k-space Interpretation of the Rose Model: Noise Limitation on the Detectable Resolution in MRI

Transcription

1 k-space Interpretation of the Rose Model: Noise Limitation on the Detectable Resolution in MRI Richard Watts and Yi Wang* Magnetic Resonance in Medicine 48: (2002) Noise limitation on the detected spatial resolution, described by the Rose Model, is well known in X-ray imaging and routinely used in designing X-ray imaging protocols. The purpose of this article is to revisit the Rose Model in the context of MRI where image data are acquired in the spatial frequency domain. A k-space signal-to-noise ratio (ksnr) is introduced to measure the relative signal and noise powers in a circular annulus in k-space. It is found that the ksnr diminishes rapidly with k- space radius. The Rose criterion that the voxel SNR 4 is translated to ksnr cutoff values was tested using theoretical derivation and experimental histogram analysis. Experiments demonstrate that data acquisition beyond this cutoff k-space radius adds little or no information to the image. In order to reduce the noise limit on spatial resolution, the signal strength must be improved through means such as increasing the coil sensitivity, contrast enhancement, and signal averaging. This finding implies that the optimal k-space volume to be sampled or the optimal scan time in MRI should be matched to the relative SNR level. Magn Reson Med 48: , Wiley-Liss, Inc. Key words: k-space sampling; noise limited resolution; signalto-noise ratio (SNR); Rose model One of the challenges to high-resolution MRI is noise: the detectable spatial resolution can be much inferior to the prescribed resolution if the image is very noisy. This noise limitation on spatial resolution is well known in X-ray imaging, where the approximate Rose Model is widely used to determine the X-ray dose (proportional to the square of the SNR) for a desired image resolution (1 4). In general, the MR signal in k-space is strongly peaked at the center and falls off rapidly with k-space radius (5) and the noise is white spread evenly over the entire frequency range (6,7). Clearly, the acquisition of data points dominated by noise is not an efficient use of scan time. The Rose Model may also provide guidance in selecting imaging parameters such as scan time for achieving a desired imaging resolution. The Rose Model describes the relationship between noise and detectable spatial resolution: the differentiation of a single voxel from the noise background requires a minimum SNR per voxel of 4 (1 4,8). In order to determine the implications of this statement on scan time or k-space size in MRI, the Rose Model must be reassessed in terms of k-space. The voxel SNR is reexpressed in terms of k-space sampling size. We further introduce a k-space SNR (ksnr) to measure the relative strength of signal and noise in k-space at a given radius and express the Rose criterion using histogram analysis. A cutoff k-space radius is found that determines the optimal scan time and the achievable spatial resolution for a given SNR. Experimental measurements are presented in k-space and image-space. MATERIALS AND METHODS Existence of a Cutoff k-space Sampling Radius In general, the SNR of a voxel (xsnr) in MRI can be expressed as (7): xsnr V N 1/2 F K [1] where V is the voxel volume, N the number of data points sampled, F the signal factor determined by the pulse sequence parameters including sampling bandwidth, and K is the coil sensitivity. For the purpose of imaging an organ of a desired volume V with a given signal strength (F K), what is the critical maximum sampling radius in k-space, beyond which the xsnr will be less than the Rose minimum? For a simple and qualitative answer to this question, assume k-space is sampled isotropically with radius k m. Then N a k m D V, with the geometric factor depending on the imaging dimensionality D: a for D 2 (circle); a 4 /3 for D 3 (sphere). Substituting this and V V/N in Eq. [1] leads to: xsnr a 1/2 F K V 1/2 k m D/2. [2] Therefore, as the k space sampling radius increases the detected resolution increases but the voxel SNR decreases; once the Rose minimum xsnr 4 is reached, the detected resolution ceases to increase. For a given imaging situation, the optimal k-space sampling size k mo or the detected resolution is: k mo (a 1/2 /4 F K V 1/2 ) 2/D. [3] Measurement of Image Space Signal and Noise vs. Resolution (k max ) Department of Radiology, Weill Medical College of Cornell University, New York, New York. Grant sponsors: Whitaker Foundation; American Heart Association; NIH; Grant numbers: RO1 HL60879; HL *Correspondence to: Yi Wang, Ph.D., MR Research Center, UPMC Presbyterian B864, 200 Lothrop Street, Pittsburgh, PA wangy3@msx.upmc.edu Received 25 May 2001; revised 26 March 2002; accepted 29 March DOI /mrm Published online in Wiley InterScience ( Wiley-Liss, Inc. 550 All measurements were made using a GE Echospeed Signa MR imager (General Electric Medical Systems, Milwaukee, WI). The original Rose criterion requires direct measurement of the voxel SNR (statistical properties) and the voxel conspicuity (human observation) (4). The former criterion can be considered as a statistical test for a hypothesis that the probability distribution of the signal intensity in one voxel is significantly different from that of the background (4,8,9). The probability distribution can be obtained from

2 Rose Model in MRI 551 FIG. 1. Increase in image noise with limiting k-space radius, k max for data acquired using the body coil and the head coil. Note that the true signal is close to constant with k max, giving SNR ratios of 4 at the positions indicated by the dashed lines. many repeated experiments. Instead of performing the same experiment repeatedly, a uniform phantom can be imaged once and the distribution can be estimated from a statistical distribution of voxel intensities over a large area of the uniform phantom. Images of a uniform spherical phantom were acquired at high ( ) resolution using a 2D fast gradient echo sequence (TR/TE 11ms/5ms, flip angle 20 ). A circular filter of radius k max was then applied to the k-space data and the images reconstructed to a matrix size of 2k max / k, such that the pixel size in each image directly represents the true resolution of the acquisition. In each reconstructed image, regions of interest (ROIs) were defined within the center of the phantom and in the background. From each ROI, a histogram of the pixel intensities was calculated. The overlap of the phantom and background histograms represents the probability of a given pixel being wrongly identified as belonging to the phantom or the background. Measurement of k-space Signal-to-Noise Ratio (ksnr) To measure the ksnr two nominally identical acquisitions were obtained from a uniform spherical phantom. The root-mean-square (RMS) noise ( ) and signal (s(k)) were determined by complex subtraction of the two measurements and averaging within an annulus of median radius k r in k-space to derive the SNR at k r : ksnr s(k r )/. [4] RESULTS Signal and Noise Characteristics in k-space The analysis of signal and noise in k-space was applied to 2D MR data from a uniform spherical phantom. The imagespace noise level vs. spatial resolution for the body and head coils are shown in Fig. 1. The signal intensity is determined primarily by the central k-space points. The resolution cutoff for each coil corresponding to the Rose criterion that SNR 4 is indicated by dashed lines. Using this measure, the detectable resolution with the head coil (k max 300/FOV) is approximately three times higher than that of the body coil (k max 100/FOV) for the same pulse sequence parameters and imaging object. The variation of RMS signal and noise with k-space radius for the same data are shown in Figs. 2a and 3a, respectively. The data shows that the RMS noise is constant with k r, demonstrating its white nature. The RMS signal is strongly peaked at the center of k-space and is found to decrease approximately as k r 1.5. The variation of signal with k r represents the spatial frequency distribution of the object being imaged. The distribution for the phantom is comparable to that of the torso of a human subject and the measurements of Fuderer (5). The k-space SNR is very poor for all but the center of k-space. For both the head coil and the body coil many of the acquired points have ksnr 0.1. The question remains as to how much useful information the acquisition of such points adds to the image. This is addressed quantitatively in terms of pixel SNR. Rose Model in Image-Space and k-space Noise-Limited Resolution The SNR of MR images are often determined by comparison of the average (magnitude) pixel values in an ROI within the object and within the background. We refer to this simple measure as the voxel SNR, xsnr. Such a measure is widely used in MRI because it is easy to perform and the noise in MR is spread evenly over the entire image (object and background). The xsnr does not provide a direct measure of the confidence level in distinguishing the object from the background. The histogram overlap is a much more direct measure of the confidence level. As expected, low-resolution (large voxel size) images display excellent SNR for acquisitions made using both the body (Fig. 2b, xsnr 11.0) and head coil (Fig. 3b, xsnr 27.5). On the intensity histograms (Figs. 2f, 3f), this is shown by the clear separation of the signal and noise intensity distributions. At a resolution of (Fig. 2c xsnr 5.4; 3c xsnr 14.7), the distributions are again separated (Figs. 2g, 3g), although for the body coil the separation is small. As the resolution is further increased to (Fig. 2d xsnr 2.8; 3d xsnr 7.4), there is a significant overlap (9.2%) in the distributions using the body coil (Fig. 2h), but still no overlap with the head coil (Fig. 3h). Finally, at resolution (Fig. 2e xsnr 1.6; Fig. 3e xsnr 3.8), there is a large overlap (28.8%) for data taken using the body coil (Fig. 2i), and a significant overlap (2.2%) with the head coil (Fig. 3i). At this high resolution using the body coil, it becomes difficult to distinguish the outline of the object (Fig. 2e). At a threshold of 95% confidence for identifying a given pixel as either phantom or background, the resolution is limited to less than for the body coil and more than for the head coil. Data from both head and body coils (Figs. 2, 3) indicate that the cutoff ksnr Data acquisition beyond this

3 552 Watts and Wang FIG. 2. The effect of increasing resolution on image quality for data taken using the body coil. a: Comparison of the RMS signal and noise amplitudes vs. k-space radius. The solid arrow indicates voxel SNR 4 orksnr b e: Simulations of stopping the acquisition at the k-space radii indicated in a. f i: Histograms of the signal (dark bars) and background (light bars) intensities measured from b e, indicating that for cases (d) ( ) and (e) ( ) the resolution is noise-limited. The background distribution approximated the Rician distribution at high resolution (h,i), but deviated from that at lower resolution (f,g) due to truncations. cutoff ksnr contributes primarily noise and can degrade image quality (Figs. 2d, 3e). Evidence for an Optimal k-space Sampling Size in a Given Imaging Situation To demonstrate the existence of an optimal sampling size, an imaging experiment using a standard gradient echo sequence (TE/TR/flip 20/600/30, body receiving coil) was used to image the low-contrast pattern in the American College of Radiology quality control phantom. Images of three different sampling sizes ( , , ) are illustrated in Fig. 4, where the low-contrast pattern is best depicted in sampling (Fig. 4b). DISCUSSION Noise limitation on spatial resolution is well known in imaging. When data are acquired in real image space, the detectable resolution is limited by the voxel SNR as characterized by the Rose Model. For MRI where data are acquired in the spatial frequency domain, the Rose Model is reformulated here in k-space to illustrate the relationship between the detectable spatial resolution and the choice of scanning parameters for a given imaging task. Our histogram (probability) analysis demonstrated that the 95% confidence level (5% histogram overlap between background and subject) for detecting a voxel corresponds to a voxel SNR 4 (Rose criterion) and the corresponding SNR in k-space ksnr With increasing k-space radius, the histograms broaden and overlap and ksnr decreases rapidly. The cutoff radius defines the detectable spatial resolution limited by noise. Data acquired beyond this point contribute little information to the final images. Accordingly, there is an optimal k-space sampling size for a given imaging situation. It should be noted that detectability is subjective and the Rose Model cannot dictate a rigorous threshold on the required SNR. Furthermore, the ksnr cutoff corresponding to the Rose criterion varies depending on the spatial frequency content of the imaged object and should be determined accordingly for a given imaging task. Previous investigations on the information content of MR images

4 Rose Model in MRI 553 FIG. 3. The effect of increasing resolution on image quality for data taken using the head coil. a: Comparison of the RMS signal and noise amplitudes vs. k-space radius. The solid arrow indicates voxel SNR 4 orksnr b e: simulations of stopping the acquisition at the k-space radii indicated in a. f i: Histograms of the signal and background intensities measured from b e. The noise-limited resolution for the head coil is around (e) ( ). indicate that the MR signal in k-space falls off rapidly in an approximate power function (5). Although the Rose ksnr cutoff value of 0.05 may not be a general result, for any MR imaging there is a cutoff ksnr value (Eq. [3]), beyond which data acquisition contribute little information, but contaminate the final image with noise. A practical implication from our analysis is that improving the detectable resolution requires an increase in the cutoff radius in k-space, which is determined by the signal strength (for a given noise level). Factors affecting the signal strength include pulse sequence design, imaging parameters, coil sensitivity, spin density, and relaxation FIG. 4. Images of the low-contrast pattern in the American College of Radiology quality control phantom with sampling size of (a), (b), and (c). For this imaging situation, the sampling size (b) was the best to depict the low-contrast pattern.

5 554 Watts and Wang properties. Using a birdcage head coil instead of a body coil can more than double the detectable spatial resolution (Fig. 1, also cf. Figs. 2, 3). For contrast-enhanced MRA (CE-MRA) using fast gradient echo sequences, the signal strength in Eq. [1] can be approximated as F K K(cR 1 TR/ BW) 1/2 e TE/T2 *, where K is the coil sensitivity, c is the contrast agent concentration, and R1 is the contrast agent relaxivity (10,11). Therefore, in addition to using a more sensitive coil, increasing contrast concentration or using agents with higher relaxivity will increase the noise limit on spatial resolution. These considerations can be used to optimize imaging protocols of CE-MRA. The above discussion applies to all areas of MRI that require high spatial resolution, such as MR microscopy and clinical ultrahigh resolution imaging. In clinical brain imaging, voxel sizes as small as 0.5 mm are used with a matrix. Such images may appear grainy because of the noise limitation. Unlike CE-MRA, signal averaging may be used to reduce the noise level, while still sampling the high spatial frequencies. CONCLUSIONS In summary, noise limits the detectable spatial resolution in MRI. Data acquisition beyond a cutoff radius in k-space (corresponding to the Rose criterion of image SNR 4) contributes little information to, and can degrade, the images. This provides a guide to selecting the optimal scan time and other imaging parameters. ACKNOWLEDGMENTS The authors thank Dr. Martin R Prince for stimulating discussion and Mr. Alexander MacKenzie for assistance in data acquisition. REFERENCES 1. Rose A. Vision: human and electronic. (Optical physics and engineering.) New York: Plenum; Shosa D, Kaufman L. Methods for evaluation of diagnostic imaging instrumentation. Phys Med Biol 1981;26: Wagner RF. Toward a unified view of radiological imaging systems. P II. Noisy images. Med Phys 1977;4: Burgess AE. The Rose model, revisited. J Optic Soc Am A, Optics, Image Sci Vision 1999;16: Fuderer M. The information content of MR images. IEEE Trans Med Imag 1988;7: Henkelman RM. Measurement of signal intensities in the presence of noise in MR images. Med Phys 1985;12: Edelstein WA, Glover GH, Hardy CJ, Edington RW. The intrinsic signalto-noise ratio in NMR imaging. Magn Reson Med 1986;3: Haacke EM, Brown RW, Thompson MR, Venkatesan R. Magnetic resonance imaging physical principles and sequence design. New York: John Wiley & Sons; Gudbjartsson H, Patz S. The Rician distribution of noisy MRI data. Magn Reson Med 1995;34: Heid O, Remonda L. Outer limits of contrast-enhanced 3D MRA. In: Proc ISMRM 5th Annual Meeting, Vancouver, BC, p Pelc NJ, Alley MT, Shifrin RY, Herfkins RJ. Outer limits of contrastenhanced 3D MRA, revisited. In: Proc ISMRM 6th Annual Meeting, Sydney, p 254.