Claudia M. Hillenbrand^{1}

Acquiring an optimal image for clinical applications often means to strike the right balance between resolution, scan time, and signal to noise (SNR) in order to achieve the desired imaging objectives. The SNR is a fundamental measure of quality and performance in MRI. This presentation will review the basic principles relevant to signal and noise, measurement of SNR, factors influencing SNR, and discuss techniques that attempt to optimize SNR.

How to Get the Optimal Signal-to-Noise?

Many research efforts in MRI are geared towards fast, quantitative imaging, ideally in high resolution; they often meet fundamental restrictions set by the signal-to-noise ratio (SNR). Through SNR measurements we can gain a better understanding of the performance and quality of the applied hardware (MR scanners, RF coils, coil positioning), and the imaging and reconstruction techniques. Through quantitative SNR analysis, protocols/sequence parameters can be optimized and objectively compared. Clinically, signal and noise considerations are, for example, important for reliable lesion detection and characterization, as well as in the context of quantitative diagnostic imaging, e.g. dynamic contrast enhanced imaging. Further, many advanced reconstruction methods and postprocessing steps require an SNR estimate: regularization in parallel imaging, signal thresholds in spectral analysis or curve fitting (relaxometry), automatic quality assessment, de-noising filters, and many more. For all these reasons SNR is probably the most essential metric of quality and performance in MRI. In this presentation, we review the basic principles relevant to SNR: sources of noise, basic noise statistics, multi-channel noise, measurement of SNR and contrast-to-noise ratio, and factors influencing SNR.

The random noise σ^{2} in the MR signal originates primarily from the thermal noise associated
with the resistance of the receiving coil and electronics, as well as from inductive
losses in the sample (1):
σ^{2}=4kTR_{e} BW, where R_{e} is the effective resistance, BW is the sampling bandwidth,
k is the Boltzman constant, and T is
the temperature. The sample losses are usually dominant for common clinical conditions
with efficient receiver coils and higher magnetic field strength (>0.5T). The
final image noise will depend on the voxel size, BW, and number of averages (2,3).

For a simple single coil scenario, it can be assumed that the noise acquired on the two quadrature channels is white, independent, and has a Gaussian distribution with zero mean (4,5). The noise is derived from the root-mean-square (RMS) of the amplitude of white noise which equals, for Gaussian distributions, its standard deviation s. Real and imaginary images are reconstructed by complex Fourier transform (FT). The FT will preserve the Gaussian distribution of the noise and the variance of the noise will be evenly distributed over the image. The SNR of a complex image can then be determined as the ratio of the mean signal intensity S in a region of interest of the object to the standard deviation s of the noise, SNR = S/σ. Alternatively two images can be acquired, one with and one without transmit pulses, the image without RF excitation contains pure noise.

If the image is formed as the magnitude of the complex image,
then both the mean and the standard deviation exhibit a bias because taking the
magnitude is a nonlinear transform of noisy data (5,6). Indeed the Gaussian distribution is not applicable
anymore, the noise distribution has no zero mean anymore, is skewed, and follows
a Rician distribution (or Rayleigh as a special case of Rice for no signal). Henkelman calculated an underestimation, 65.5%
of the true noise standard deviation (5), and suggested numerical correction factors for the bias, while Gudbjartsson et al. (6) suggested a simpler, effective scheme to obtain a corrected signal Ā as a measure for the true magnitude image intensity Ā=√|M^{2}-σ^{2}| where M is the measured magnitude signal.

If data is acquired with multiple coils and especially if parallel imaging is used, noise is generally not homogeneously distributed over the image anymore and the noise of individual channels may be correlated because of coupling between the receive coils. Noise decorrelation adjustments are available in modern systems which can be used to estimate the noise level (7,8). Spatial variations of noise and noise amplification (9) across the image now also depend on coil placement and selection, as well as coil combination and image reconstruction methods used. Therefore, coil channels need to be optimally combined to avoid unnecessary amplification and spatial variations in the noise. Common multi-coil combination methods on clinical systems are the “sum-of-squares” (10) and “adaptive combine” (11) algorithms.

Many current imaging protocols use parallel imaging (9,12-14). Parallel imaging acceleration will degrade the SNR by the square root of the acceleration factor because of reduced data sampling. Further, the signal synthetization step of parallel imaging unavoidably amplifies noise, and noise in the images becomes spatially varying (9). The effect is described by the g-factor, which can be estimated using phantoms for SENSE and for a given coil setup (9). This is not the case for the GRAPPA g-factor, which is dependent on the object and the image contrast, and therefore must be estimated for each individual measurement data set (15).

When quantifying noise for multi-coil and parallel acquisitions, simple SNR measurement methods as the ones introduced above are not sufficient. A better method would be to perform two acquisitions and subtract the images to obtain a noise-only image (16,17). Even more accurate would be to perform multiple acquisitions and estimate the SNR pixel-wise as the mean over the standard deviation of the image series. This requires the object to be completely still, which makes this approach mostly unsuitable for human subjects (8).

It is possible to calculate SNR scaled images based on the measurements of absolute noise levels and perform image reconstruction in SNR units (8). This can be used for basic SNR measurements and quality control, and also in the context of optimum multi coil combinations. This methods requires a pre-scan without RF excitation. However, it also requires careful analysis of the processing steps and applies only to linear operations. For parallel imaging reconstruction in SNR units, the g-factor has to be taken into account separately. Analytical approaches for determining the g-factor have been derived for SENSE (9), SMASH (12), and GRAPPA (15).

A pragmatic and promising alternative approach to measure SNR is to use the raw data of a single acquisition and to feed it through the image reconstruction multiple times, each time with a different noise realization added. The method has been proposed with use of natural noise (18), and synthetic noise (19). The latter work introduced the term pseudo-replicas. The pseudo-replicas can then be used to derive the SNR. This measure of SNR does not only include the basic noise characteristics of the patient and the hardware, but also incorporates how noise is handled by the signal processing. This method is therefore independent of the actual sequence, reconstruction, and postprocessing implementation. Later work introduced refined data processing for efficiency (20).

1. Hoult DI, Lauterbur PC. Sensitivity of the Zeugmatographic Experiment Involving Human Samples. Journal of Magnetic Resonance 1979;34(2):425-433.

2. Edelstein WA, Glover GH, Hardy CJ, Redington RW. The Intrinsic Signal-to-Noise Ratio in Nmr Imaging. Magnet Reson Med 1986;3(4):604-618.

3. Brown WB, Cheng Y-CN, Haacke EM, Thompson MR, Venkatesan R. Signal, Contrast, and Noise. In: Brown WB, Cheng Y-CN, Haacke EM, Thompson MR, Venkatesan R, editors. Magnetic Resonance Imaging: Physical Principles and Sequence Design, 2nd Edition. Hoboken, New Jersey: Wiley-Blackwell; 2014. p 325-374.

4. McVeigh ER, Henkelman RM, Bronskill MJ. Noise and filtration in magnetic resonance imaging. Med Phys 1985;12(5):586-591.

5. Henkelman RM. Measurement of signal intensities in the presence of noise in MR images. [Published erratum in: Magn Reson Med. 1986;13:544]. Med Phys 1985;12(2):232-233.

6. Gudbjartsson H, Patz S. The Rician distribution of noisy MRI data. Magn Reson Med 1995;34(6):910-914.

7. Pruessmann KP, Weiger M, Bornert P, Boesiger P. Advances in sensitivity encoding with arbitrary k-space trajectories. Magn Reson Med 2001;46(4):638-651.

8. Kellman P, McVeigh ER. Image reconstruction in SNR units: a general method for SNR measurement. Magn Reson Med 2005;54(6):1439-1447.

9. Pruessmann KP, Weiger M, Scheidegger MB, Boesiger P. SENSE: sensitivity encoding for fast MRI. Magn Reson Med 1999;42(5):952-962.

10. Roemer PB, Edelstein WA, Hayes CE, Souza SP, Mueller OM. The NMR phased array. Magn Reson Med 1990;16(2):192-225.

11. Walsh DO, Gmitro AF, Marcellin MW. Adaptive reconstruction of phased array MR imagery. Magn Reson Med 2000;43(5):682-690.

12. Sodickson DK, Manning WJ. Simultaneous acquisition of spatial harmonics (SMASH): fast imaging with radiofrequency coil arrays. Magn Reson Med 1997;38(4):591-603.

13. Jakob PM, Griswold MA, Edelman RR, Sodickson DK. AUTO-SMASH: a self-calibrating technique for SMASH imaging. SiMultaneous Acquisition of Spatial Harmonics. MAGMA 1998;7(1):42-54.

14. Griswold MA, Jakob PM, Heidemann RM, Nittka M, Jellus V, Wang J, Kiefer B, Haase A. Generalized autocalibrating partially parallel acquisitions (GRAPPA). Magn Reson Med 2002;47(6):1202-1210.

15. Breuer FA, Kannengiesser SA, Blaimer M, Seiberlich N, Jakob PM, Griswold MA. General formulation for quantitative G-factor calculation in GRAPPA reconstructions. Magn Reson Med 2009;62(3):739-746.

16. Sijbers J, den Dekker AJ, Van Audekerke J, Verhoye M, Van Dyck D. Estimation of the noise in magnitude MR images. Magn Reson Imaging 1998;16(1):87-90.

17. Reeder SB, Wintersperger BJ, Dietrich O, Lanz T, Greiser A, Reiser MF, Glazer GM, Schoenberg SO. Practical approaches to the evaluation of signal-to-noise ratio performance with parallel imaging: application with cardiac imaging and a 32-channel cardiac coil. Magn Reson Med 2005;54(3):748-754.

18. Riffe MJ, Blaimer M, Barkauskas KJ, Duerk JL, Griswold MA. SNR Estimation in Fast Dynamic Imaging Using Bootstrapped Statistics Proceedings 15th Scientific Meeting, International Society for Magnetic Resonance in Medicine. Berlin2007. p 1879.

19. Robson PM, Grant AK, Madhuranthakam AJ, Lattanzi R, Sodickson DK, McKenzie CA. Comprehensive quantification of signal-to-noise ratio and g-factor for image-based and k-space-based parallel imaging reconstructions. Magn Reson Med 2008;60(4):895-907.

20. Wiens CN, Kisch SJ, Willig-Onwuachi JD, McKenzie CA. Computationally rapid method of estimating signal-to-noise ratio for phased array image reconstructions. Magn Reson Med 2011;66(4):1192-1197.