A new single acquisition, two-image difference method for determining MR image SNR

Transcription

1 A new single acquisition, two-image difference method for determining MR image SNR Michael C. Steckner a Toshiba Medical Research Institute USA, Inc., Mayfield Village, Ohio Bo Liu MR Engineering, GE Healthcare, Waukesha, Wisconsin, Leslie Ying Department of Electrical Engineering and Computer Science, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin, Received 5 February 2008; revised 6 November 2008; accepted for publication 6 November 2008; published 27 January 2009 A new method for computing the signal-to-noise ratio SNR of magnetic resonance images is presented. The proposed method is a difference of images based technique where two images are produced from one acquisition in which the readout direction field of view FOV and matrix size are doubled compared to the phase encode direction. Two normal unaliased FOV images are produced by splitting undersampling the even versus odd data points in the read direction into two separate raw data sets. After image reconstruction, conventional difference of images SNR computations are applied. Signal defined as mean within signal producing region of interest ROI in one image, noise defined as standard deviation of the difference between the two images using the same signal ROI position and size, divided by sqrt 2 to account for the subtraction process. This method combines the desirable minimal acquisition time of a single image acquisition technique and the superior noise quantification characteristics of the difference of images methodology. The proposed method is more robust against system drift than existing SNR difference of images methods because the two images are effectively acquired nearly simultaneously in time. This method is compatible with phased array coils and is useful for parallel image reconstruction analysis because it is very stable. This method produces results that can be made equivalent to, and compared with, other existing SNR methods with simple known scale factors, assuming the image noise follows theoretical expectations American Association of Physicists in Medicine. DOI: / Key words: MRI, SNR, image quality I. INTRODUCTION The signal-to-noise ratio SNR is a fundamental measure of magnetic resonance image MRI quality. Many methods have been presented in the literature for measuring MR image SNR. 1 9 All of these methods compute estimates for the signal and noise components separately, and divide these estimates to produce a measure of image SNR. All of the methods estimate the signal component as the average signal value within a region of interest ROI, averaging out any image intensity nonuniformities. The differences between the various methods relate to how the noise estimates are determined. The various noise estimates are tradeoffs between speed of acquisition, simplicity of measurement and computation, accuracy, and robustness of measurement. In all cases, it is assumed that the noise is spatially invariant, otherwise the noise statistics would vary as a function of measurement location. The most common methods for noise estimation can be broadly categorized as follows: 1 measurements in a background region of the image well removed from any signal and ideally free of artifacts, 2 measurements in the signal producing region where the signal component has been eliminated by subtracting two identically collected images, and 3 measurements from a noise-only image. All of these methods have well known benefits and weaknesses. The single image based methods are the fastest, excluding one dimensional 1D projection methods, 10 but it has been demonstrated that subtle image artifacts, even those artifacts that might not be visible, can contaminate the noise measurements and produce anomalous results. 11 It is also possible that any filter roll-off will change the edge of the image, containing the noise measurement region, differently than the central region of the image, containing the signal measurement region. These problems are some of the primary motivations for this new method. The collection of two images for subtraction based methods doubles the total measurement time and these longer measurements can be sensitive to system drift, resulting in anomalous results. For this reason, the NEMA MS1 SNR standard 9 stipulates a maxi- 662 Med. Phys. 36 2, February /2009/36 2 /662/10/$ Am. Assoc. Phys. Med. 662

2 663 Steckner, Liu, and Ying: Computing the SNR of MR images 663 mum time between scans to minimize drift induced anomalous results. In the absence of system drift issues, this method is probably the most reliable, well known, SNR measurement technique. 11 Other methods attempt to eliminate system drift by collecting one signal scan and one noise scan, as discussed in the NEMA MS1 standard, method No While this eliminates system drift issues, it may not capture the full range of system noise sources found in a typical MR image. In addition, noise-only scans are potentially sensitive to the anomalous results already noted for single image SNR methods and any image reconstruction e.g., parallel imaging or postprocessing algorithms noise suppression algorithms that may modify the background noise regions. In addition to specifying where in the images the noise measurements are taken, or how the images are acquired to produce the best image noise data, there have also been efforts to determine how to best compute the noise statistics 1,4 7,12 robustly. The generation of a magnitude MR image may alter the noise distribution from the expected Gaussian distribution to a Rician distribution 1,4,12 depending on the signal level bias. It was recognized early 12 that the mean and standard deviation of noise-only regions are biased in a consistent fashion and that computing SNR with these biased estimates is acceptable 1 as long as the bias was appropriately compensated per the number of receive channels. 6 To enable comparisons between the various SNR computation methods, this article uses the appropriate bias factors. However, there are other methods for computing estimates. 5 7 As noted in Ref. 7, some alternate methods for computing noise require systematic errors to be consistent in dual image acquisition methods. In the authors experience system drift can generate erroneous negative noise estimates when computing such noise estimates. The method presented in this article effectively stabilizes and eliminates system drift so that it is possible to compute such noise estimates for nonparallel imaging applications. Starting on the premise that the difference of images method is generally the most reliable general purpose method for SNR determinations, 11 in the absence of system drift, the proposed new method modifies a single acquisition method to behave as a difference of images method, where the time difference between the two resultant images has been reduced to the absolute minimum possible thereby eliminating one of the few limitations of difference of images methods. The rest of this article describes and demonstrates the characteristics of the new method 13 by comparing it to other commonly used SNR calculation methods e.g., Refs. 1, 2, 8, and 9 and variants. First, 1D numerical simulations are shown that demonstrate equivalence between all of the eight of the primary SNR methods used in this article. These simulations are also used to demonstrate that the proposed method does not have a bias mechanism that anomalously improves SNR results. The article then demonstrates the utility of the new method with simulated two dimensional 2D artifacted images and experimental data. Last, the article finishes with a demonstration of the suitability of the technique for parallel imaging applications. II. THE PROPOSED NEW METHOD FOR COMPUTING SNR The proposed new method reduces the effective time separating the two image acquisitions to the minimum possible in order to eliminate any system stability and/or drift factors. One possible solution is to acquire each phase encode line twice, in direct succession, but rather than averaging the two acquisitions together, two separate and complete k-space data sets are created. In this case, the time separation between the two images is reduced to the TR interval. This alternative is not demonstrated here, but is a method used internally by at least one commercial vendor. 14 Clearly such a technique is an improvement over two serial image acquisitions where the time separation between images is the total scan time of one image. The proposed method reduces the time interval between the two acquisitions to the minimum possible: the time separation between successive data points in the readout direction. To create two images from a single acquisition, with equivalent field of view FOV, bandwidth, and matrix size or resolution, the number of data points in the read direction must be doubled, the read direction FOV doubled, thereby doubling the total sampling time. Since most single image SNR measurement methods use low duty cycle spin echo sequences, the total scan time should not change. There is no need to increase the FOV in the phase encode direction. Most MR scanners have a feature to acquire a rectangular FOV where the phase encode direction FOV and matrix size are reduced by an equal percentage, reducing total scan time yet retaining the same resolution. Given the read direction FOV is twice the size required to acquire a normal unaliased image of the signal phantom, and there are twice as many points as normally required, it is possible to produce two images with a square FOV by decimating the data in the read direction. The even numbered data points and odd numbered data points are separated into two distinct data sets. The resultant two magnitude reconstructed images will have the same characteristics bandwidth, number of raw data points, FOV, etc. as the original magnitude reconstructed normal single image acquisition. The magnitude operator also eliminates the slight phase difference between the two images introduced by the slight time shift between even and odd data points Fourier transform shift theorem. After the data decimation step, the proposed method is identical to the standard dual image difference of images method. Requiring access to the raw data is a potential implementation limitation of this technique. It is not possible to recreate the k-space data from the single original rectangular FOV, magnitude reconstructed image for later decimation. However, if the MR scanner provides a complex domain output of the original rectangular FOV image, the necessary k-space data can be recreated with a simple 2D inverse Fourier transform and decimated as described, assuming that the image recon-

3 664 Steckner, Liu, and Ying: Computing the SNR of MR images 664 struction process has no special processing algorithms, etc. As the complexity of manufacturers reconstruction algorithms increase, such assumptions may not be good. Since the number of data points, pixel dimensions, FOV, bandwidth, amplitude, etc. of the decimated data set images are identical to the original single image acquisition, the SNR measurements should be identical to perfectly well behaved, serially acquired two-image data, if we make two reasonable assumptions: 1 halving the readout gradient amplitude required to double the FOV, and maintaining resolution by doubling the readout matrix size, has no impact on the SNR measurement and 2 signal decay during the lengthened data readout has no impact on image SNR, assuming T2* is much longer than the length of the data acquisition window. If the SNR measurement sequence in question is a high duty cycle sequence, the increase in data sampling time may force an increase of TE/TR, and consequently, slightly lower SNR and increase total scan time. Alternatively, if sequence considerations demand that TE/TR and total scan time remain fixed, or the phantom T1/T2 require shorter data readouts, it is possible to restore the gradient amplitude to the original magnitude, double the sampling rate, double the readout matrix size, and maintain the same total data sampling time. However, the SNR measurement will be sqrt 2 lower than the original single image SNR measurement because the image bandwidth was increased by a factor of 2. This extra factor must be accounted for, depending on how the user wishes to compare the results of this technique with another method using a different sampling bandwidth. Note that this alternate implementation assumes the total frequency domain response of the receiver channel due to any filters or coil Q is flat within the sampling bandwidth. Other combinations of sequence parameters can also be used, and appropriate correction factors applied, to produce SNR measurements equivalent to a normal square FOV SNR acquisition. This method is also compatible with any phased array coil reconstruction whereas single image SNR measurements are not necessarily compatible, depending on details of the reconstruction process, and/or postprocessing algorithms and other issues such as artifacts, etc. 11 While there is research 6 which describes the statistical characteristics of background noise in phased array coil images, the statistical relationships do not necessarily hold if the individual coil combinations are not root-mean-square with appropriate weighting factors for equal noise contributions, or if other specialized reconstruction and postprocessing algorithms are used. Since details of the image reconstruction process are generally unknown, it is helpful to avoid situations where assumptions can cause errors. Conversely, the same unspecified and unknown reconstruction and/or postprocessing algorithms are typically not a factor when the noise is substantially biased away from zero signal levels by high signal levels because the noise distribution is not rectified 4,12 and any modifications to the noise distribution will be applied equally to both images in the proposed new method. Last, since a normal FOV SNR image is never aliased, the doubled FOV image will also be unaliased and the decimated data sets should therefore be compatible with parallel imaging techniques where aliasing is deliberately introduced in the orthogonal phase encode direction. III. THE EIGHT SNR METHODS The characteristics of the new technique are best illustrated by comparison with existing well known SNR computation methods. If two images are acquired with a double sized read FOV and matrix size images A and B and can be decimated images A1, A2, B1, B2 then it is possible to compute SNR by the following eight methods: 1 single image SNR of the full two times FOV image, using mean of a background region as a noise estimate two estimates: one from image A, another from image B, method reference: Ref. 1, 2 single image SNR of the full two times FOV image, using the standard deviation of a background region as a noise estimate two estimates, same as No. 1, method reference: Ref. 1, 3 difference of images SNR of the full two times FOV image one estimate: from A B, method reference: Ref. 9, 4 single image SNR of the decimated image, using mean of a background region as a noise estimate four estimates: A1, A2, B1, B2, method reference: Ref. 1, 5 single image SNR of the decimated image, using the standard deviation of a background region as a noise estimate four estimates, same as method No. 4, method reference: Ref. 1, 6 difference of images SNR of the decimated image two estimates: A1 A2, B1 B2 the recommended new SNR measurement technique, 7 difference of unrelated images SNR of the decimated image four estimates: A1 B1, A1 B2, A2 B1, A2 B2, method reference: Ref. 9, and 8 NEMA alternate. Similar to No. 3, but using difference between neighboring pixels in noise image to reduce any drift, motion etc. issues one estimate. method reference: Ref. 9. Each pair of images thus produces 20 SNR estimates. Since no special processing was applied to the raw data and all image reconstructions were done with simple 2D magnitude Fourier transforms, it was possible to compute the various SNR estimates and use various correction factors to compensate for Rician distribution characteristics of background noise measurements, and apply a correction factor to compensate for the subtraction step required in difference of images measurements. The basis of comparison is the single, normal FOV image. The following correction factors were applied to bring all eight different methods together to a common basis:

4 665 Steckner, Liu, and Ying: Computing the SNR of MR images 665 a b c d for SNR measurements using the full two times FOV data sets, SNR is sqrt 2 higher because there are twice as many data points, for SNR measurements using difference of images, the subtraction process increases the standard deviation noise estimate by sqrt 2, for SNR measurements using the mean of a background region free of artifacts beside the phantom in the read direction, the mean value is 2.74 times larger than the standard deviation for a four channel coil, according to Rician distribution statistics, 6 and for SNR measurements using the standard deviation of a background region free of artifacts beside the phantom in the read direction, the measured standard deviation is times the true standard deviation, according to Rician distribution statistics. 6 TABLE I. Average SNR, and standard deviation, over 1000 simulations for high and low SNR conditions. All methods give nearly identical SNR results. Method No. High SNR results Low SNR results SNR Standard deviation SNR Standard deviation new method Therefore, the eight SNR methods were compensated as follows: 1 SNR/sqrt 2 * SNR/sqrt 2 *0.695 =SNR *1.937 factors Nos. a, c, =SNR *0.983 factors Nos. a, d, 3 SNR/sqrt 2 *sqrt 2 =SNR factors Nos. a, b, 4 SNR *2.74 =SNR *2.740 factor No. c, 5 SNR *0.695 =SNR *0.695 factor No. d, 6 SNR *sqrt 2 =SNR *1.41 factor No. b, 7 SNR *sqrt 2 =SNR *1.41 factor No. b, 8 SNR/sqrt 2 *sqrt 2 =SNR factors Nos. a, b. While method Nos. 4 and 5 appear identical after corrections because of identical ROI, in experimental practice different results may occur because of the relative robustness of standard deviation less robust versus mean more robust in the authors experience for estimating the noise statistics. There can also be slight experimental differences between method Nos. 1 and 4 mean of background, and Nos. 2 and 5 standard deviation of background because the ROIs are not shared. IV. 1D SIMULATION RESULTS AND DISCUSSION The eight SNR measurement methods were simulated in MATLAB Mathworks, Natick, MA to verify equivalence. In order to simulate the downsampling from the analog-todigital A/D sampling rate to the final sampled output a point sinc function was generated and Gaussian noise added to both the real and imaginary channels. This simulated A/D stream was then decimated by a factor of 50 decimate data, 50, fir to create a 1D data set with 512 points. This 512 point data set was then split into two data sets even data points versus odd data points as required by the new SNR computation method. The simulation of the decimation process was added to ensure that the even versus odd data points did not contain some residual correlation that would contaminate the noise determination process of the proposed SNR measurement method. One thousand iterations were performed at both high and low SNR levels as a further check of signal decimation on SNR calculations. The results are summarized in Table I after all appropriate correction factors were applied. All SNR methods produced similar SNR results. The low standard deviation for method Nos. 1 and 2 is due to the large number of background pixels used to compute the noise statistics. The SNR values produced by the new technique appear no different than the other methods, suggesting that the decimation and sharing process required by the new SNR method does not introduce some noise correlation that results in anomalously different SNR. The potential for noise correlations due to the decimation and sharing process was also further investigated by computing the correlation coefficient of the noise produced in the two full images correlation test No. 1: image A versus B, the decimated images correlation test No. 2: image A1 versus A2, correlation test No. 3: image B1 versus B2 and the unrelated decimated images correlation test Nos. 4 7, respectively: images A1 versus B1, A1 versus B2, A2 versus B1, A2 versus B2. The average correlation coefficients of these seven comparisons and the standard deviation of all the iterations are shown in Table II. Table II shows that the noise is not correlated. The standard deviation of the full data set correlation iterations is lower than all other results because the number of data points used for the calculations was larger. Given that the last four correlations in Table II come from unrelated images, and that those four correlations are no different than the correlations found from the two related images Correlation tests Nos. 2 and 3 this suggests that the decimation and sharing process does not produce a special noise correlation that causes the new SNR measurement technique to produce anomalously different SNR results. V. 2D SIMULATION RESULTS AND DISCUSSION All eight of the SNR computation techniques were then tested with a simulation of system drift mechanisms to observe their relative performance. The A/D decimation process was not simulated because the 1D results demonstrated that decimation does not cause any noise correlations. The simulated system instability had two components:

5 666 Steckner, Liu, and Ying: Computing the SNR of MR images 666 TABLE II. The average correlation coefficient and standard deviation over 1000 simulations for high and low SNR conditions for those SNR measurement methods that require two images. High SNR results Low SNR results Measurement No. Correlation coefficient Standard deviation Correlation coefficient Standard deviation a random normal 5% amplitude fluctuation on each phase encode line and 2 a random normal 2.5 phase fluctuation on each phase encode line. The applied errors were constant for each point in the entire phase encode line. These fluctuations produced ghosting artifacts both within and outside of the phantom. Therefore, all signal average estimates for all SNR measurement methods were corrupted. Noise estimates based on difference of images were also corrupted. The authors do not suggest that these are typical operational fluctuation levels or artifact producing mechanisms, but rather, produced a level of artifact that looked about right for the purposes of this article. The background noise measurement regions were in the read direction, well removed from the expected phase encode direction ghosts. However, the simulated amplitude variations caused the noise characteristics to no longer be perfectly white or spectrally flat. For the purpose of this article, it is assumed that this small deviation from white is statistically insignificant. Furthermore, all of the SNR methods would suffer equally. Consequently, the single image SNR measures were used as an internal benchmark. Twenty image pair simulations were done at each of five different SNR levels, both with and without the artifact generating mechanism. The results are shown in Table III and graphed in Fig. 1. The top half of Table III shows the results of all eight SNR computation methods with no system instability. All methods produce comparable SNR results. As before, the full image background statistic measures have the lowest standard deviation across the 20 iterations because those images had the largest background noise measurement regions greatest number of data pixels. The bottom half of Table III shows the results of all eight SNR computation methods with TABLE III. Twenty image pair simulation of all eight SNR methods with, and without a system instability artifact, at five different arbitrary SNR levels. The new SNR computation method is the only difference of images method that is insensitive to system instability. All background noise measurements methods work in these simulations because the measurements were made in the undisturbed read direction. The SNR results are plotted in Fig. 1. No artifact applied SNR No. 1 Standard deviation SNR No. 2 Standard deviation SNR No. 3 Standard deviation SNR No. 4 Standard deviation SNR No. 5 Standard deviation SNR No SNR No SNR No SNR No SNR No SNR No SNR No SNR No Artifact model applied SNR No SNR No SNR No SNR No SNR No SNR No SNR No SNR No

6 667 Steckner, Liu, and Ying: Computing the SNR of MR images 667 FIG. 1. This graph plots the 16 rows of SNR data from Table I and shows all eight different SNR estimation methods, using simulated data, with and without simulated artifacts at five different target SNR levels. Note that 13 rows of SNR measurements have collapsed on to the same line. Only the results from artifact method Nos. 3, 7, 8 diverge from the expected line. All methods work similarly well when no artifacts are present. instability artifact. Note that those SNR measurement methods using background regions to produce the noise statistics actually produced SNR results that are essentially unperturbed by the system instability. This is an expected result because the background regions were not appreciably contaminated. With the exception of the new SNR computation technique, the difference of images based method Nos. 3, 7 results are badly corrupted by system instability. Surprisingly, method Nos. 3 and 7 also report a lower SNR at the highest SNR level than at the next lower SNR level. This unusual oscillation has also been seen in the experimental data. The results of method No. 8 were also slightly degraded by the presence of artifacts. These three artifact corrupted results are clearly seen in Fig. 1, all other methods producing nearly identical results. While these results suggest that single image based SNR methods are superior, any image/ reconstruction processing operation that alters background noise characteristics invalidates any single image SNR based result. VI. EXPERIMENTAL SNR RESULTS AND DISCUSSION The comparison of the various SNR computation methods is demonstrated experimentally by varying slice thickness. It has been the authors experience that most SNR measurement methods work well at low SNR, provided the signal level does not drop to approximately the noise level where magnitude reconstruction alters the expected Gaussian distribution of the noise, 4,12 because artifacts that corrupt noise statistics are masked by the high noise level. Therefore, low SNR images can provide an internal, consistent reference point as a basis of comparison. It has also been the authors experience that as SNR increases, computed SNR values can oscillate such that further increases in expected SNR can lead to measured SNR values that unpredictably decrease or increase, as shown earlier in the 2D simulation results. The authors believe that as slice thickness increases, and/or bandwidth decreases, SNR increases and the sensitivity of SNR measurements to any system drift, or other artifact source increases. Such a sensitivity mechanism would explain why further increases in SNR conditions results in a decrease in measured SNR, but does not seem to explain why SNR results can oscillate. Figure 2 shows how the new method tracks with the expected linear response whereas the other methods start to fail. All data were collected on a Hitachi Medical Corp. Airis FIG. 2. SNR corrected vs slice thickness for eight different methods of SNR estimation. Difference of images methods suffer in the high SNR regime where subtle system instabilities produce anomalous image differences. By modifying the data acquisition and reconstruction process it is possible to produce an SNR estimate which is more robust, and higher, than other commonly used methods.

7 668 Steckner, Liu, and Ying: Computing the SNR of MR images 668 Elite Kashiwa, Chiba, Japan 0.3 T permanent magnet scanner using a four channel head coil and a single slice spin echo sequence FOV= mm, TE/TR=140/800 ms, 4 khz bandwidth, matrix size. Two acquisitions were collected at each slice thickness with no delay between matched acquisitions. In the case where SNR is very low, the high noise levels mask any subtle artifacts and all measurements are in close agreement. As slice thickness increases and SNR increases, the results from the various methods start to diverge. The most significant divergence occurs with the full image difference of images method No. 3, where the individual images are separated in time and have the highest intrinsic SNR, before correction factors are applied. Since the images have the highest SNR, they are the most sensitive to any system drift. Since the artifact signal does not conform to the expected Gaussian signal distribution the SNR measurements are anomalously skewed. The second most significant divergence is produced by the difference of images SNR measurement No. 7. Since there are fewer data points decimated data sets it does not have the SNR sensitivity to drift issues as No. 3 but the time between the two images is the same as in method No. 3. By eliminating the time difference between images No. 6 the recommended method the results improved dramatically and in this particular example produced the highest SNR estimate. Two difference of images noise images 25 mm slice thickness from method Nos. 7 and 6 are shown in Figs. 3 a and 3 b, respectively. These two images are shown at the same window and level. Otherwise, it would be difficult to appreciate the differences without introducing subjective changes in window and level settings. While the printed reproduction of Fig. 3 b appears to show a completely flat and uniform noise field, it is just possible to estimate the location of the phantom on close inspection of the image with a computer monitor. All of the other methods use background region statistics that are mostly artifact free, within the range of SNR tested here, and SNR changed linearly with slice thickness. The slopes SNR per mm of slice thickness are from highest to lowest slope : SNR method No. 6 slope=4.44 SNR method No. 1 slope=4.22 SNR method No. 2 slope=4.07 SNR method No. 4 slope=4.03 SNR method No. 5 slope=3.93 SNR method No. 8 slope=3.82 SNR method No. 3 slope=3.55 SNR method No. 7 slope=2.23 recommended method, full single image, background mean, full single image, background standard deviation, decimated single image, background mean, decimated single image, background standard deviation, alternate NEMA, difference of full image, and difference of unmatched single images. FIG. 3. a A difference of images method No. 7 noise image from a 25 mm slice image, after decimation and subtraction from two unrelated full images window/level identical a b. b A difference of images method No. 6 noise image from a 25 mm slice image, after decimation and subtraction of one full image window/level identical for a b. The correlation coefficients r 2 range from to for the seven best linear fits and for the lowest slope line. The slopes are also presented in Fig. 4 as a bar chart. Note that a comparison of method Nos. 1 and 2, then method Nos. 4 and 5 shows that using the background mean as a noise estimate produces a slightly higher SNR. Since the standard deviation measurement formula has a squared term,

8 669 Steckner, Liu, and Ying: Computing the SNR of MR images 669 FIG. 4. A comparison of the line slopes shown in Fig. 2. The asterisk above method No. 6 denotes the recommended method of this article. experience suggests that any slight increase in the tail of the noise distribution will be magnified, resulting in higher noise lower SNR, less robust measurement. VII. PARALLEL IMAGING APPLICATION RESULTS AND DISCUSSION Calculating the SNR of parallel reconstructed images is difficult. One of the characteristics of parallel image reconstruction is the spatially variant noise. 15 Any SNR method that measures or fits noise statistics within a subregion of the image cannot be used e.g., Refs. 5 and 16 with parallel reconstructed MR images because the measured noise statistics will vary as a function of noise measurement position. In the worst case a parallel reconstructed image can go from aliased in one pixel to no aliasing in the neighboring pixel, and hence, have dramatically different noise characteristics. One novel and complex method 17 shows how with a single image and sample noise data it is possible to compute a SNR image. The powerful but complex method of Kellman et al. was not implemented for this article. A simple statistical bootstrapping technique has also been proposed 18 which collects a single image and synthesizes additional collections by randomly reordering sample noise data. It is also possible to extend the basic method described in this article to parallel reconstructed images, as discussed below. Unlike most image quality metrics and standards which can be simply computed on typical and easily available scanner reconstructed images, parallel imaging SNR measurements require access to raw data, at a minimum, and some multiple parallel reconstructions or relatively complex computations. In the absence of direct access to raw data, it might be possible to recreate the raw data if the scanner provides a complex image output. Given the increasing complexity and black box nature of commercial MRI reconstruction algorithms, SNR measurements relevant to the particular scanner require the native scanner reconstruction processing. Ideally, SNR metrics also use acquired not synthesized data and images be fully available for simple offline SNR calculation and/or verification purposes. The new parallel image SNR computation method presented below comes closer to reaching such potential user requirements, but is not ideal. The new SNR computation method is also well suited for parallel imaging applications. The simplest experimental solution to this spatial variance problem is collecting a stack of many identically acquired images , preferably many more as others have noted. 11 With such a stack of images, SNR can be computed on a pixel-by-pixel basis: SNR one pixel location = mean of pixel stack / standard deviation of pixel stack. 1 Such a large stack of images compounds the previously noted MR system stability issues. There is a lot of scatter in these SNR estimates because there are relatively few data points when compared to the large number of pixels in a typical SNR measurement ROI. Acquiring more images in order to improve the SNR estimate increases the likelihood of system drift. Pooling of neighboring pixels to increase the number of measured points is not a viable solution because neighboring noise pixels may be spatially correlated e.g., GRAPPA Ref. 20 k-space interpolations, thus further corrupting the noise statistics. The new SNR computation method is sufficiently robust against system drift that it can be successfully applied to parallel imaging methods. Each oversized FOV image required by the new SNR computation method produces two normal parallel reconstructed images, providing two estimates of the signal level and one robust estimate of the noise level from the difference of the two images at the same pixel location. The standard deviation of the stack of noise pixels provides the noise level estimate. The new SNR computation method cannot, however, resolve the statistical problems caused by a limited number of data points. To demonstrate the suitability of the new SNR method for parallel imaging SNR determinations, a stack of N = 100 oversized images, with fully acquired k-space sampling, were collected on a Hitachi Medical Corp. Airis Elite Kashiwa, Chiba, Japan 0.3 T permanent magnet scanner using a four channel head coil and a single slice gradient echo sequence FOV= mm, TE/TR=75/250 ms, 4 khz bandwidth, matrix size. Given that the pur-

9 670 Steckner, Liu, and Ying: Computing the SNR of MR images 670 pose of this comparison is to demonstrate that the stack of images variant of the new SNR method produces the same results as the conventional SNR methods, parallel image reconstructions cannot be used. The eight previously described conventional SNR computation methods were compared with another set of seven stack of images SNR methods. The same ROI position was used for both the conventional SNR calculations and the stack of images SNR calculations. The conventional SNR results were aggregated by computing the mean and standard deviation of all the individual results. The stack of images results were aggregated by computing the average and standard deviation of the SNR values for all the pixels within the same ROI. If the images were perfectly well behaved we would expect that the average SNR, whether averaged first within an ROI than over a stack e.g., the aggregation of the conventional methods, or averaged per pixel within the same ROI than averaged across all pixels e.g., the aggregation of the stack of images methods, to produce the same result. However, the standard deviation of the aggregated SNR results will be larger because image nonuniformity will cause the SNR to vary between pixels whereas the image nonuniformity is eliminated with conventional SNR methods by averaging over the entire area of nonuniformity. The seven stack of images methods are: 1 stack of full images using Eq. 1, 2 stack of full images but the noise estimated from the N 1 successive difference of images, 3 stack of decimated images X using Eq. 1, 4 stack of decimated images Y using Eq. 1, 5 stack of decimated images X using Eq. 1, but the noise estimated from the N 1 successive difference of images, 6 stack of decimated images Y using Eq. 1, but the noise estimated from the N 1 successive difference of images, and 7 stack of decimated images averaged together X +Y /2 for the signal average and the stack of images X Y for the noise statistics the recommended method, where X denotes the images produced using only the even numbered raw data points, and Y are the images from the odd numbered raw data points. The results are shown in Table IV and Fig. 5. The top half of Table IV presents the conventional SNR methods. Again, the proposed method No. 6 produces the highest average SNR over the 100 images, but the lowest standard deviation occurred with method No. 1 because the full image background region statistics are produced with many pixels. The range of results is not particularly large, excluding method No. 3 which suffers significantly from system drift between images. These results provide a useful benchmark for the stack of images results. The bottom half of Table IV shows the results from the seven stack of images methods. With the exception of the proposed new SNR computation method No. 7 which TABLE IV. A comparison of conventional SNR methods and stack of images SNR methods. Conventional SNR methods Stack of images SNR methods Method No. SNR Standard deviation matches the conventional SNR results, all of the SNR results are significantly lower than the conventional SNR results because of system drift issues. These results suggest that the proposed new SNR method is sufficiently robust against system drift that parallel image SNR can be measured on a pixel by pixel basis with good accuracy. As noted earlier, the higher standard deviations of the stack of images results are due to image nonuniformity. If we assume that the spatial variation of SNR is slowly varying, consistent with the assumption that parallel imaging spatial sensitivity maps are also slowly varying, it might be acceptable to smooth the stack of images SNR map with a low-pass filter. Given the ability to compute a parallel image SNR map, it would seem logical to compute the SNR degradation factor caused by parallel image reconstruction e.g., the g factor for SENSE parallel imaging 15. However, since the g-factor map is the ratio of the fully acquired versus parallel reconstructed image SNR maps, the g-factor map noise will be even FIG. 5. A comparison of results produced by the eight conventional SNR methods, and seven stack of images SNR methods, in a parallel imaging-like application from Table IV. The recommended technique is the only stack of images method that produces results that match the expected conventional SNR method results. The asterisks above method No. 6 single image SNR methods and method No. 7 stack of images SNR methods denote the recommended methods of this article.

10 671 Steckner, Liu, and Ying: Computing the SNR of MR images 671 noisier. Unless a very large number of images are acquired and/or heavy filtering of the SNR or g-factor maps is used, the accuracy of the experimental g-factor maps may be poor. VIII. CONCLUSION A new robust method for computing image SNR has been developed. The data acquisition are as fast as single image based methods, assuming that SNR is typically measured with low duty cycle sequences, and is less sensitive than single image based methods to anomalous disturbances and artifacts because it is based on a dual image subtraction methodology. Dual image subtraction methods avoid issues arising from reconstruction or postprocessing algorithms that modify the background noise region of an image by using noise statistics generated within the phantom signal ROI, after a difference of signal images has been computed. The new method is more robust than typical dual image acquisition SNR methods because the two images are produced from one acquisition and therefore does not suffer from the interscan system drift associated with conventional dual image based SNR methods. It is also less sensitive to intrascan system drift issues because the phase encodes for each image are collected simultaneously. Simulations demonstrate that the SNR results produced by the new method match the SNR results of existing methods when system drift and/or background noise artifacts are not relevant. Simulations and experimental results also demonstrated the robustness of the new method by producing SNR measurements that were comparable with base line reference measurements, unlike any of the other SNR measurement methods used in this article. The method was also demonstrated to be appropriate for parallel image SNR determinations where SNR must be computed on a pixel-by-pixel basis. Given the large number of images that must be acquired to compute SNR on a pixelby-pixel basis, robustness against system drift becomes very important. This new nonparallel imaging method for computing SNR will become one of the allowed methods in the next edition of the NEMA MS-1 SNR standard. ACKNOWLEDGMENTS The author thanks colleagues on the NEMA MR technical committee, the IEC MR performance standards committee WG35/MT40, and Labros Petropoulos for their useful comments. a Author to whom correspondence should be addressed. Telephone: ; Fax: Electronic mail: msteckner@tmriusa.com 1 L. Kaufman, D. M. Kramer, L. E. Crooks, and D. A. Ortendahl, Measuring signal to noise ratios in MR imaging, Radiology 173, B. W. Murphy, P. L. Carson, J. H. Ellis, Y. T. Zhang, R. J. Hyde, and T. L. Chenevert, Signal-to-noise measures for magnetic resonance imagers, Magn. Reson. Imaging 11, A. J. Miller and P. M. Joseph, The use of power images to perform quantitative analysis on low SNR MR images, Magn. Reson. Imaging 11, G. McGibney and M. R. Smith, An unbiased signal to noise ratio measure for magnetic resonance images, Med. Phys. 20, J. Sijbers, P. Scheunders, N. Bonnet, D. Van Dyck, and E. Raman, Quantification and improvement of the signal-to-noise ratio in a magnetic resonance image acquisition procedure, Magn. Reson. Imaging 14 10, C. D. Constantinides, E. Atalar, and E. R. McVeigh, Signal to noise measurements in magnitude images from NMR phased arrays, Magn. Reson. Med. 38, J. Sijbers, A. J. den Dekker, E. Raman, and D. Van Dyck, An unbiased SNR measure for magnitude MR Images, ESMRMB M. J. Firbank, A. Coulthard, R. M. Harrison, and E. D. Williams, A comparison of two methods for measuring the signal to noise ratio on MR images, Phys. Med. Biol. 44, N261 N National Manufacturers Electrical Association, MS determination of signal-to-noise ratio SNR in diagnostic magnetic resonance images, NEMA, Washington, DC. 10 W. A. Edelstein, P. A. Bottomley, and L. M. Pfeifer, A signal to noise calibration procedure for NMR imaging systems, Med. Phys. 11, O. Dietrich, J. G. Raya, S. B. Reeder, M. F. Reiser, and S. O. Schoenberg, Measurement of signal-to-noise ratios in MR images: Influence of multichannel coils, parallel imaging, and reconstruction filters, J. Magn. Reson Imaging 26, R. M. Henkelman, Measurement of signal intensities in the presence of noise in MR images, Med. Phys. 12, M. C. Steckner, A new single acquisition, two-image difference method for determining MR image SNR, International Society of Magnetic Resonance in Medicine, 2006, p Miha Fuderer, Philips Medical Systems, Best, The Netherlands personal communication. 15 K. P. Pruessmann et al., SENSE: Sensitivity encoding for fast MRI, Magn. Reson. Med. 42 5, J. Sijbers, D. Poot, A. J. den Dekker, and W. Pintjens, Automatic estimation of the noise variance from the histogram of a magnetic resonance image, Phys. Med. Biol. 52, P. Kellman and E. R. McVeigh, Image reconstruction in SNR units: A general method for SNR measurement, Magn. Reson. Med. 54, M. J. Riffe, M. Blaimer, K. J. Barkauskas, J. L. Duerk, and M. A. Griswold, SNR estimation in fast dynamic imaging using bootstrapped statistics, International Society of Magnetic Resonance in Medicine, 2007, p D. K. Sodickson et al., Signal-to-noise ratio and signal-to-noise efficiency in SMASH imaging, Magn. Reson. Med. 41 5, M. A. Griswold et al., Generalized autocalibrating partially parallel acquisitions GRAPPA, Magn. Reson. Med. 47 6,