To derive and implement a method for correcting spatial distortion caused by in vivo inhomogeneous static magnetic fields in echo-planar imaging (EPI).
To derive and implement a method for correcting spatial distortion caused by in vivo inhomogeneous static magnetic fields in echo-planar imaging (EPI).
The reversed gradient method, which was initially devised to correct distortion in images generated by spin-warp MRI, was adapted to correct distortion in EP images. This method provides point-by-point correction of distortion throughout the image. EP images, acquired with a 3 T MRI system, of a phantom and a volunteer's head were used to test the correction method.
Good correction was observed in all cases. Spatial distortion in the uncorrected images ranged up to 4 pixels (12 mm) and was corrected successfully.
The correction was improved by the application of a nonlinear interpolation scheme. The correction requires that two EP images be acquired at each slice position. This increases the acquisition time, but an improved signal-to-noise ratio (SNR) is seen in the corrected image. The local SNR gain decreases with increasing distortion. In many EPI acquisition schemes, multiple images are averaged at each slice position to increase the SNR; in such cases the reversed gradient correction method can be applied with no increase in acquisition duration. J. Magn. Reson. Imaging 2004;19:499–507. © 2004 Wiley-Liss, Inc.
BECAUSE OF ITS INHERENTLY low bandwidth per point in the phase-encoding direction, echo-planar imaging (EPI) exhibits more spatial distortion than spin warp magnetic resonance imaging (MRI) techniques (1). This can be problematic if measurements of distance are required from the EP images, or if registration between EP images and spin warp MR or x-ray computed tomography (CT) images is attempted (2).
MR image production requires that the relationship between magnetic field strength and position be known throughout the volume being imaged. If an image is produced with the use of an incorrect relationship, it may display artifacts, including spatial distortion. The difference between the assumed and actual magnetic field, ΔB, results from magnetic field inhomogeneity, which may have a number of sources (3). Both the main static magnetic field and the magnetic field gradients are unlikely to be entirely uniform throughout the volume containing the sample being imaged. The magnetic field may also be perturbed due to the sample itself. Differences in magnetic susceptibility of various structures within the sample, as well as between the sample and the surrounding air, modify the local magnetic field (4). These all contribute to ΔB.
Many methods for correcting the spatial distortion present in spin warp images have been presented. Two steps are involved in the majority of correction methods: 1) the difference between the assumed and actual magnetic field, ΔB, is measured, and 2) the effect of ΔB is accounted for in the process of image reconstruction so that an undistorted, or spatially corrected, image is produced. The methods differ in the way in which the inhomogeneous field map is measured. ΔB may be derived from the measurement of the effect it has on the evolution of phase throughout the NMR signal acquisition. Such methods are termed phase map correction methods (5–8, 12, 14–16). The field map also may be derived directly from the spatial distortion it produces. This method, which was proposed by Chang and Fitzpatrick (9), is termed the reversed gradient correction method in the present study (9).
The methods mentioned above all derive pixel-by-pixel magnetic field maps that are dependent on the actual object being imaged. A number of other methods have been described that rely on information gathered from MR images of phantoms of known physical dimensions, and the use of mathematical transformations (17, 18). While adequate distortion correction can be achieved with these methods under some circumstances, since they incorporate smoothly varying functions, one would not expect the correction to be as accurate as that obtained by methods that correct each point independently.
Phase map correction methods have been applied to EPI as well as to spin warp MRI (7, 8, 12, 14–16). The reversed gradient correction method may also be used to correct spatial distortion in EP images (10). This approach constitutes the subject of this study. In particular, we describe the implementation and use of the reversed gradient method in correcting spatial distortion in EP images generated with a 3 T MRI system.
The effects of an unknown static magnetic field inhomogeneity on a two-dimensional MR image acquired in its presence have been described mathematically (4, 9). The addition of a spatially varying static magnetic field inhomogeneity, ΔB(x, y, z), results in the spins experiencing a modified local magnetic field that changes their frequency of precession by γΔB(x, y, z), where γ is the gyromagnetic ratio. This modulates the NMR signal with the term exp(iγΔB(x, y, z)t), where t is time relative to the application of the RF pulse. Ignoring T2 and T2* decay, it has been shown that in a general case the addition of ΔB(x, y, z) causes spatial distortion of (ΔB(x, y, z)/Gr) along the rth axis, where r can be x, y, or z, and Gr is the strength of the effective magnetic field gradient along that axis. The coordinates (x, y, z) describe the location of the source of the NMR signal, which in the presence of ΔB(x, y, z) is assigned to position (x1, y1, z1) in the MR image. Hence, the transform between actual and distorted image space is r1 = r + (ΔB(x, y, z)/Gr). In this work, the x, y, and z axes are taken to correspond to the frequency-encoding, phase-encoding, and slice-selective axes, respectively.
In the case of a two-dimensional spin-warp MRI acquisition, each frequency-encoding line is acquired at the same time, TE, after the RF excitation pulse. Hence, in k-space, there is no time evolution along each line parallel to the ky axis, and there is no variation in the modulation of the signal due to field inhomogeneity. Consequently, there is no spatial distortion in this direction and y = y1.
In a single-slice MRI acquisition in which the voxel dimensions are of similar magnitude along all axes, Gz, the effective gradient in the slice direction, is usually several times larger than either Gx or Gy. Hence, the spatial distortion along the slice-selective axis will be significantly smaller than that occurring in-plane. Its effect is therefore generally ignored in the reversed gradient correction method for single slices (9), as it is in phase map-based correction methods (7).
Spatial distortion in single-slice, spin-warp MRI acquisitions may therefore be considered to occur only along the frequency-encoding (x-) direction. The reversed gradient correction method relies on the fact that if a second acquisition is performed under exactly the same conditions (except that the polarity of the frequency-encoding gradient, Gx, is reversed), the spatial shifting of signal in the second image will occur in the opposite direction along the x-axis. Whereas in the first image, i1, the spatially shifted signal from x appears at
in the second image, i2, the spatially distorted signal from x appears at
Corresponding pairs can be found by performing line integrals of pixel intensity in both distorted MR images, i1 and i2, along each frequency-encoding line. Corresponding values of x1 and x2 are identified as those that equalize the two integrals (9). In practice, instead of calculating x-values from pairs of x1 and x2 coordinates, it is simpler to work backwards by stepping through pixel locations in the corrected image matrix (x) for which the computer then searches for corresponding values of x1 and x2 that have equal line integrals and satisfy Eq. . Edge detection of the objects in the distorted MR images is usually required prior to calculation of the line integrals, in order to avoid integrating over excess amounts of background noise.
However, correcting the spatial distortion alone does not yield a correct image, i. The Jacobian J(x1/x) of the coordinate transform from distorted to undistorted space must be used to compensate for the effect of the transform on the pixel intensity. For the case of one-dimensional distortion considered here, and described in more detail by Chang and Fitzpatrick (9), this results in
One can generate a corrected image using either Eq.  or , by calculating the Jacobian term from differentiation of either Eq.  or , respectively. However, it is more convenient to differentiate Eq.  with respect to x and substitute from Eqs.  and  to eliminate the differentials, giving
This equation allows one to calculate the correct intensity at each pixel in the undistorted image. The corrected image is generated from the combination of both distorted images, and as such would be expected to exhibit a signal-to-noise ratio (SNR) that is increased compared to that of a single, undistorted image. The SNR in a region of the corrected image actually depends on the local gradient in ΔB, with higher gradient values causing reduced SNR. It can be shown (see Appendix) that the ratio of the SNR in a region of the corrected image to that in the same region of a single image unaffected by distortion is given by
where f = 1/Gxd(ΔB(x))/dx. This function is plotted in Fig. 1, indicating that there is a gain in SNR in the corrected image for |d(ΔB(x))/dx| ≤ 0.4Gx..
In order for Eqs.  and  to be meaningful, the Jacobian must be greater than zero. Calculating the Jacobian term directly by differentiating Eqs.  and , and applying the condition that the Jacobian must be greater than zero gives
when Gx > 0 and
Equations  and  set limits on the maximum ΔB that may be corrected using this method, for a particular gradient strength. If the inequalities are violated, “piling over” of signal intensity from one pixel to the next occurs (4) in the distorted image and a complete correction cannot be performed (9). It can also be seen from Eqs.  and  that this reversed gradient scheme will only correct for a field inhomogeneity that remains invariant during reversal of the polarity of Gx. Any inhomogeneity in the magnetic field gradient that reverses with Gx will not be identified. With a structured phantom of known dimensions, it is possible to correct for the effects of gradient field nonlinearities as well as static field inhomogeneities (25), but it is impractical to apply this routinely in vivo. Correction can also be performed when an overall uniform frequency shift has been applied—for example, when the field of view (FOV) has been offset from the central axis of the magnet. As long as the edges of the object can be found in images acquired with both polarities of gradient, the line integration can be performed and a corrected image constructed. However, in this case the pixel shifts will reflect both ΔB and the frequency offset.
For a single-slice EPI acquisition, spatial distortion would be expected along both the x and y in-plane axes (7, 8). Each phase-encoded step occurs with increasing time after the single RF excitation pulse, and hence the data are modulated in both the kx and ky directions by a term exp(iγΔB(x, y, z)tr). However, in a typical EPI acquisition of N phase-encoded lines, the time taken to span k-space in the phase-encoding direction will be at least N times the time taken to span k-space in the frequency-encoding direction. This results in the bandwidth per point in the phase-encoding direction being approximately (1/N)th of that in the frequency-encoding direction. Therefore, for a given field offset, the spatial distortion along the phase-encoding direction will be N times that along the frequency-encoding axis, where N is likely to be at least 32 and more typically 64–128. Spatial distortion along the phase-encoding axis will therefore be the dominant effect to the extent that distortion along the frequency-encoding direction may be ignored in comparison.
The dominant distortion along the phase-encoding axis in an EP image may be considered to be analogous to distortion along the frequency-encoding axis in a spin warp acquisition (10). Hence, one can correct spatial distortion in EP images by acquiring two images that are identical except that the second is acquired in the presence of a phase-encoding gradient with opposite polarity to the first. Only the phase-encoding gradients associated with the actual EPI acquisition are reversed; other gradients applied along the phase-encoding axis, such as diffusion or spoiling gradients, are not reversed. Clearly, if oblique imaging planes are selected, the actual phase-encoding gradient may be a combination of the physical gradients in the MR scanner, all of which must be reversed. The reversed gradient correction may then be applied along the phase-encoding direction to produce an undistorted EP image. In most current implementations of EPI, the phase-encoding gradient takes the form of a series of short-duration pulses, or “blips” (each of duration Δ, and strength G), which are applied at the zero crossings of the orthogonal, switched magnetic field gradient (period, 2τ). In this situation, the effective gradient, Ge, which should be employed in Eqs. – to calculate the spatial distortion, is scaled by the ratio of the time for which the gradient is applied to the time over which evolution under the action of magnetic field inhomogeneity occurs, giving
We acquired the EP images using a 3 T whole-body MRI system fitted with a head RF coil and head gradient set (19). Single-shot spin-echo modulus blipped EPI (MBEST) acquisitions (13) were performed and modulus images were constructed. Frequency-selective fat saturation was applied to eliminate the shifted fat signal in the images, which otherwise may have interfered with the correction method. Great care was taken to eliminate Nyquist ghosts, which also would have interfered with the correction. The polarity of the phase-encoding gradient was reversed by inverting the input to the gradient amplifier.
Images were acquired of a structured water phantom and a volunteer's head, in a 64 × 64 matrix. The voxel dimensions were 3 × 3 × 5 mm3. Each frequency-encoding line was acquired in 472 μs. Hence the total acquisition time was 30.2 msec, resulting in a bandwidth per point of 33 Hz in the phase-encoding direction. The effective TE was 44 msec. The effective frequency- and phase-encoding (broadening) gradient strengths were 16.6 mTm–1 and 0.26 mTm–1, respectively.
For each object, two experiments were performed. In the first, the shims were adjusted to their optimum settings. For the second experiment, the x2-y2 shim was deliberately offset so as to introduce visible spatial distortion in the EP image. This allowed us to assess the performance of the correction method in the presence of large magnetic field inhomogeneities. Offsetting the x2-y2 shim was chosen as a means of producing a nonlinear inhomogeneous magnetic field quickly and reproducibly, which could also be recognized in the distortion maps as a check on the success of the correction.
The reversed gradient correction method, as described by Chang and Fitzpatrick (9), was implemented as a computer program. Prior to correction, the edges of objects in the images were identified to ensure that line integration was performed across the object only, and not over the background noise. Poor edge detection could result in suboptimal correction, and a number of modifications to simple intensity thresholding were made to improve edge-finding. To avoid including signal artifacts, which are often present at the edge of images, the center of mass of intensity in the image was found. If the center of mass was calculated to be in a low signal region (e.g., a signal void in the object), a radial search was performed to find the nearest region of high signal intensity. A thresholded mask was grown from this seed point, which was assumed to be located within the object. This mask should isolate the object from background noise and other artifacts, including any residue Nyquist ghosts. Within this mask, we defined the edges more precisely by calculating the fractional pixel position at which the signal intensity fell below a predefined threshold value, using linear interpolation between adjacent pixels at the edge of the mask. To check for potentially erroneous edge detection, we tested whether the coordinate of each edge in turn was more than a predefined distance from the edge coordinates found in neighboring lines. An algorithm to detect edges defined as the maximum gradient of signal intensity was also implemented. This algorithm worked well for fairly homogeneous phantoms, but it did not perform so well on in vivo images, and hence was not used for the experiments presented herein. For in vivo acquisitions, care must be taken to orientate the imaging plane so that the edges of the object are visible along the axis for which the correction is being performed. For EPI acquisitions, this is the phase-encoding direction; therefore, if edges are to be visualized, no phase wrap can be present. For spin-warp acquisitions, all edges of the object must be visible in the frequency-encoding direction. This limits the application of the correction to certain imaging planes (e.g., a correction cannot be performed on sagittal spin-warp brain images with the frequency-encoding axis orientated in the head–feet direction).
The total integrals of pixel intensity between edges along pairs of phase-encoded lines from the two images were normalized before the line integrals were performed. A corrected image was produced, and an image map of spatial distortion, x-x1, was also generated.
The numerical application of Eqs.  and  requires an estimation of the image intensity at fractional pixel positions. To improve the calculated intensity at fractional pixel positions, we implemented a number of nonlinear interpolation schemes, as well as a simple linear scheme using the neighboring pixels. Sinc interpolation using all points along a line would be expected to produce the best results (20), but it also significantly increases the calculation time of the correction. Therefore, we implemented computationally faster schemes: the cubic spline technique and two truncated sinc methods (Gaussian- and Welchian-weighted) (24). These were performed with five points around the required coordinate. The corrected images produced with each scheme were subtracted from the corrected image, calculated by full sinc interpolation, and visually assessed for agreement.
We also performed a conventional, spin-warp, T1-weighted MRI acquisition on the volunteer using an inversion recovery three-dimensional fast low-angle shot (FLASH) magnetization prepared rapid gradient echo (MPRAGE) sequence on a 1.5 T SP MRI scanner (Siemens Medical, Erlangen, Germany), which was better optimized for high-resolution three-dimensional spin warp imaging than the 3 T system. Other acquisition parameters were as follows: matrix = 256 × 256 × 128, voxel size = 0.98 × 0.98 × 1.41 mm3, TE = 4 msec, TR = 10 msec, TIset = 300 msec, and bandwidth = 195 Hz per point in the frequency-encoding direction. With a much higher bandwidth per point (in the frequency-encoding direction) compared to that in the phase-encoding direction of the EPI acquisition used, the resulting data set exhibited significantly less spatial distortion. This set was acquired to provide a comparison with the EP images.
Figure 2 shows the transaxial images of the structured phantom, which were acquired with reasonable shim settings. Figure 3 shows an image of the same phantom acquired with an offset x2-y2 shim so as to introduce increased spatial distortion. Figures 4 and 5 show data from similar experiments performed on a volunteer's head. Figure 4 shows a fat-suppressed transaxial image at the position of the lateral ventricles acquired with well-adjusted shim settings, and Fig. 5 shows an image acquired at the same position with the x2-y2 shim offset. In Figures 2–5, “a” is the image acquired with normal imaging parameters, “b” is the image acquired with reversed polarity of the horizontal phase-encoding gradient, “c” is the corrected image produced by the method described in this work, and “d” is a distortion map. The distortion map depicts pixel displacements: mid-gray values represent no distortion, and black and white represent positive or negative shifts. For all figures, the grayscale bar spans shifts ranging from –4 to +4 pixels, equivalent to –12 to +12 mm or –132 to +132 Hz. In addition, the distortion maps were scaled to units of Tesla and numerically differentiated to yield values of d(ΔB(x))/dx.. Hence, values of f, as used in Eq. , could be calculated. For all four experiments, modulus values of f were in the range of 0–0.38, which is well within the limits for correction set by Eqs.  and . A typical SNR in the uncorrected phantom image was 20, as compared to a typical SNR in the corrected phantom image of 26.
It should be noted that the head coil used to acquire these data was known to exhibit a nonuniform RF field resulting in a reduction of image intensity to the left and right of the images. This is particularly noticeable in Figs. 2 and 3, which show images of the phantom. However, with the reversed gradient method, as long as the reduction in signal does not interfere with the edge detection for the line integrals, it will not affect the correction of static magnetic field inhomogeneities, as the RF nonuniformity will remain constant on gradient reversal.
Figure 6 shows a conventional T1-weighted spin warp slice through the volunteer's brain at a similar position to the EP image. It provides a comparison with the EP images, and in particular demonstrates the asymmetrical shape of the volunteer's head.
The results generated by applying different interpolation schemes to the images of the volunteer's head are shown in Fig. 7. Compared to the full sinc interpolation scheme, the truncated Gaussian- and Welchian-weighted sinc methods worked the best, and the linear interpolation between the nearest neighbors clearly had the worst performance. As such, Welchian-weighted sinc interpolation was used for the corrected images presented in Figs. 2–5. For the data presented here, correction using Welchian weighted sinc interpolation took just under one second per image using a Sun Ultra 1 UNIX workstation (Sun Microsystems Inc., Palo Alto, CA). It took just over four times longer to correct the same data by full sinc interpolation. Streaks in the subtraction images are a result of different edge positions being identified with different interpolation methods.
A modified version of the reversed gradient distortion correction method was used to correct spatial distortion in the phase-encoding direction in EP images. Spin-echo MBEST EP images of a structured phantom and a volunteer's head were acquired with a 3 T MRI system to test the correction method. In both cases, good correction was observed. The relevant structures within the phantom were restored to be square and circular in shape. The corrected image of the volunteer's brain was in good visual agreement with a conventional spin warp image acquired at a similar slice position.
No artifacts are seen in the corrected phantom images. Offsetting the shim to increase the spatial distortion in the head images, as shown in Fig. 5, reduces the effectiveness of the fat saturation, and a fat artifact is present in the distorted images of Fig. 5a and b. The nonoptimal fat saturation resulted in a dark vertical strip in both the distorted and corrected images because the shifted fat signal overlaid the white matter. While this did not cause global artifacts in the corrected image (Fig. 5c), it highlights the importance of removing shifted fat signal from images prior to correction with this technique. All corrected images show an improved SNR, as expected from the level of distortion and Eq. . The structures in the corrected images remain sharp and well defined, and visual assessment shows there was no blurring of the structures as a result of the correction.
This study shows that a nonlinear interpolation method, such as a Welchian-weighted truncated sinc scheme, to calculate the image intensity at fractional pixel positions, can improve the corrected image (compared to simple nearest-neighbor linear interpolation) while still allowing rapid image correction. The use of appropriate interpolation methods other than nearest-neighbor linear interpolation should be investigated not just for the reversed gradient correction method for EPI, as in the current study, but also for other distortion correction methods that require the calculation of image intensities at fractional pixel positions.
Other groups have investigated the use of phase map correction methods to correct spatial distortion in EPI (6–8, 12). Corrected images produced by phase map methods are less susceptible to artifacts arising from low SNR or ghosting in the raw images (7). This is because the reversed gradient correction method is more sensitive to a low SNR as errors accumulate along each line integral, affecting the correction of all points along that particular line. A low SNR affects each point in the phase map correction, but the errors it introduces are not accumulated from one point to the next. However, a recently introduced method using dynamic time warping (11) shows promise for reducing the sensitivity of the reversed gradient correction method to low SNR. In the images acquired for the current work, the SNR was sufficient to prevent any visible artifacts in the corrected images. It may be beneficial to use the dynamic time warping method when one applies the reversed gradient correction method to EP images with a lower SNR, such as those acquired at a lower main magnetic field strength or higher spatial resolution.
On other counts, the reversed gradient correction method performed as well as the phase map correction. Both methods require two (or more) images to be acquired at each slice position, and therefore are both susceptible to the effects of patient movement between these two acquisitions. Both methods require that the magnetic field inhomogeneity be static through time, and hence both perform suboptimally under the influence of time-varying eddy currents. The phase map method has been adapted to correct for the effects of time-varying magnetic field inhomogeneities (8) through an EPI acquisition, but this requires a separate EPI reference scan for each phase-encoding line in the corrected image. Both methods suffer from effects of shifted fat signal, especially in regions of high spatial distortion where incomplete fat saturation may occur. In the reversed gradient correction method, signal from fat interferes with the line integrals, while in phase map methods, the superposition of fat and water signals results in a shift in the phase compared to that from water alone.
A feature of the reversed gradient correction method is that the corrected image can be constructed as the combination of the two raw images, resulting in a corrected image with an improved SNR compared to that of an individual image. Furthermore, for many imaging protocols, more than one image per slice would be acquired routinely and subsequently averaged to improve the SNR. For these protocols, the reversed gradient correction method may be implemented with no penalty in acquisition time, with the additional benefit of obtaining averaged distortion-free images.
Interestingly, the EPI reversed-gradient method also corrects image distortion resulting from concomitant fields, which effects nonaxial, EP images acquired at low magnetic field (21, 22). Such distortion results from the change in the effective magnetic field experienced by the spins due to the presence of the x- and y-components of the spatially varying magnetic field produced by the gradient coil. These field components must be present in order to satisfy Maxwell's equations, and are often known as Maxwell terms or concomitant fields. Consider, for example, the case in which a coronal EP image is acquired with a switched frequency-encoding x-gradient of strength, g. The effective field under these conditions is (22)
where B0 is the main magnetic field. Since in most circumstances, gz ≪B0 for all z-values within the FOV, the modification to the effective field is usually very small (for the same reason, Eq.  also omits other terms of the form gn/B with N >2). However, at low magnetic field, B0, the third term in Eq.  may have some effect. This is a particular problem in EPI because this term is unaffected by the polarity reversal of the switched gradient, and thus it produces a continuous phase evolution over the echo train, leading to a distortion of spatial magnitude g2z2/GB0 in the phase-encoding direction (where G is the effective phase-encoding gradient strength). However, reversing the polarity of the phase-encoding gradient changes the sense of this distortion, in exactly the same way as happens for distortion produced by field inhomogeneity. Therefore, the method of Chang and Fitzpatrick (9) could be used to produce an undistorted image. Currently employed phase-map methods would not correct this effect, because varying the time of application of the EPI gradients relative to that of the RF pulses does not affect the phase accumulated due to the third term in Eq. . The method described here will also correct the uniform image shifts that affect axial EP images acquired with large slice offsets (23). However, it will not correct the increased Nyquist ghosting that the concomitant fields cause in such images and those acquired in nonaxial planes.
The edge detection method described here was set to find only the extreme edges for each line. If more than one discrete object is represented in the image, or the object contains signal voids, these will not be correctly identified. Edge detection algorithms could be employed to find multiple pairs of edges along each line. However, when our method was extended to detect multiple edges, artifacts arose in the corrected image due to an occasional mismatch between pairs of edges in the two images acquired with opposing gradient polarity, as well as a different number of edge pairs occasionally being found between images. As such, detection of multiple edge pairs along a line was not performed in this work, which may have resulted in errors when line integration over internal voids in the object was performed. However, to overcome this limitation in the future, more complicated edge detection algorithms could be employed to find multiple pairs of edges along each line.
The SNR in a region of an MR image corrected using the reversed gradient method depends on the local magnetic field inhomogeneity, with higher local gradient values causing a reduction in the local SNR of the corrected image. In the following, an expression is derived for the relationship of the SNR in the corrected image (SNRcorr) that depends on the SNR in the uncorrected base image (SNRbase), the local inhomogeneous magnetic field gradient, d(ΔB(x))/dx, and the magnetic field gradient, Gx, where x is the direction with lowest bandwidth per point in which the distortion is being corrected.
The expression for a corrected image, i, is given by Eq. , where i1 and i2 are the distorted images acquired with opposing polarity of Gx. If the noise in i1 and i2 is Δi1, and Δi2, respectively, then the noise in the corrected image, Δi, is
Differentiating Eq.  with respect to i1 and i2 gives
Assuming that the expectation value of the noise in both i1 and i2 has the same mean-squared value, n2, then 〈δi〉 = 〈δi〉 = n2. It follows from Eq.  that the expectation value of the squared noise in the final image is
Let f = 1/Gx (dΔB/dx), which is proportional to the gradient of the local magnetic field inhomogeneity and inversely proportional to the imaging magnetic field gradient. Substituting Eq.  into Eq.  gives
which reduces to
Hence the root mean square noise in the corrected image is given by
Equation  shows that the ratio between the SNR in the corrected image to the SNR in the uncorrected base image depends on the local inhomogeneous magnetic field gradient, d(ΔB(x))/dx,, and the imaging magnetic field gradient, Gx. If d(ΔB(x))/dx = 0, then f = 0 and Eq.  gives SNRcorr = SNRbase√2, as expected. Regions of higher d(ΔB(x))/dx result in a larger value of f, which yields a smaller increase in SNRcorr relative to SNRbase. For large values of d(ΔB(x))/dx that give f > 0.4, the local SNR in the corrected image will be less than the SNR in the uncorrected image.