An autofocus algorithm for the automatic correction of motion artifacts

An Autofocus Algorithm for the Automatic Correction of Motion Artifacts in M R Images David Atkinson 1, Derek L. G. Hill 1, Peter N. R. Stoyle 2, Paul E. Summers 1, and Stephen F. Keevil 1. 1 UMDS, Radiological Sciences, Guy's Hospital, London SE1 9RT, UK. e-mail: d.atkinson~umds.ac.uk 2 Defence Research Agency, St. Andrew's Road, Great Malvern. WPd4 3PS, UK. A b s t r a c t . We present the use of an entropy focus criterion to enable automatic focussing of motion corrupted magnetic resonance images. Our technique can determine unknown patient motion or use knowledge of motion from other measures as a starting estimate. The motion estimate is used to compensate the acquired data and is iteratively refined using the image entropy. Our entropy criterion focuses the who!e~image principally by favoring the removal of motion induced ghosts and blurring from otherwise dark regions of the image. Using only the image data, and no special hardware or pulse sequences, we demonstrate correction for arbitrary rigid-body translational motion in the imaging plane and for a single rotation. Extension to 3D and more general motion should be possible.

1

Introduction

Magnetic Resonance Imaging (MRI) is capable of sub-millimeter resolution, shows excellent contrast between soft tissues and does not subject the patient to ionising radiation. However, ghosting and blurring caused by patient motion can either reduce the diagnostic usefulness of the scan, necessitate a repeat scan, or in many paediatric studies require the use of general anaesthetics (with an increase in both cost and risk to the patient). Existing techniques to reduce the effects of motion include navigator echoes, fast or modified acquisitions and post-processing. Navigator correction of translational displacements in the frequency encode direction is well established [1,2]. Correction of more complicated motion has been less successful and requires either further radio frequency (RF) pulses [3], phase retrieval algorithms [1], patient preparation with special markers [4] or the gradient control necessary for orbital navigator echoes [5]. Alternatively, the sensitivity to motion can be reduced by decreasing the total scan time although this often compromises the image resolution and quality. Furthermore, rapid acquisitions using single shot echo planar imaging (EPI) require specialized hardware. The effects of unknown motion can also be reduced by acquiring data using a spiral k-space trajectory. This requires compliant hardware and complicated gradient programming (not always available). Radial-scan with projection

342

reconstruction is an alternative to 2D Fourier reconstruction and can reduce motion-induced ghosting [6]. However, ghosting still exists and motion between read-outs leads to shifted projections and blurring. Motion can be deduced using post-processing techniques that examine the kspace data or its 1D Fourier transform (hybrid space). Zoroofi et al. [7] combined phase retrieval and edge-detection to correct for motion in the phase-encode and read-out directions. Alternatively, interleaved spiral acquisitions may provide phase information about motion [8] at certain times during a scan. Both techniques suffer from the need to retrieve phase information from the data. As a further alternative, segments of k-space acquired while the patient is stationary between discrete movements can be identified from discontinuities in k-space [9]. This technique presumably will not work with continuous motion. Post-processing of the image data has been demonstrated. Two acquisitions can be combined using a weighted average to reduce motion artifacts [10]. Alternatively, if two interleaved acquisitions are taken, ghosts due to quasi-periodic motion can be removed [11] by assuming that the motion is approximately periodic with respect to the acquisition order. The method of generalized projections uses a predefined region of image support prior to iterative image improvement [12]. This requires phase retrieval and a prior knowledge of the region of support, such as a motion-free image. Recently, a minimum energy method has been proposed [13] that iteratively determines and corrects for rigid body motion. This technique minimises the MR image energy outside of the object and also requires the image boundary to be found prior to motion correction. Although motion degrades images, the mechanism is principallythrough the corruption of the phase of the received signal.No information is lost during rigid body translations and information is lost only at certain times during rotations. Following from the autofocus of radar images [14],the objective of this work was to develop an automatic post-processing technique that autofocuses an image using only the original real and imaginary data from a conventional scanner. No patient preparation, extra pulse sequences, edge detection, phase retrieval, or region of image support are needed. The algorithm developed can detect step, cyclicor generalized aperiodic motion to sub-pixel resolution.The methods presented here provide a new tool that can be used in isolation,or in combination with many of the motion correction techniques mentioned previously.

2 2.1

Method O u t l i n e of M e t h o d

Patient motion is modeled as a series of displacements or rotation angles as a function of time - the motion trajectory. Time is discretized into nodes with the timing of each node corresponding to the acquisition of a k-space line. In this way, continuous movement is approximated in a piecewise constant or piecewise linear fashion. The starting estimate of the trajectory can be zero motion, or the output of some other motion measure such as navigator echoes. This estimate is modified with a trial motion, the acquired data is corrected for this motion

343

and the resultant image quality is assessed using a focus criterion. Using an optimisation algorithm, the patient motion parameters that yield the best quality image are determined iteratively. 2.2

C o m p e n s a t i o n for Translational Motion

The effects of translational motion in the frequency encoding (FE) and phase encoding (PE) directions are compensated by applying phase shifts to the acquired k-space data [1,2]. The relation between acquired data Saq(U, v) and the corrected data Sc(U, v) for the U TM k-space point in the v TM readout-line is, sc(

,v) = FOVpE

(v

,1,

where dFE and dps are the displacements, FOVFE and FOVpE are the fields of view and NFE and NpE are the number of points in the FE and PE directions respectively. The indexing ofv is from 0 to NpE--1 and v = NpE/2 corresponds to zero PE gradient, similarly for u. Only motion occurring between the acquisition of phase encode lines is compensated in the current implementation. 2.3

The Entropy Focus Criterion

Improvements to images degraded by incomplete and noisy data have been made using entropy maximisation, for example, see [15]. Entropy maximisation attempts to produce the smoothest image consistent with the data, this can prevent low brightness objects from being obscured by noise. The corruption of MR data by patient motion is not directly analogous t o these situations [16] and following radar autofocussing work [14], we use entropy minimisation as a focus criterion. Entropy minimisation favors high contrast and we demonstrate its use to remove motion induced ghosts from low intensity regions of an image. The entropy focus criterion E, used here is, M

z ,o[ B, ] j-=l Smax

(2)

where M is the number of image pixels (1282 here) and Bj is the modulus of the complex value of the j t h image pixel, referred to in this work as the pixel 'brightness'. The image energy is constant under motion induced phase shifts and if all the image energy were in one pixel, we would have the largest possible pixel brightness Bmax, given by;

Bma~ =

B~. j=l

(3)

344

Using this scheme, E = 0 when all the image energy is located in a single pixel and the remaining pixels are black. When the 128 • 128 'image' is uniformly gray, Bj/Bma= -= 1/128 for all pixels and the image entropy has the value E = 621. This entropy criterion thus favors alterations to the data that tend to increase the number of dark pixels. Motion often has the opposite effect, creating ghosts and blurring in image regions that would otherwise be dark. Thus one might expect entropy minimisation to aid the search for a motion free image, though it is by no means clear that this will always be the case. In Fig. 1 we plot the variation of entropy with simulated sinusoidal and step motion on a good quality image. In these examples, entropy always increases monotonically with the motion amplitude, indicating that it is a good focus criterion.

520.0

i

500.0

FE .... PE .... FE ........ PE

i

step step oscillation oscillation

i

.,...--"" .....""

"

. ..-'" .,.,,.,-" d~" j J....--" .o..- ....... j ~ ' ...oO

O

480.0

LU

460.0 8

440.0 0.0

S ~

'

210

'410 ' Amplitude of simulated motion (ram)

'

,0.0

Fig. 1. The effect of simulated translational motion on image entropy. The step motions occur half way through the acquisition of k-space, oscillatory motion covers 3 cycles over the whole scan time.

2.4

O p t i m i s a t i o n S t r a t e g y for I n - P l a n e T r a n s l a t i o n a l M o t i o n

The role of the optimisation strategy is to allow the iterative determination of the patient motion trajectory using the focus criterion as a measure of image quality. Minimisation of the focus criterion is multi-dimensional with translational motion possible in two dimensions and at any time during the acquisition. Our preliminary work indicates that finding the gross temporal features of motion before refining these features is a good search strategy.

345

The 128 readouts within the acquisition time are initially grouped into 16 intervals, each of 8 nodes. The interval nearest scan center (zero phase encode gradient) has trial displacements of { - N s , - ( N - 1)s,..., - s , s , . . . , (N - 1)s, Ns} in both the FE and PE directions applied, where s is a step size. Typically N = 3, giving 36 combinations of displacement to try for this interval. For each combination, the data is Fourier transformed to the image domain and the entropy calculated using equation (2). The combination giving the lowest image entropy is accepted and the motion curves are updated. The algorithm then similarly considers each interval in turn to the end of the scan, after which it considers the remaining intervals from scan center to the start of the scan. Working outwards from the center of k-space (tow image spatial frequency) is analogous to the use of multi-resolution search schemes in other areas of image processing. This process is repeated a second time and then the interval length is halved and the whole iteration procedure repeated. In this way the gross temporal features of the patient motion are determined first and subsequently refined. The algorithm halts after considering all the intervals with lengths of two nodes. 2.5

C o m p e n s a t i o n for Rotational M o t i o n

Rotational motion part way through a scan causes a rotation of the corresponding section of k-space, for example, see [4]. If the image is to be reconstructed from k-space using the fast Fourier transform (FFT), the rotated data must be interpolated or 're-gridded' to lie on a regular Cartesian grid. To interpolate data we must chose a convolution kernel that is a balance between computational speed, accuracy and the introduction of extra image artifacts. We take from Marschner et al. [17] the separable windowed sinc kernel h given by;

h(k) = [1 + cos(~k/k~)] sinc(4k/k~)

(4)

where k is the number of k-space points along the FE or PE direction from the k-space position being interpolated to the data point and kr, the kernel 'radius' is 4.78. Because our focus criterion is sensitive to ghosts, we oversample the data (i.e. zero pad the image) to reduce post-aliasing effects. Using this kernel it takes less than 30 seconds to rotate an oversampled image (256 • 256) on a Sparc Ultra 1-140 using un-optimised code. Our rotation of k-space rotates the image about its center, however, the true center of rotation will be nearer the point of contact of the head with the bed. Also, as the subject nods, this point of contact will move. To account for these effects, the rotated sections of k-space must have an unknown translational correction applied after the rotation correction. For rotational motion, we have not yet developed an optimisation strategy other than to use the prior knowledge that the volunteer made a single "yes" type nod of between -10 ~ and +10 ~ within =t=10nodes of k-space center. All possible combinations of rotation angle (to an accuracy of 1~ and time are applied to the data, followed by a simple gradient descent algorithm to find the translational correction for the unknown center of rotation.

346

2.6

Image Acquisition

All images were acquired on a Philips Gyroscan ACS-II MRI scanner operating at 1.5 T. The acquisition parameters were: spin-echo, TR=500 ms, TE=10 ms with a navigator echo at TE=50 ms, used as a control to determine motion, slice thickness 5 mm, sagittal orientation. In all cases, the PE gradient was decremented sequentially. Each volunteer was first asked to lie still and imaged with the bed stationary. Images degraded by translational motion were then obtained by manually moving the scanner bed during a scan; this moves the volunteer in the cranio-caudal direction. Navigator echoes were taken to provide an independent estimate of the volunteer motion. The estimate was obtained using a least squared method that detects shifts in the navigator hybrid space [18]. Normally this gives motion in the FE direction only and there is no information about phase encode motion. Here we have designed the experiment such that the motion in the FE direction is the same as that in the PE direction. This is achieved by orienting both of the imaging gradients at 45 ~ to the direction of bed motion. For rotational motion, the volunteer was asked to count the gradient pulses (clearly audible) and perform a step nod at the 64th imaging pulse. The nod moved his chin away from his chest. In all cases the algorithm had no prior knowledge about the direction of translational motion. The autofocus algorithm treats motion in the FE and PE directions as independent. Except for the rotation case, the algorithm had no prior knowledge of whether the motion is step-like or more general. Some prior knowledge of the magnitude of the motion is currently required to set the step size s if N is to be kept low in order to constrain the amount of computer time required. However, with more complex optimisation strategies, this a priori knowledge may not be required. Only motion in the imaging plane is considered.

3

Results

Figure 2 presents the results for step motion at 45 ~ to the imaging gradients. We show images of the volunteer stationary, corrupted by motion, after autofocussing and after navigator correction. The step size used was 0.63 mm (0.3 of a pixel size) and N = 9. The step motion at a time near the acquisition of the center of k-space has caused blurring, ghosting and a double image to appear. After autofocussing, the image entropy reduces by 6.1% and after navigator correction, it decreases by 4.6%. Visually the navigator corrected, autofocussed and stationary images appear to be of similar quality, indicating that autofocus is effective. In Fig. 3 we plot the navigator and autofocus determined motion. In the optimisation process, all intervals are allowed to vary freely and the origin of displacement will be determined by some average position of the initially blurred image. Hence the origins of the FE, PE and navigator displacements will all be different for a given image. In Fig. 3, the origins of the FE and PE autofocus determined

347

Fig. 2. Volunteer stationary, with step motion near the center of k-space, after autofocus and after navigator correction. The read-out direction is horizontal in the images.

348

displacements have been shifted (but not scaled) to minimise the mean absolute difference (MAD) between the algorithm and navigator displacements 9 For motion in the F E direction the MAD is 0.2mm and in the P E direction it is 0.9mm. These compare well with a search step size s of 0.63mm. The program has tracked the F E motion very well. For the P E direction, there is some inaccuracy at times near the motion step and near the outer regions of k-space (near nodes 0 and 127).

20.0 Nay ........ FE .... PE

f E

g

10.0

E u

0,0

-10 9

I

64

128

Node

Fig. 3. Navigator and autofocus determined motion corresponding to Fig. 2.

Results on data from other volunteers (not shown here), moved in the P E direction only, indicate that the algorithm can fail to detect P E motion near the edges of k-space when this causes ghosts in the brain but not outside the head. By dividing the image into small regions and calculating the entropy in each region, we have observed that the presence or absence of ghosts in the brain has relatively little effect on the entropy. The algorithm performs the bulk of its focusing by removing ghosts and blurring from regions of the image that would otherwise be dark (i.e. the region outside the head, in the paranasal sinuses, oesophagus and bone). Figure 4 is similar to Fig9 2 except that the motion is more general, the bed was moved throughout the scan. The step size s was 0.63 mm and N = 3. Again clear improvements can be seen in image quality for both antofocus and navigator corrected images. The entropy reduction is 3.8% after autofocus and 3.7% after navigator correction.

349

F i g . 4. Volunteer stationary, with general motion over the whole acquisition, after autofocus a n d after navigator correction.

350

Figure 5 shows that again the program determined FE motion follows the navigator well (MAD of 0.52mm), as has the PE motion (MAD 0.83ram) although there are some deviations near the edges of k-space.

3.0

NavJ 0,0

g E

-3.0

O.

.2 r~

-6.0

-9.0

I

64

128

Node

Fig. 5. Navigator and autofocus determined motion corresponding to Fig. 4

When searching phase encode line v for the optimum displacement dpE in equation (1), the phase correction applied to the acquired data is equivalent for any value of dpE that yields the same phase correction, modulo 2~. This means that the more extreme variations of PE displacement in Figs. 3 and 5 are actually equivalent to displacements closer to the navigator determined motion. The use of entropy to correct for rotational motion is demonstrated in Fig. 6 with entropy being reduced by 1.5%. Clear focussing can be seen in the brain and around the nose. The neck still appears blurred and ghosted - presumably because the motion is not rigid-body in this region. The examples presented above illustrate the operation of our algorithms. Navigator echo data was taken to validate the output from the algorithm. In practice, navigator echo information might be used to correct images for motion detected in the FE direction and an autofocus approach could be used to determine any additional, unknown, motion. To investigate this, we used the known navigator motion in the case of general motion (Fig. 4) to correct only for FE direction motion; the algorithm then searched for motion in the PE direction only. The detected PE motion trajectory was similar to before (MAD now 0.9 mm) and the image visually appears similar to the fully navigator corrected output. This demonstrates that autofocus can be used alone or in conjunction

351

Fig. 6. Volunteer stationary, with a single nod near the center of k-space and after focussing using the entropy focus criterion.

352

with navigator echoes to compensate for motion. Prior knowledge from other motion tracking sources might also be used. Searching for motion in only one direction reduces the search space and this took only 20 minutes on a Sparc Ultra 1-140 using un-optimised code, compared with 3.5 hours for the full 2D search.

4

D i s c u s s i o n and C o n c l u s i o n

We have demonstrated the use of an entropy focus criterion to compensate for motion induced artifacts in an MR image. Our illustrative examples show reduced blurring and ghosting but in some cases there is a residual, un-corrected ghosting that is not present in the image of the volunteer lying still. These ghosts are also present in the navigator corrected data and hence we infer they are caused by uncorrected out of plane motion, or motion between an RF excitation and the subsequent readout. The examples presented here demonstrate the concept of using entropy as a focus criterion in the improvement of MR images. The algorithms should be extendable to 3D volume acquisitions and more general rotational and translational motion. The amount of computer time will increase with the higher dimensional search space and we are currently devoting effort to reduce the optimisation time. The loss of data upon rotation is unavoidable with this type of acquisition sequence and may ultimately limit the types of motion for which correction is possible. The autofocus method should also be applicable to other MR imaging sequences such as T2-weighted images, "fast" or "turbo" sequences, segmented EPI and half k-space acquisitions. Navigator correction of diffusion weighted images [20] is most effective when diffusion is measured in the phase .encode direction. Autofocus techniques may be used to complement these navigator corrections for diffusion measurements in other directions. Correction of image artifacts due to other physical causes, such as timing errors in echo planar imaging that result in Nyquist ghosts, might also be compensated using autofocus methods. The entropy focus criterion we chose to use in this work should be applicable whenever ghosts or blurring occur in regions of the image that would otherwise be dark. Studies in which Bmax of equation (2) is varied locally within an image indicate that we can make the focus criterion sensitive to contrast changes both within tissue regions and outside. A number of other contrast measures and focus criteria exist in the literature [14, 19, 21, 22] and these may also be beneficial to improving the effectiveness of autofocus techniques. This is the subject of on-going work. In conclusion, we have demonstrated the use of an autofocus algorithm based on the minimisation of an entropy focus criterion to reduce motion induced blurring and ghosting in MR images. The algorithms were validated using independent navigator echo information and comparison with volunteers lying still. In practice, other motion measures such as navigator echoes can be used to quicken

353

the search and improve its accuracy. We envisage t h a t this type of autofocus algorithm will be used as a new tool in improving the quality of patient imaging. The technique does not require patient preparation, extra pulse sequences, complex gradient p r o g r a m m i n g or specialized hardware. All that is required is the real and imaginary d a t a from a conventional MRI scanner to achieve a reduction in motion induced blurring and ghosting.

5

Acknowledgements

We would like to acknowledge Dr. K. Hanson of Los Alamos National Laboratory and Dr. J.V. Hajnal of H a m m e r s m i t h Hospital, UK, for useful discussions. We are also grateful to the anonymous volunteers, to Mr J.D. Cox and Mr D. Greenhaugh of the DRA for their support, and for the encouragement of Dr. D. Hawkes as Director of the Radiological Sciences Image Processing Group.

References 1. R. L. Ehman and J. P. Felmlee: Adaptive Technique for High-Definition MR Imaging of Moving Structures. Radiology 173 (1989) pp. 225-263 2. H. W. Korin, F. Farzaneh, R. C. Wright, and S. J. Riederer: Compensation for Effects of Linear Motion in MR Imaging. Magn. Resort. Med. 12 (1989) pp. 99113 3. K. Butts, A. de Crespigny, J. M. Pauly, and M. Moseley: Diffusion-Weighted Interleaved Echo-Planar Imaging with a Pair of Orthogonal Navigator Echoes. Magn. Reson. Med. 35 (1996) pp. 763-770 4. H. W. Korin, J. P. Felmlee, S. J. Riederer, and R. L. Ehman: Spatial-FrequencyTuned Markers and Adaptive Correction for Rotational Motion. Magn. Reson. Med. 33 (1995) pp. 663-669 5. Z. W. Fu, Y. Wang, R. C. Grimm, P. J. Rossman, J. P. Felmlee, S. J. Riederer, and R. L. Ehman: Orbital Navigator Echoes for Motion Measurements in MagneticResonance-Imaging. Magn. Reson. Med. 34 (1995) pp. 746-753 6. G. H. Glover and J. M. Panly: Projection Reconstruction Techniques for Reduction of Motion Effects in MRI. Magn. Reson. Med. 28 (1992) pp. 275-289 7. R. A. Zoroofi, Y. Sato, S. Tamura, H. Naito, and L. Tang: An Improved Method for MRI Artifact Correction Due to Translational Motion in the Imaging Plane. IEEE Trans. Med. Imag. 14 (1995) pp. 471-479 8. R. Khadem and G. H. Glover: Self Navigation Correction for Motion in Spiral Scanning. Proc. Soc. Magn. Reson. (1994) pp. 346 9. M. L. Wood, M. J. Shivji, and P. L. Stanchev: Planar-Motion Correction Using kspace Data Acquired by Fourier MR Imaging. J. Magn. Reson. Imag. 5(1) (1995) pp. 57-64 10. B. Madore and R. M. Henkelman: A New Way of Averaging With Applications to MRI. Med. Phys. 23(1) (1996) pp. 109-113 11. Q. S. Xiang, M. J. Bronskill, and R. M. Henkelman: Two-Point Interference Method for Suppression of Ghost Artifacts Due to Motion. J. Magn. Reson. Imag. 3 (1993) pp. 900-906

354

12. M. Hedley, H. Yan, and D. Rosenfeld: Motion Artifact Correction in MRI Using Generalized Projections. IEEE Trans. Med. Imag. 10(1) (1991) pp. 40-46 13. R. A. Zoroofi, Y. Sato, S. Tamura and H. Naito: MRI Artifact Cancellation Due to Rigid Motion in the Imaging Plane. IEEE Trans. Med. Imag. 15(6) (1996) pp. 768-784 14. P~. P. Bocker, and S. A. Jones: ISAR Motion Compensation Using the Burst Derivative Measure as a Focal Quality Indicator. Int. J. Imaging Systems and Technol. 4 (1992) pp. 285-297 15. S. F. Gull, and G. J. Daniell: Image Reconstruction from incomplete and noisy data. Nature 272 (1978) pp. 686-690 16. R. T. Constable and R. M. Henkelman: Why MEM Does Not Work in MR Image Reconstruction. Magn. Reson. Med. 14 (1990) pp. 12-25 17. S. R. Marschner and R. J. Lobb: An Evaluation of Reconstruction Filters for Volume Rendering. In: R. D. Bergeron and A. E. Kaufman. (eds.) Proc. Visualization '94. IEEE Computer Society Press, 1994, pp. 100-107 18. Y. Wang, R. C. Grimm, J. P. Felmlee, S. J. Riederer, and R. L. Ehman: Algorithms for Extracting Motion Information from Navigator Echoes. Magn. Reson. Med. 36 (1996) pp. 117-123 19. D. C. Noll, J. M. Pauly, C. H. Meyer, D. G. Nishimura, and A. Macovski: Deblurring for Non-2D Fourier Transform Magnetic Resonance Imaging. Magn. Reson. Med. 25 (1992) pp. 319-333 20. A. W. Anderson and J. C. Gore: Analysis and Correction of Motion Artifacts in Diffusion Weighted Imaging. Magn. Reson. Med. 32 (1994) pp. 379-387 21. A. Buffington, F. S. Crawford, S. M. Pollaine, C. D. Orth, and R. A. Muller: Sharpening Stellar Images. Science 200 (1978) pp. 489-494 22. F. Berizzi, and G. Corsini: Autofocusing of Inverse Synthetic Aperture Radar Images Using Contrast Optimization. IEEE Trans. Aerospace and Elec. Systems 32(3) (1996) pp. 1185-1191