Dietrich O, Raya JG, Reiser MF MR noise measurements at very low noise levels MR noise measurements and signal-quantization effects at very low noise levels Olaf Dietrich, José G. Raya, Maximilian F. Reiser Josef Lissner Laboratory for Biomedical Imaging, Department of Clinical Radiology – Grosshadern, LMU Ludwig Maximilian University of Munich, Munich, Germany ELECTRONIC PREPRINT VERSION: Not for commercial sale or for any systematic external distribution by a third party Final version: Magn Reson Med 2008; 60(6): 1477–1487. <URL:http://dx.doi.org/10.1002/mrm.21784> Abstract The well-known noise distributions of MRI data quantization error can be accurately estimated (Rayleigh, Rician, or non-central chi-distribution) with the proposed maximum-likelihood ap- describe the probability density of real-valued proach. (i.e. floating-point) signal intensities. MR image data, however, is typically quantized to integers before visualization or archiving. Depending on Keywords: the scaling factors applied before the quantiza- Magnetic resonance imaging, Signal quantization, tion and the signal-to-noise ratio (SNR), very low Statistical noise distribution, Signal-to-noise ratio noise levels with substantial artifacts due to the quantization process can occur. The purpose of Corresponding author: this study was to analyze the consequences of Olaf Dietrich, PhD the signal quantization, to determine the theo- Josef Lissner Laboratory for Biomedical Imaging retical absolute lower limit for noise measure- Department of Clinical Radiology – Grosshadern ments in discrete data, and to evaluate an im- LMU Ludwig Maximilian University of Munich proved method for noise and SNR measurements Marchioninistr. 15 in the presence of very low noise levels. Image 81377 Munich data were simulated with original noise levels Germany between 0.02 and 2.00. Noise measurements Phone: +49 89 7095-3623 were performed based on the properties of back- Fax: +49 89 7095-4627 ground and foreground data using the conven- E-mail: [email protected] tional approach, which exploits the standard de- viation or mean value of the signal, and a maxi- mum-likelihood approach based on the relative (Parts of this study were presented on the ISMRM frequencies of the observed discrete signal in- annual meeting 2008, abstract #1529.) tensities. Substantial deviations were found for the conventionally determined noise levels, while noise levels comparable to or lower than the Magn Reson Med 2008; 60(6): 1477–1487 (accepted 21 July 2008) Page 1 of 15 Dietrich O, Raya JG, Reiser MF MR noise measurements at very low noise levels zation of the signal intensity is ignored. Conse- Introduction quently, the original noise standard deviation as well as the SNR determined from these parameters The properties of the MR signal intensity in the with the methods mentioned above will be over- or presence of noise, such as the statistical distribu- underestimated. The purpose of this study was to tion, mean value, and standard deviation, have analyze the consequences of the signal quantization been extensively discussed in several publica- and to evaluate an improved method for the exact tions (1–11). It is commonly assumed that noise in determination of very low noise levels in discrete MRI raw data is normally distributed in each re- image data. ceiver channel. The statistical signal distribution in the final image depends on the reconstruction and channel-combination technique. Signal statistics Theory are described, e.g., by the Rayleigh or Rician dis- tribution for single-channel magnitude data after Continuous signal distributions conventional Fourier-transform reconstruction (1– If we assume normally distributed noise superim- 8) and by the non-central χ-distribution in the case posed on the signal of the real and imaginary part of a root-sum-of-squares (RSS) reconstruction of of raw data in each receiver channel, then real and multi-channel data (9,12,13). These statistical dis- imaginary part of the complex image data of each tributions play an important role for the measure- channel reconstructed by Fourier transform are ment of the signal-to-noise ratio (SNR) or the con- also superimposed by white Gaussian noise (3,4). trast-to-noise ratio (CNR) in MR images, both of The statistical distribution of the signal, x, is de- which are based on the exact determination of the scribed by the probability density image noise level. Several methods for noise meas- 1 − ()x − µ 2 urements have been proposed depending on the PGauss ()x;µ,σ = exp [1] 2π σ 2σ 2 noise properties. If a spatially uniform noise distri- with original standard deviation σ and mean value µ bution can be assumed, then the noise level can be (the mean values, µ, of the real and imaginary part determined by analyzing the background noise (i.e. will in general be different; in background areas, the MR signal in air) of an MR image (2,14,15). If this mean value is zero). We call σ the “original” the noise level is variable over the image (e.g., as a standard deviation, since the actual standard devia- consequence of parallel-imaging reconstruction), tion measured in the final (e.g., magnitude) image then the noise level should be determined at the will be different from σ, in general. same location of the image foreground as the sig- nal, for instance by analyzing the signal in a differ- MR images are usually presented as magni- ence image (10,15–17) or by pixelwise evaluation tude data. The most important probability densities of the signal time-course in a large number of re- for magnitude image signals in MRI are the Rician peated acquisitions (10,15,17). distribution (2,3,7), which describes signal statis- tics after magnitude calculation of complex data, The statistical distributions mentioned above P x;µ,σ = describe the probability density of real-valued sig- Rice ( ) nal intensities, i.e. signal intensities represented by 1 − x 2 + µ 2 xµ , [2] x exp ()I 2 2 0 2 floating-point numbers. Image data, however, are σ 2σ σ commonly quantized to integers before visualiza- and the non-central χ-distribution of signal intensi- tion or archiving in the DICOM format (18). De- ties after RSS reconstruction of n channels (9), pending on the signal-to-noise ratio and the scaling Pncχ (x; µ,σ ,n) = factors applied before the quantization, very low n noise levels with substantial artifacts due to the µ x − x 2 + µ 2 xµ , [3] exp ()I 2 2 n−1 2 quantization process can occur. In this case, the σ µ 2σ σ evaluation of the signal mean value or the standard where In(z) is the modified Bessel function of the deviation in a background region, in subtraction first kind of nth order. In the image background data, or in a pixel time-course will lead to inaccu- with zero original signal, µ = 0, Eq. [2] simplifies to rate estimations of the true noise level if the quanti- Magn Reson Med 2008; 60(6): 1477–1487 (accepted 21 July 2008) Page 2 of 15 Dietrich O, Raya JG, Reiser MF MR noise measurements at very low noise levels the Rayleigh distribution (1): difference between the original value, x, and the 1 − x 2 quantized value, k, i.e., e = x − Q()x ; for truncation, PRayleigh ()x;σ = PRice (x;0,σ )= x exp [4] σ 2 2σ 2 the quantization error is between 0 and 1; and for and Eq. [3] can be simplified to (9) symmetric rounding, it is between –0.5 and 0.5. It is also useful to define quantization boundaries, bk, Pncχ ()x;0,σ ,n = which separate intervals that are quantized to dif- n−1 1 x − x 2 , [5] ferent values, i.e., two real numbers x , x are quan- x n exp 1 2 2 2 2 Γ()n σ 2σ 2σ tized to the same value k = Q(x1) = Q(x2), if and only where (n) is the gamma function. If, on the other if they lie between two successive quantization hand, the signal-to-noise ratio is sufficiently high, boundaries, bk ≤ x1,x 2 < bk+1. For instance, the then both the Rician and the non-central χ- quantization boundaries for truncation of non- distribution can be well approximated by Gaussian negative numbers are 0, 1, 2, 3, …, while the quan- distributions with the original standard deviation, σ tization boundaries for arithmetic rounding are 0, (Eq. [1]). This is important for noise measurements 0.5, 1.5, 2.5, … based on the pixel time-course in repeated acquisi- If the original image noise level, σ, is suffi- tions or on data of a difference image. In both ciently large in comparison to the quantization er- cases, the SNR in the analyzed region should be ror, i.e. σ >> 1, then quantized MR image data are sufficiently high to provide approximately normally well described by the continuous probability den- distributed signal intensities. The standard devia- sity, P(x; µ, σ, r); here, r shall denote any other pa- tions of the considered data are either the original rameters such as number of channels. However, standard deviation, σ, if a single pixel time-course the continuous description by P(x; µ, σ, r) is not is considered or the original standard deviation longer appropriate if σ is in the order of 1 or below, increased by a factor of 2 if a region of subtrac- i.e. if image noise and quantization noise become tion data is evaluated. comparable. In this case, the signal intensity cannot longer be considered as (almost) continuous quan- Discrete signal distributions tity. Instead of characterization by a probability The last step of the image reconstruction com- density, the distribution of the discrete signal in- monly includes the transition from floating-point to tensities, k, is much better described by the relative (non-negative) integer signal intensities, i.e.