Magnetic Resonance in Medicine 64:15–22 (2010)

Image Restoration and Spatial Resolution in 7-TeslaMagnetic Resonance Imaging

Gabriele Lohmann,1* Stefan Bohn,2 Karsten Müller,1 Robert Trampel,1 and Robert Turner1

A good spatial resolution is essential for high precision seg-mentations of small structures in magnetic resonance images.However, any increase in the spatial resolution results in adecrease of the signal-to-noise ratio (SNR). In this article, thisproblem is addressed by a new image restoration technique thatis used to partly compensate for the loss in SNR. Specifically,a two-stage hybrid image restoration procedure is proposedwhere the first stage is a Wiener wavelet filter for an initialdenoising. The artifacts that will inevitably be produced bythis step are subsequently reduced using a recent variant ofanisotropic diffusion. The method is applied to magnetic res-onance imaging data acquired on a 7-T magnetic resonanceimaging scanner and compared with averaged multiple mea-surements of the same subject. It was found that the effectof image restoration procedure roughly corresponds to averag-ing across three repeated measurements. Magn Reson Med64:15–22, 2010. © 2010 Wiley-Liss, Inc.

Key words: image restoration; high-field magnetic resonanceimaging; spatial resolution

In principle, acquisition parameters such as the spatialresolution can be altered almost at will. However, anyincrease in the spatial resolution results in a decrease ofthe signal-to-noise ratio (SNR). This is simply caused bythe fact that the number of nuclear spins present in a voxelvolume decreases with its size. For instance, improvingthe spatial resolution from 1 × 1 × 1 mm3 voxel size to0.5 × 0.5 × 0.5 mm3 results in a signal loss by a factorof 8. Theoretically, this can be compensated by repeatedmeasurements and subsequent averaging. In practice, how-ever, this is often impossible due to unacceptably longacquisition times and subject motion.

The SNR increases with magnetic field strength. Stan-dard clinical scanners typically have field strengths of1.5 or 3 T, permitting high-quality structural imaging atabout 1 × 1 × 1 mm3 resolutions in vivo. Higher spatialresolutions can only be achieved using longer acquisitiontimes followed by sophisticated image processing tech-niques (1). Clark et al. (2) achieved a resolution of 0.4 ×0.4×3 mm3 at 1.5 T using a Spin-echo sequence. However,their acquisition time was 48 min.

Recently, high-field scanners of 7 T and more havebecome available opening the way to much better image

1Max Planck Institute for Human Cognitive and Brain Sciences, Stephanstr.1a, 04103 Leipzig, Germany.2Innovation Center Computer Assisted Surgery (ICCAS), University of Leipzig,Semmelweisstr. 14, 04103 Leipzig, Germany.*Correspondence to: Gabriele Lohmann, Max Planck Institute for Human Cog-nitive and Brain Sciences, Stephanstr. 1a, 04103 Leipzig, Germany. E-mail:lohmann@cbs.mpg.deReceived 11 August 2008; revised 5 March 2010; accepted 30 March 2010.DOI 10.1002/mrm.22488Published online in Wiley InterScience (www.interscience.wiley.com).

quality. However, when spatial resolution is taken to thelimit, the SNR is bound to degrade nonetheless.

High spatial resolutions are crucial for high-precisionmorphometric segmentations such as cortical thicknessestimation. In healthy subjects, the cortex has an averagethickness of about 3 mm. It tends to be thinnest in the cal-carine cortex (about 2 mm) and thickest in the precentralgyrus (about 4 mm) (3). Currently used in vivo magneticresonance imaging (MRI) methods have a spatial resolu-tion of about 1 × 1 × 1 mm3 so that the cortex appears asa sheet with a thickness of three to four voxels. Segmen-tation errors consisting of not more than one voxel can,therefore, produce a relative error of 30–40 percent at thatlocation. A good estimate of cortical thickness requires spa-tial resolutions far better than 1×1×1 mm3. Obtaining suchresolutions at reasonable scan times are a challenge even inhigh-field imaging. Even without segmentation errors, mor-phometric measurements based on low-resolution imagescan lead to significant discretization errors.

Therefore, our goal was to explore the limits of spa-tial resolution on a 7-T MRI scanner by acquiring veryhigh resolution measurements with a predictably poorSNR. However, the approach is also suitable for dataacquired at lower field strength. We propose a novel imagerestoration/denoising technique based on a combination ofWiener wavelet filtering and speckle reducing anisotropicdiffusion to compensate for the loss in SNR and checkthe results of the denoising against averaged multiplemeasurements of the same subject.

METHODS

Generally, data in MRI are represented as the magnitude of

a real and an imaginary component√

x2real + x2

imag, where

the noise is gaussian distributed in the real and imagi-nary component independently. The transformation intothe magnitude image causes the noise to become Riciandistributed (4,5) with

p(x | a, σ) = xσ2 e(x2+a2)/(2σ2) I0(a · x/σ2) [1]

where x denotes the measured signal value including noise,a the idealized noise-free value, I0 the zero-order Besselfunction of the first kind, and σ the standard deviation ofthe noise. For large SNR (a/σ > 2) the Rician distributionapproaches a gaussian distribution, while for small SNRs,we obtain a distribution that is far from a gaussian. In theextreme case of a = 0 (absence of a signal), we obtain theRayleigh distribution.

In the approach presented in the following, filtering isapplied after combination of the individual array channelimages so that noise becomes noncentral chi distributed(6,7).

© 2010 Wiley-Liss, Inc. 15

16 Lohmann et al.

Many algorithms for reducing noise in magnetic res-onance images have been proposed in recent years (8)and wavelet filtering has become one of the most widelyused methods (9). In wavelet decompositions, frequenciesof signal components are expressed by wavelet and scal-ing coefficients. The signal is generally concentrated in asmall number of high amplitude wavelet coefficients whilethe noise is spread out across the entire wavelet domainwith relatively small coefficients. In wavelet filtering, smallamplitude coefficients are suppressed using various tech-niques. The simplest one is to set a hard threshold belowwhich all coefficients are set to zero. This may, however,cause severe artifacts, so that hard thresholding is gener-ally avoided. Here, we propose to use a Wiener waveletfilter in which each wavelet coefficient is dampened by aweighting factor 0 ≤ αjk ≤ 1 such that

d̂jk = αjk djk [2]

where djk is the wavelet coefficient at the j-th scale and kis an index for a voxel address (9). The weighting term αjk

depends on the noise variance:

αjk = max

d2

jk − γσ̂2jk

d2jk

, 0

[3]

where σ̂2jk is an estimate of the noise variance so that high

noise levels lead to stronger smoothing. The user-definedparameter γ is used to regulate filter strength. Differentmethods for estimating the noise variance exist (9). In ourimplementation, we used the method proposed by Donohoet al. (10) where the noise variance σ2 is estimated solelyfrom the wavelet coefficients of the finest scale using themedian absolute deviation (MAD)

σ̂ = 1β

median(|d11 − m1|, . . . , |d1k − m1|, . . .) [4]

where m1 denotes the median of the coefficients d1k . Theparameter β depends on the distribution of the noise. ForGaussian noise, Donoho et al. (10) obtained β = 0.67.As noise in MRI data is known to be Rician distributed(see above), we use a different estimate. By empiricallycontaminating noise-free images with Rician distributednoise using different standard deviations σ, we obtainedan estimate of β = 0.455.

Thresholding wavelets generally produces so-called“ringing” or pseudo-Gibbs artifacts reminiscent of alias-ing in frequency filters. In our implementation, we chosethe Haar wavelet basis functions as they are known to pro-duce less ringing (11). However, even the Haar transformproduces artifacts to some degree (Fig. 1).

Several approaches for dealing with this problem havebeen proposed in the literature. As shift-dependence is oneof the causal factors, a common method is to make thefilter shift invariant. Shift invariance can be obtained byapplying the filter to all possible shifts, filtering the shiftedimages separately, and averaging after shifting back (9,12).Because of the extremely high computational demand ofthis procedure, we use a simplified version called “Shift4”

FIG. 1. A synthetically generated test image artificially corrupted byRayleigh and gaussian distributed noise using σ = 70 (a); after firststage of noise cleaning using wavelet filtering (b); result of secondstage noise cleaning using 40 iterations of classical anisotropic diffu-sion ADF (c); result of second stage noise cleaning using 20 iterationsof SRAD (d). The parameters ρ and κ used in ADF and SRAD weredefault choices as explained in the text. Note that SRAD does muchbetter in reducing pseudo-Gibbs artifacts than ADF even though onlyhalf the number of iteration were used.

(13) in which only four shifts by one pixel in vertical, hor-izontal, and diagonal directions are applied. This methodis computationally feasible and helps to reduce artifacts tosome extent but not completely.

To further reduce pseudo-Gibbs artifacts, Durant et al.(14) proposed to replace the thresholded coefficients byvalues minimizing the total variation. We found that thismethod works well in low-noise situations and generallyhelps to reduce artifacts but does not completely solvethe problem. An interesting connection between the shift-invariant Haar wavelet shrinkage and nonlinear diffusionwas observed by Mrazek et al. (15). They exploit the struc-ture of two-dimensional diffusion filters to use a coupled,synchronized shrinkage of the individual wavelet coeffi-cient channels. The motivation for their work was primar-ily theoretical in nature and not aimed at reducing artifacts.

We pick up the idea of combining anisotropic diffusionand wavelet filtering. Unlike Mrazek et al., however, wepropose a two-stage procedure in which anisotropic diffu-sion and wavelet filtering are uncoupled. Here, anisotropicdiffusion serves the role of reducing artifacts after waveletfiltering.

We first applied standard anisotropic diffusion filtering(ADF) (16,17) to the wavelet-filtered image to check if it issuitable for removing pseudo-Gibbs artifacts. In ADF, filter-ing is modelled as a diffusion process in which a flow existsbetween adjacent regions. As the process evolves over time,the homogeneity with homogeneous regions increaseswhile the boundaries between them remain intact. Let

Image Restoration in 7-Tesla Magnetic Resonance Imaging 17

I (x, t) denote the image intensity at voxel location x atiteration t. Then classical ADF can be described by

∂I (x, t)∂t

= div(c(x, t)∇I (x, t)) [5]

where c(x, t) is the diffusion coefficient that describes theamount of interchange between adjacent cells. We wanta strong flow within homogeneous regions and little orno flow across region boundaries. Therefore, in classicalADF, the function c is defined as a monotonically decreas-ing function of the image gradient. We found that ADFis feasible for reducing pseudo-Gibbs artifacts providedthey are small in amplitude so that they do not producestrong gradients which would stop the diffusion process.For high amplitude artifacts however, anisotropic diffusioncan even be detrimental (Fig. 1).

Therefore, we propose to use a speckle-reducing variantof anisotropic diffusion (18), which is aimed at reducinghigh-amplitude but spatially small noise that is characteris-tic both of speckle and of pseudo-Gibbs artifacts. The mainidea is to replace the simple gradient ∇I (x, t) of classicalADF with an edge detector for speckled images using thefollowing term (18)

q =

√√√√√√12

( |∇|I

)2 − 116

(∇2

I

)2

[1 + 1

4

(∇2

I

)]2 [6]

At speckle peaks, the Laplacian ∇2I (x, t) approaches zerowhile the gradient term ∇I (x, t) dominates so that largeredges remain while speckle is reduced. In the following, wewill use “SRAD” as an abbreviation for speckle-reducinganisotropic diffusion.

Both classical ADF and SRAD depend on a diffusionfunction. For classical ADF, we used

c(x, t) = 11 + (|∇|/κ)1+a

as suggested in Ref. 16. The parameter κ should be chosento reflect the noise level in the image. In our experiments,we estimated the noise level and adjusted κ accordingly.The parameter α was empirically chosen to be 0.3.

For SRAD, we used

c(q) = 11 + [q2 − q2

0]/[q20(1 + q2

0)]as in Ref. 18. They suggest to use

q0(t) = s0 exp(−ρt)

with s0 the speckle coefficient, and ρ a free parameter whichaccording to Ref. 18 may be set to a standard value ρ = 1/6.In our experiments, we set the speckle coefficient to s0 = 1which corresponds to fully developed speckle.

RESULTS

In the following, we present several test images. To betterdemonstrate the capability of the method, we chose imageswith extremely poor SNRs. In real applications, noise levelsshould not be taken to such extremes.

Synthetic Test Images

We first generated a 64 × 64 test image (Fig. 1) simulating asignal void (a = 0) and a signal presence (a = 100) whichwas artificially corrupted using a Rayleigh distribution forthe signal void and a Gaussian distribution for the signalpresence with σ = 70.

This image illustrates the effect of noise cleaning usingwavelet filtering alone (b), and the effects of the secondstage processing using both ADF (c) and SRAD (d). Asshown in image b, wavelet filtering reduces noise but alsoproduces pseudo-Gibbs artifacts. The purpose of the sec-ond stage is to reduce these artifacts. Images c and d showthat SRAD clearly outperforms ADF even though we usedonly half the number of iterations for SRAD than for ADF.The wavelet filtering was done using Haar basis functionswith γ = 1 (γ as in Eq. [3]). In SRAD, ρ was set to 0.1666which is the default value that was used in all subsequentexperiments.

To quantitatively assess the difference between the orig-inal and the restored image, we computed the mean squareerror (MSE) and also the so-called Q-index introduced byWang et al. (19). The Q-index was designed to remedysome of the limitations of the MSE in estimating imagequality. It incorporates loss of correlation, luminance andcontrast distortion as relevant features. Its value is in therange of (−1, 1) with a value of ‘1’ corresponding to aperfect match between the two images. The comparisonbetween the uncorrupted test image and the noise cleanedimage yielded Q = 0.51, MSE = 90.99 for SRAD andQ = 0.49, MSE = 91.60 for ADF. In addition, we alsochecked whether the estimated noise (i.e., difference ofthe cleaned images vs. the noisy image) correlated withthe “true” noise which was artificially added and is knownexactly. This correlation was better for SRAD than for ADF(97.7 versus 94.7%).

The second test image of size 100 × 100 (Fig. 2a,b) alsosimulates a signal void (a = 0) and a signal presence (a =100) again using a Rayleigh distribution for the signal voidand a gaussian distribution for the signal presence withσ = 40 (Fig. 2a) and σ = 30 (Fig. 2b). The image containsseveral different spatial frequencies. The grid lines at thetop of the image are spaced one pixel apart. Noise cleaningusing wavelet filtering followed by 10 iterations of SRADwas applied.

Despite the excellent noise reduction with the filtering,contrast loss is observed for the highest resolution bars,suggesting some level of resolution loss.

The Q-index between the original noisefree image andthe corrupted image was 0.33, and MSE equalled 67.53,while the Q-index between the noisefree image and thecleaned image was 0.38, with MSE = 62.95. In other words,the noise cleaning produced an improvement of both Q-index and MSE. The correlation between the estimatednoise and the “true” noise was 89.3%.

MRI Data 1

Several single-slice magnetic resonance images wereacquired on a 7-T MRI scanner (Magnetom 7T, SiemensHealthcare Sector, Erlangen, Germany) using an eightchannel–phased array coil (Rapid, Rimpar, Germany).

18 Lohmann et al.

FIG. 2. a: A synthetically generated testimage artificially corrupted by noise andthe result of noise cleaning using waveletfiltering followed by 10 iterations of SRAD(right). The vertical bars at the top ofthe synthetic image are spaced one pixelapart. Gray value profiles along two rowsof this image are shown below. Their posi-tions are indicated by red lines. Notethat the noise cleaning process does notimpair spatial accuracy. However, imagecontrast is somewhat reduced. b: Thesame image as in Fig. 2a with less noisetogether with a line profile. The position ofthe line is marked by a red bar.

Image Restoration in 7-Tesla Magnetic Resonance Imaging 19

FIG. 3. MRI data at 7 T using aFLASH sequence. A single mea-surement with a spatial resolu-tion of 0.215 × 0.215 × 2 mm3

(left); the result after noise clean-ing (middle); and an average ofthree measurements (right).

The test subject was a healthy male volunteer (age 32)who gave informed consent. A FLASH sequence was usedwith repetition time (TR) = 200 msec; echo time (TE) =4.82 msec; Flipangle = 10◦; field of view (FoV) = 200.0 ×171.88 mm2; Slice thickness = 2.0 mm. The image matrixcontained 1024 × 800 voxels with an in-plane spatialresolution of 0.215 × 0.215 mm2.

The image data and the noise cleaning results are shownin Fig. 3 and Table 1. The same slice was acquired fivetimes using the same acquisition parameters. Acquisitiontime for a single image was 160 sec, and 800 sec for all fivemeasurements.

We applied Wiener wavelet filtering with subsequentSRAD to a single slice using γ = 0.75 for the Wienerwavelet filtering and 20 iterations of SRAD. We comparedthe denoising results with averages obtained from multiplemeasurements (Fig. 3). To assess the difference, we com-puted the MSE and also the Q-index described above (seeTable 1). The 3-average showed the closest match to thecleaned image. In addition, we found that the similaritybetween the cleaned image and the 3-average is greater thanthe similarity between the 3-average and all other averages.In particular, the comparison between the 3-average andthe 2-average yielded MSE = 84.50, Q = 0.76, while thecomparison between the cleaned image and the 3-averageyielded MSE = 67.59, Q = 0.86.

MRI Data 2

To push the limits of voxel size even further, we acquired amagnetic resonance image with five axial slices on the 7-TMRI scanner using a FLASH sequence with TR = 211 msec;TE = 25 msec; flipangle = 20◦; FoV = 210 × 164 mm;

slice thickness = 0.6 mm. The image matrix contained640 × 500 voxels with an in-plane spatial resolution of0.328×0.328 mm2. The subject was a female volunteer aged23 who gave informed consent. The scan time for the sin-gle measurement was 106 sec. Five repeated measurements

Table 1Differences Between the Noise Cleaned Image and theMeasurement Data of Fig. 3

MSE 1 Q-index 1 MSE 2 Q-index 2

Single measurement 83.71 0.77 107.73 0.672-average 69.69 0.82 84.50 0.763-average 62.44 0.86 0 15-average 67.59 0.84 75.38 0.82

Noise cleaning was applied only to the single measurement dataand compared to the unprocessed single measurement and to theaveraged data. The results of this comparison are shown in columns“MSE 1” and “Q-index 1.” The Q-index has values in (−1, 1) with avalue of “1” indicating a perfect match. “MSE” denotes the meansquared error. The columns “MSE 2” and “Q-index 2” show theresults of a comparison between the three-fold average and theother averages and the single measurement. Note that the similaritybetween the cleaned image and the 3-average (MSE = 62.44, Q =0.86) is greater than the similarity between the 3-average and allother averages (see main text). In addition, both the mean squareerror (MSE 1) and the Q-index show that the noise cleaning corre-sponds most closely to a three-fold average. The average over fivemeasurements was still closer to the cleaned image than the origi-nal data, but not quite as close. Both indices were computed aftermasking out non-brain voxels. Note that both MSE 1 and Q-index 1decline slightly at the five-fold average. This may be due to motionartifacts caused by a long acquisition time. An alternative explanationmay be that the 5-average is significantly less noisy than the cleanedimage leading to a larger discrepancy between the two images.

20 Lohmann et al.

FIG. 4. MRI data at 7 T using a FLASH sequence with 0.328 × 0.328 × 0.6 mm3 resolution. Average across two acquisitions (left); the resultafter noise cleaning applied to the 2-average (middle); and the 5-average (right). The result of noise cleaning applied to a single acquisitionis shown in Fig. 5.

were taken. It should be noted that whereas the previousdata set was obtained using a slice thickness of 2 mm theslice thickness here was only 0.6 mm.

The result of noise cleaning is shown in Fig. 4. As thenoise level was extremely high, we applied noise clean-ing first to an average of two acquisitions (Fig. 4) and laterto a single acquisition (Fig. 5). Figure 6 shows the differ-ence between the single (1-average) image of Fig. 5 and theresult of the noise cleaning. Note that MSE and Q-index aregiven in Table 2. Note that the MSE decreases with eachadditional averaging.

For comparison, we acquired an additional dataset of thesame subject using a lower in-plane resolution of 0.65 ×0.65 mm2. The other imaging parameters were identical.Figure 5 shows a comparison of this acquisition with theresult of the noise cleaning. To facilitate visual comparison,the images are shown scaled to the same size. However, aquantitative comparison using MSE or Q-index is not pos-sible as the two images do not have the same image matrixand hence have a different number of voxels.

DISCUSSION

We have introduced a new method for denoising MRIimages using a two-stage procedure consisting of Wienerwavelet filtering followed by speckle reducing anisotropicdiffusion. The purpose of the second stage is to reduceartifacts of the pseudo-Gibbs type incurred by waveletfiltering.

The method we have proposed here has commonalitieswith the compressed sensing (CS) technique (20,21). BothCS and our method aim at reducing acquisition time usingassumptions about sparsity implicit in MR images. In thecase of CS, acquisition time is reduced by undersamplingk-space and sparsity assumptions enter the reconstructionprocess. In our method, acquisition time is reduced byrequiring less averaging and sparsity assumptions relateto wavelet coefficients. The relative merits of these meth-ods probably depend on image content. But more work isneeded is investigate this further.

Using two different imaging protocols, we obtained a spa-tial resolution of 0.2 × 0.2 × 2 mm3 and 0.328 × 0.328 ×0.6 mm3 with reasonably good image quality after noisecleaning.

The noise cleaning appeared to have an effect compara-ble to a three-fold repeated measurement with subsequentaveraging (Fig. 3). This is supported by the fact that thesimilarity between the cleaned image and the 3-average wasgreater than the similarity between the 3-average and the 2-average. At very high noise levels (Fig. 4), the effect seemedto be even stronger. However, it should be noted that someloss of contrast and spatial resolution is possible.

As noted earlier, our method is applied after combinationof the individual array images in which case the data are nolonger Rician distributed but have a noncentral chi distri-bution. Nonetheless, our experimental results appear quitegood in spite of this problem suggesting that our methodseems to be relatively robust against violations of the dis-tribution assumptions. The head coil we have used in our

Image Restoration in 7-Tesla Magnetic Resonance Imaging 21

FIG. 5. Image a shows a single acquisition of the data set of Fig. 4 with an in-plane resolution of 0.328 × 0.328 mm2, slice thickness0.6 mm (a); and the result of noise cleaning applied to this data (b). For comparison, image c shows the same subject imaged with identicalparameters as before except for a lower in-plane resolution of 0.65×0.65 mm2. Image c is an average of two repeated measurements scaledto the same size as images a, b to make visual comparison easier. Some slight differences between images a, b and image c may be dueto subject motion. The scans took place during the same session with about 15 min between scans. The scan time for the high-resolutionsingle acquisition was the same as for the averaged low-resolution data (106 sec). Note that the poorer resolution in image c leads to somesusceptibility artifacts which are reduced at higher resolution due to the smaller within-voxel dephasing. The bottom row shows a zoomedportion focusing on the ventricles. Note the step edges in image c along the border of the ventricles that are due to the poorer spatialresolution.

experiments was an eight-channel coil. It remains to beinvestigated whether this problem becomes more pressingfor coils with more channels.

Imaging at spatial resolutions of this scale using nomore than 160 sec of acquisition time represents a major

step forward. A comparable spatial resolution would nor-mally require much longer acquisition times. For example,Christoforidis et al. (22) describe gradient-echo imaging at8 T using 0.2×0.2×2.0 mm3 but needed 13 min of acquisi-tion time. Other researchers have used very high in-plane

FIG. 6. A zoomed-up portion of Fig. 5 (1-average) (left), the result of noise cleaning as in Fig. 5 (right), and the difference between thetwo images (middle). Strong edges at the outer contour of the brain and skull are vaguely visible in the difference image suggesting thatthose edges have a slightly lower contrast in the cleaned image. Interior parts of the brain show almost no structure in the difference imagesuggesting that the loss of information incurred by the cleaning process is small.

22 Lohmann et al.

Table 2Differences Between the Noise Cleaned Image and theMeasurement Data of Fig. 4

MSE Q-index

2-average 1044.93 0.8153-average 897.39 0.8484-average 832.97 0.8645-average 814.62 0.872

The 5-average shows the closest match to the noise cleaned image.Noise cleaning was applied to the 2-average.

resolution but a larger slice thickness (7 T, 0.4 × 0.4 ×2.0 mm3 (23), or isotropic 3D imaging with 0.4 × 0.4 ×0.4 mm3 (24). Very high resolution imaging at 7 T andsegmentation issues are disussed in Ref. 25.

Using advanced imaging coils, higher resolutions can beachieved with good SNR (26). In these cases, even betterresults might in the future be achievable if noise cleaningis applied.

We envisage our method as a preprocessing step forhigh-precision morphometric segmentations such as cor-tical thickness estimation as one of the main areas ofapplication.

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