRLS:

Iterative reweighted leastsquares
 MRI:

Magnetic resonance imaging
 NMSE:

Normalized mean squared error
 pMRI:

Parallel magnetic resonance imaging
 RF:

Radio frequency
 SAR:

Specific absorption rate
 SENSE:

Sensitivity encoding
 SMASH:

Simultaneous acquisition of spatial harmonics
 SNR:

Signaltonoise ratio
 SoS:

Sum of squares
References
D.W. McRobbie, E.A. Moore, M.J. Graves, M.R. Prince, MRI from Picture to Proton (Cambridge University Press, 2017)
E.M. Haacke, R.W. Brown, M.R. Thompson, R. Venkatesan, Magnetic resonance imaging: physical principles and sequence design, vol 82 (WileyLiss, New York, 1999)
G. Masselli, G. Gualdi, MR imaging of the small bowel. Radiology. 264(2), 333–348 (2012)
D.K. Sodickson, W.J. Manning, Simultaneous acquisition of spatial harmonics (SMASH): Fast imaging with radiofrequency coil arrays. Magn. Reson. Med. 38(4), 591–603 (1997)
K.P. Pruessmann, M. Weiger, M.B. Scheidegger, P. Boesiger, SENSE: Sensitivity encoding for fast MRI. Magn. Reson. Med. 42(5), 952–962 (1999)
M.A. Griswold, P.M. Jakob, R.M. Heidemann, M. Nittka, V. Jellus, J. Wang, B. Kiefer, A. Haase, Generalized autocalibrating partially parallel acquisitions (GRAPPA). Magn. Reson. Med. 47(6), 1202–1210 (2002)
J. Park, Q. Zhang, V. Jellus, O. Simonetti, D. Li, Artifact and noise suppression in GRAPPA imaging using improved kspace coil calibration and variable density sampling. Magn. Reson. Med. 53(1), 186–193 (2005)
R. Nana, T. Zhao, K. Heberlein, S.M. LaConte, X. Hu, Crossvalidationbased kernel support selection for improved GRAPPA reconstruction. Magn. Reson. Med. 59(4), 819–825 (2008)
F. Huang, G. Duensing, in Proceedings of the Fourteenth Annual Meeting of the International Society of Magnetic Resonance in Medicine. A theoretical analysis of errors in GRAPPA (2006)
P.J. Shin, P.E. Larson, M.A. Ohliger, M. Elad, J.M. Pauly, D.B. Vigneron, M. Lustig, Calibrationless parallel imaging reconstruction based on structured lowrank matrix completion. Magn. Reson. Med. 72(4), 959–970 (2014)
M. Uecker, P. Lai, M.J. Murphy, P. Virtue, M. Elad, J.M. Pauly, S.S. Vasanawala, M. Lustig, ESPIRiT—An eigenvalue approach to autocalibrating parallel MRI: Where SENSE meets GRAPPA. Magn. Reson. Med. 71(3), 990–1001 (2014)
C. Liao, Y. Chen, X. Cao, S. Chen, H. He, M. Mani, M. Jacob, V. Magnotta, J. Zhong, Efficient parallel reconstruction for high resolution multishot spiral diffusion data with low rank constraint. Magn. Reson. Med. 77(3), 1359–1366 (2017)
D. Liang, B. Liu, J. Wang, L. Ying, Accelerating SENSE using compressed sensing. Magn. Reson. Med. 62(6), 1574–1584 (2009)
P. Qu, C. Wang, G.X. Shen, Discrepancybased adaptive regularization for GRAPPA reconstruction. J. Magn. Reson. Imaging 24(1), 248–255 (2006)
F. Lin, in Proc 14th Annual Meeting ISMRM, Seattle. Priorregularized GRAPPA reconstruction (2006)
D. Huo, D.L. Wilson, Robust GRAPPA reconstruction and its evaluation with the perceptual difference model. J. Magn. Reson. Imaging 27(6), 1412–1420 (2008)
Z. Chen, J. Zhang, R. Yang, P. Kellman, L.A. Johnston, G.F. Egan, in 2009 IEEE International Conference on Control and Automation. 2D IIR filter for parallel magnetic resonance image reconstruction (IEEE, 2009), pp. 1792–1795
Z. Chen, J. Zhang, R. Yang, P. Kellman, L.A. Johnston, G.F. Egan, IIR GRAPPA for parallel MR image reconstruction. Magn. Reson. Med. 63(2), 502–509 (2010)
V. Volterra, Theory of functionals and of integral and integrodifferential equations (Courier Corporation, 2005)
S.P. Boyd, Volterra series: Engineering fundamentals (1985)
R.D. Nowak, B.D. Van Veen, Random and pseudorandom inputs for Volterra filter identification. IEEE Trans. Signal Process. 42(8), 2124–2135 (1994)
J. ChaniCahuana, C. Fager, T. Eriksson, Lower bound for the normalized mean square error in power amplifier linearization. IEEE Microwave Wireless Compon. Lett. 28(5), 425–427 (2018)
H. Wang, L. Tam, E. Kopanoglu, D.C. Peters, R.T. Constable, G. Galiana, Experimental Ospace turbo spin echo imaging. Magn. Reson. Med. 75(4), 1654–1661 (2016)
H. Wang, L. Tam, E. Kopanoglu, D.C. Peters, R.T. Constable, G. Galiana, Ospace with high resolution readouts outperforms radial imaging. Magn. Reson. Imaging 37, 107–115 (2017)
Acknowledgements
The authors would like to acknowledge all the participants for their significant contributions to this research study.
Funding
This work was supported in part by the grant from the National Natural Science Foundation of China (nos. 61871373 and 81729003), Natural Science Foundation of Guangdong Province (no. 2018A0303130132), Sanming Project of Medicine in Shenzhen (no.SZSM201812005) and the Natural Science Foundation of Shenzhen (no. JCYJ20160331185933583).
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YZ, JL and YC conceived the idea of the study. YC designed the framework of the study. HW, YZ and SS collected the data. HW, YZ and YC carried out part of the algorithms. SS, ZH and JL analyzed the data and the results. HW and YC wrote the manuscript. YZ, SS, ZH and JL revised the manuscript. All authors read and approved the final manuscript.
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Haifeng Wang, Yihang Zhou, and Shi Su are cofirst and equal authors.
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Wang, H., Zhou, Y., Su, S. et al. Adaptive Volterra Filter for Parallel MRI Reconstruction. EURASIP J. Adv. Signal Process. 2019, 34 (2019). https://doi.org/10.1186/s1363401906335
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DOI: https://doi.org/10.1186/s1363401906335
Keywords
 Parallel MRI
 GRAPPA
 Secondorder nonlinear noise
 Volterra series
 Nonlinear filter