MRI Artifact Cancellation due to Rigid Motion in the

Imaging Plane 1

Reza Aghaeizadeh Zoroo�, Yoshinobu Sato, Shinichi Tamura and Hiroaki NaitoDivision of Functional Diagnostic Imaging

Osaka University Medical SchoolSuita, Osaka 565, JAPAN

E-mail: reza@image.med.osaka-u.ac.jp

AbstractA post-processing technique has been developed to suppress the MRI artifact arising from object planar rigid

motion. In 2-DFT MRI, rotational and translational motions of the target during MR scan respectively imposenon-uniform sampling and a phase error on the collected MRI signal. The artifact correction method introducedconsiders the following three conditions: (i) For planar rigid motion with known parameters, a reconstructionalgorithm based on bilinear interpolation and the super-position method is employed to remove the MRI artifact.(ii) For planar rigid motion with known rotation angle and unknown translational motion (including an unknownrotation center), �rst, a super-position bilinear interpolation algorithm is used to eliminate artifact due to rota-tion about the center of the imaging plane, following which a phase correction algorithm is applied to reduce theremaining phase error of the MRI signal. (iii) To estimate unknown parameters of a rigid motion, a minimumenergy method is proposed which utilizes the fact that planar rigid motion increases the measured energy of anideal MR image outside the boundary of the imaging object; by using this property, all unknown parameters ofa typical rigid motion are accurately estimated in the presence of noise. To con�rm the feasibility of employingthe proposed method in a clinical setting, the technique was used to reduce unknown rigid motion artifact arisingfrom the head movements of two volunteers.

Key words: MRI, rigid motion, artifact correction, rotational artifact, non-uniform sampling reconstruction.

I IntroductionMagnetic resonance imaging (MRI) is rapidly becoming a major diagnostic modality because of its many

promising and substantial capabilities for investigating various body organs, especially the brain, spine and ex-tremities [1]. With respect to two-dimensional Fourier transform (2-DFT) MRI [3], [2], acquiring an image witha standard spin echo sequence [4]-[6] takes several minutes. Body movements of restless, disoriented or injuredpatients, especially children and infants, are virtually unavoidable during conventional MR data acquisition. Asa consequence, the resultant image quality is degraded by the imposition of a ghost-like artifact, blurring, andreducing the intensity of moving structures [6]. For this reason, several recent papers have reported on methodsof reducing the MRI artifact arising from undesired patient rigid motion (such as the rotating or nodding of thehead) and/or physiological movements (respiratory, cardiac, gastrointestinal and vascular motions, for example)[7]-[21]. Because of the complexity and variety of possible patient movements, most previous works have dealtwith some limited types of motion. A review of computer post-processing methods that have been proposed tosuppress motional artifacts is given in [21]. Some recent techniques have sought to reduce the MRI artifact result-ing from object rigid motion (translational and/or rotational) in the imaging plane [13] - [21]. In previous work,we developed techniques to reduce the MRI planar artifact resulting from rotational [20] and translational [21]motions. However, rigid motion of the head during MR data acquisition remains a serious problem. Frequent headmovement can limit the quality of brain images, which thus provide only vague diagnostic information. Hence,the development of a substantial computer post-processing technique to suppress the MR brain image artifact dueto rigid motion was one of the main motivations of undertaking the present research. The e�ect of planar rigidtranslational motion is to impose a phase error on the MRI signal [10], [17]. On the other hand, rigid rotationalmotion about a �xed point in the imaging plane corrupts the MR image with non-uniform sampling of the relatedspatial frequency (k-space) data [19], [20]. Thus, the general planar rigid motion (translational and rotational)of the object during data acquisition imposes both a phase error and non-uniform sampling on the MRI signal.In this paper, we attempt to combine and improve on the strong points of our previous work [20], [21], with theaim of reducing the MRI artifact caused by object rigid motion, particularly head movement, during MR scanning.

The paper is organized as follows: In Section II, we model the problem of an MRI artifact resulting from 2-Drigid motion. The formulae introduced in Section II are derived under the assumption of the inter-view e�ect - i.e.in a similar manner to that given in our previous paper [21], we assume that 2-D rigid motion does not occur veryrapidly. We can thus neglect the e�ect of the read-out axis or intra-view e�ect, which takes several milliseconds,and assume that unknown in-plane rigid motion parameters are those of the inter-view e�ect or only functions

1IEEE Transaction on Medical Imaging, 15(6), 768-784, 1996.

of the phase-encoding steps [8], [9]. In Section III, we propose an algorithm using bilinear interpolation andthe super-position method to reconstruct an MRI artifact due to target rigid motion in the imaging plane. Forestimating motion unknown parameters, Ehman el al. [12] used navigator echoes to determine the displacement ofthe imaged object. However, limitations due to hardware, acquisition time, or other considerations may prohibitthe acquisition of motion information for every image echo of clinical interest [14]. Another alternative whichhas been investigated in the previous studies [13] - [21] is to use the phase encoded image itself rather thanphase-encoded navigator echoes for measurement of view-to-view object displacement. In this study, we arefollowing the latter strategy. In Section IV.A, we assume that only the planar rotational angle is known and thatother rigid motion parameters (the center of the 2-D rotation or translational motions) are unknown during dataacquisition. To remove the artifact, �rst we apply super-position bilinear reconstruction to eliminate the artifactarising from rotational motion about a �xed center (origin) in the imaging plane. The mathematical model ofthe problem shows that in comparison with the original MRI signal, the remaining MRI signal only contains anadditional phase error. Hence, we use an improved phase correction technique [21] to suppress the remainingphase error of the MRI signal. The method outlined in Section IV.B is concerned with estimating unknownplanar rigid motion parameters using the MR artifacted image. To solve this problem, we employ minimumenergy method. The computed energy of an ideal MR image is minimum outside the boundary of the imagingobject. Undesired in-plane rigid motion increases the measured energy outside the object boundary. Using thisproperty (as an evaluation function) and a non-linear search technique, we succeeded in accurately estimatingunknown parameters of planar rigid motion in the presence of noise. In section V, we demonstrate the robustnessof the proposed method by experiments using both simulated and actual MR data. Concluding remarks are madein Section VI.

II Model of the problemUsing a spin warp imaging sequence [3], [2], the relation between the MRI signal and the density distribution

of the target in the imaging plane is given by [10]

f(kx; ky) =

Z +1

�1

Z +1

�1

p(x; y) exp[�j2�(kxx+ kyy)]dx dy; (1)

where f(kx; ky) is the MRI signal, kx and ky are spatial frequency coordinates related to the read-out and phase-encoding directions, p(x; y) is the density distribution of the non-moving imaging target, and x and y are horizontaland vertical coordinates in the imaging plane. In (1), it is seen that the MRI signal is the 2-D Fourier transformof p(x; y), the density distribution of the object in the imaging plane. Motion of the object during the MRI dataacquisition can be shown as a function of kx and ky [16]. However, as each intra-view e�ect (ky = constant) indata acquisition time occurs so rapidly (in a matter of milliseconds), it is acceptable for most types of motionto neglect the e�ect of kx [8], [9]. Thus, planar rigid motion parameters during conventional 2-DFT MRI can beregarded as a function of ky, the so called inter-view e�ect.

For a planar, inter-view e�ect rotational motion, the relation between the points of a rotated image (x�r ; y�r ),and its unrotated counterpart (x; y) can be expressed by [20]

�x�r � xcy�r � yc

�=

�cos �r � sin �rsin �r cos �r

��x� xcy � yc

�; (2)

where �r = �r(ky) and (xc = xc(ky), yc = yc(ky)), respectively, are the counter-clockwise rotation about the zaxis and the center of rotation. (2) can be rearranged as

�x�ry�r

�=

�cos �r � sin �rsin �r cos �r

��xy

�+

�1� cos �r sin �r� sin �r 1� cos �r

��xcyc

�: (3)

Hence, assuming

��x(ky)�y(ky)

�=

�1� cos �r sin �r� sin �r 1� cos �r

��xcyc

�; (4)

and using (3), (4) can be rewritten as

�x�ry�r

�=

�cos �r � sin �rsin �r cos �r

��xy

�+

��x(ky)�y(ky)

�: (5)

2

On the other hand, a planar rigid motion is the combination of translational and rotational motions. Thus,the rotated point, (xrgd; yrgd), can be written as

�xrgdyrgd

�=

�cos �r � sin �rsin �r cos �r

��xy

�+

��x�y

�; (6)

where �x = �x(ky) and �y = �y(ky) , respectively, are the general 2-D translational motions (including �x and�y) of an object along the read-out and phase-encoding axes. Hence, the displaced object, prgd(x; y), can bedescribed as a function of its unrotated counterpart, p(x; y), with the following equation:

prgd(x; y) = p(x cos �r(ky) + y sin �r(ky)� �x(ky);�x sin �r(ky) + y cos �r(ky)� �y(ky)): (7)

It is known that rotation of an object around the origin of the imaging plane by an angle �r, causes the imagespatial frequency components to rotate by the same angle [31]. Using this property, we can write the followingrelation between the MRI corrupted signal, f�r (kx; ky), because of 2-D rotational motion around the origin of theimaging plane and its original value, f(kx; ky):

f�r(kx; ky) = f(kx cos �r(ky) + ky sin �r(ky);�kx sin �r(ky) + ky cos �r(ky)): (8)

In MR imaging, as is seen in (8), the above e�ect imposes non-uniform sampling on the produced k-space data.Moreover, referring to the mathematical model described in our previous work [21], translational motion imposesphase error on the MRI signal. Thus, the MRI artifacted signal arising from planar rigid motion, frgd(kx; ky) canbe described as follows:

frgd(kx; ky) = exp[�j2�(�x(ky)kx + �y(ky)ky)]f�r (kx; ky): (9)

Equation (9) shows that the only di�erence between the signals f�r (kx; ky) and frgd(kx; ky) is their phase.Finally, substitution of (8) in (9) gives the desired mathematical relation between the MRI signal corrupted dueto planar rigid motion, frgd(kx; ky), and the original MRI signal, f(kx; ky)

frgd(kx; ky) = exp[�j2�(�x(ky)kx+�y(ky)ky)]f(kx cos �r(ky)+ky sin �r(ky);�kx sin �r(ky)+ky cos �r(ky)): (10)

III MRI Artifact Correction with Given Rigid Motion

In practice, and working with the discrete MRI signal components, we have the values of the integer points ofthe corrupted MRI signal in k-space. Using these values and known motion parameters, we suggest estimating thevalues of the integer points of the original MRI signal by means of a 2-D interpolation method. Reconstructionof an original signal from non-uniform sampled data has been studied previously. In 1956, Yen [26] proved thatthe reconstruction of a band-limited signal from non-uniform samples is possible if there is a su�cient number ofdata. Trussell et al. [27] proposed a mathematical method to reconstruct the image from non-uniform sampleddata by sinc interpolation. However, their method is impractical for 2-D reconstruction; as an example, for a(512 by 512) image, the dimension of the sinc function interpolation matrix is (262144 by 262144). Trussell etal. concluded that for 2-D distortion the computation burden using a sinc function is so great that a fasterinterpolation technique must be used. Atalar et al. [28] also tested di�erent interpolation techniques to solve theproblem of non-uniform sampling. Their best technique, using 1-D sinc interpolation (for the kx and ky axes,respectively) and singular value decomposition (SVD) (for calculation of pseudo-inverse matrices), for a (256 by256) image using a Sun-3 160 system, takes approximately 56 hours. A faster method, composite interpolation(cubic spline in the kx direction and SVD in the ky direction), takes only about 15 minutes, but the interpolationerror imposes an artifact on the reconstructed image. Hence, Atalar et al. pointed out that a trade-o� needs tobe made between the cost of computation and the quality of the output image.

The above review of previous works shows that reconstruction of an MR image using non-uniform samples ofthe k-space is far from a simple problem. Nevertheless, in medical imaging the reconstruction procedure mustbe both speedy and preserve the quality. Assuming that rigid motion is an inter-view e�ect [8], [9] and known,two previous studies [19], [20] independently proposed similar algorithms for recovering the MRI signal from itsnon-uniform sampled k-space data. We can say, the explicit algorithm which separately was proposed in [20] hasgeneralized the re-sampling method of [19] for all phase-encoding steps of the K-space: MRI data acquisitiontime contains N di�erent phase-encoding steps. Assuming planar rigid motion is an inter-view e�ect, motionparameters are �xed during each phase-encoding step. Thus, the MRI spatial frequency components can beassumed as the superposition of the N di�erent images. In each of the N images, only one line, correspondingto the phase encoding step, is non-zero and the other lines are zero. It is assumed that the zero lines have the

3

motional parameters of the non-zero line. Hence, the planar rigid motion parameters are �xed for all lines of eachof the N di�erent images. On the other hand, using the superposition property, the inverse 2-D Fourier transformof the MRI signal can be obtained by adding the inverse 2-D Fourier transform of the previously discussed Nimages. Using the k-space data, the reconstruction algorithm is as follows [19], [20]:

� Step 0: Correct the phase error of the k-space data using the translational motion parameters (which areknown or have been estimated using the techniques of Section 4).

� Step 1: Divide the MR k-space data into the N di�erent images so that each image contains one non-zeroline (the corresponding phase-encoding step) and the other lines are zero.

� Step 2: Calculate the inverse 2-D Fourier transform of the N di�erent images.

� Step 3: Using an interpolation method, rotate each of the N di�erent images with its estimated (see Section4.2) rotation angle (corresponding to the phase-encoding step of non-zero lines in each of the N-images).

� Step 4: Calculate the 2-D Fourier transform of the N di�erent images.

� Step 5: To obtain the corrected MR signal, copy the above mentioned non-zero line (see Step 1) of eachof the N images to the corresponding lines of a zero image. The accumulated data is the recovered MRIsignal (the K-space data).

In our previous study [20], the 2-D Fourier transform of the sum of the images had been applied instead ofsteps four and �ve. However, it would cause the corrected-image data corresponding to each phase encoding stepto overlap with all other phase encoding steps. This e�ect would degrade the quality of the resultant MR image.The mentioned steps four and �ve that originated from [19] do not have such a problem. Then, the algorithm ofthis Section (comparing with that of [19]) has been proposed to cope with all corrupted phase-encoding steps ofthe K-space. However, as the above algorithm interpolates data on one line of the K-space using only the rotatedversion of the same line of the K-space data, it gives accurate results around the region where the lines intersect.We can say, the above algorithm works very well if the rotation angle is very limited.

It is seen that in the method, steps 1-5 are dealing with re-sampling of the K-space non-uniform sampleddata. However, interpolation has been done in the spatial domain. In this way, the drawback of the previouswork resulting from doing the interpolation in the k-space [27], [30] is by-passed. To increase the speed of thealgorithm, in Step 2, instead of computing the inverse 2-D Fourier transform for each image, we propose thefollowing: First, the inverse 1-D Fourier transform of the non-zero line (see Step 2) along the kx axis is computedfor each image. Then, to calculate the inverse 1-D Fourier transform of all lines along the ky axis (for each image),as there is only one non-zero value at each line along ky, the inverse 1-D Fourier can be obtained very rapidly.Moreover, as Step 3 of the algorithm must be repeated N times, the selected interpolation method should beboth accurate and fast. Floyd et al. [30] showed that for image re-sampling on a cylindrical sector grid, such asrotational motion, bilinear and overlap weighting were the best interpolation methods. In comparison, overlapweighting was more accurate but bilinear interpolation was faster. To preserve both accuracy and speed, we usedthe following bilinear interpolation in the algorithm:

p(x; y) = (1� yd)[x:w3 + (1� xd)w1] + yd[x:w4 + (1� xd)w2]; (11)

where p(x; y) is the rotated image, xi � x � xi + 1; yi � y � yi + 1, w1 = p(xi; yi); w2 = p(xi; yi + 1); w3 =p(xi + 1; yi); w4 = p(xi + 1; yi + 1); xd = x � xi, yd = y � yi. In a Sun Sparc 10 system, the reconstructiontime required for N = 256 was approximately 13 minutes. Experimental results (see Section V) con�rmed thee�ectiveness of the proposed algorithm both for simulated and real MR data.

IV Estimation of Unknown Planar Rigid MotionIV.A Phase Correction Methods

Referring to (9), it is seen that the only di�erence between the MRI artifacted signal arising from planarrigid motion, frgd(kx; ky), and the MRI artifacted signal resulting from 2-D rotational motion about the origin ofthe imaging plane, f�r (kx; ky), is in their phase. If we reduce the artifact of the image caused by rotation aboutthe origin using the algorithm introduced in Section III, the remaining artifact of the MR image is due to thephase error. The recovered MRI signal after reducing the artifact due to the rotational motion about the originof the imaging plane , fc(kx; ky), can be shown by:

fc(kx; ky) = exp[�j2�(kx�x(ky) + ky�y(ky))]f(kx; ky); (12)

4

where f(kx; ky) is the original MRI signal. In (12), the interpolation error of reconstruction algorithm is neglected.

In our previous work [21], we developed an improved phase correction method based on spectrum shift [14], [18]and phase retrieval [13] techniques. We proved the robustness of the method by reducing the artifact of an actualMR image resulting from planar translational motion. In this study, the same techniques as discussed in [21] areemployed to correct the phase error of the MRI signal, fc(kx; ky), due to unknown X-directional and Y-directionaltranslational motions. The spectrum shift algorithm is used for an MRI artifact caused by X-directional motions.Then, the phase retrieval algorithm suppresses the remaining artifact due to sub-pixel motion of the X-directionand the entire motion of the Y-direction.

IV.B Minimum Energy MethodIn conventional Fourier imaging, MR data acquisition takes N phase-coding steps. As we showed in (6),

for the inter-view e�ect in-plane rigid motion, �r(ky), �x(ky) and �y(ky) are unknown but �xed at each phase-encoding step. Hence, in general, the maximum number of unknown motion parameters is 3N . With a givenrotational motion, �r(ky), the method explained in Section IV.A was e�ective in correcting the MRI artifact dueto unknown �x(ky) and �y(ky). In this section, we introduce a method for estimating all unknown planar rigidmotion parameters.

In an ideal MR imaging, most of the energy of the acquired image is located inside the boundary of the imagingobject. In other words, the produced intensity levels of the resultant image is from the magnetization distributionof the object. The energy of an MR image, E0, resulting from a stationary object outside the region of interest(ROI) (here, the boundary of the imaging object) can be expressed as follows:

E0 =X

(i;j)=2ROI

b2ij ; (13)

where bij is the intensity of the image pixel (i; j). Expressing E0 in absolute term can be a complicated matterbecause of the multitude of in uential parameters in MRI [34]. We can say, E0 represents the noise of the MRimage outside the ROI. When the rigid object moves during the MRI data acquisition, its location changes, andas a result, the energy outside the ROI increases: For example, if the energy of an ideal MRI image is assumedE0 outside the ROI (see (13)), 2-D rotational motion will change it to Et so that E0 < Et. In general, the totalenergy of the MR image outside the ROI, Et, can be written by

Et = E0 +

NXi=1

�Ei; (14)

where N is the number of phase-encoding steps and �Ei is the increasing factor of the energy due to planar rigidmotion at the phase-encoding step i. In (14), due to the many factors which in uence E0, we have assumed thatthe cross-correlation terms are relatively small and thus have been neglected. However, for large values of noise,the cross-correlation terms may not remain relatively small and hence can not be neglected. In other words,Equation (14) can be used as an evaluation function to estimate the unknown rigid motion parameters �r(ky),

�x(ky), and �y(ky) as long as the term E0 �PN

i=1 �Ei is valid.

In our previous work [20], by applying a linear search method, the e�ectiveness of the above-mentioned methodwas con�rmed for a unit step type rotational motion occurring at phase-encoding step kr with a rotation angleof �r. Unknown motion parameters (kr and �r) were estimated accurately for a simulated MRI artifact in thepresence of additive noise. In this section, following an explanation, we present an algorithm for �nding rigidmotion with more unknown parameters using the above-mentioned method and a multivariate, non-linear search.

If the rigid motion parameters change M times (M < N) during MR data acquisition, the k-space containsM + 1 segments with di�erent motion parameters where one of the segments can be assumed to be the referencesegment. Hence, in addition to �r, �x and �y, the phase-encoding step (ky) in which the rotation occurred is thefourth unknown parameter of each k-space segment. For example, if a rigid object moves four times (M = 4)during MR data acquisition, the unknown motion parameters number 16. Wood et al. [19], using the paired ttest and k-space data, identi�ed ky values where subject movement occurred. In Section V, we illustrate anothertechnique using a spectrum shift algorithm to �nd the unknown kys. Hence, for the above example the number ofremaining unknown parameters decreases to 12. In practice, rigid motion of the head is unknown, but is boundedduring data acquisition time. Therefore, it is reasonable to assume that: j�rj < �m, j�xj < �1 and j�yj < �2.Using these assumptions, we minimize Et with 3M (for the above example: M = 4) unknown parameters using adirect search Complex algorithm [32], [33]. The above-mentioned assumption causes the non-linear minimizationmethod [32], [33] to converge toward the actual global minimum. To compute Et at each iteration, knowingthe ROI is necessary. However, detecting the boundary of the imaging object (ROI) from the MRI artifacted

5

image is very di�cult due to the presence of motion ghosts. To �nd the ROI accurately, we copied one of theM +1 segments of the k-space to the corresponding segment of a zero image. We then computed the inverse 2-DFourier transform of the image obtained and binarized the result with a threshold value. For a typical k-spacesegment which contained at least 10 phase-encoding steps, we succeeded in obtaining an acceptable ROI. Toachieve convergence and to speed up the non-linear search, �nding suitable initial points for unknown parameterswas necessary. To overcome this di�culty, we �rst applied the direct search Complex algorithm [32], [33] locallyto estimate 3 unknown parameters, �r, �x and �y, of each segment. In this step, zero was selected as the initialvalues of all unknown parameters. After several iterations, zero values were replaced by suitable initial values ofunknown rigid motion parameters corresponding to each segment. By using the values obtained in the previousstep, the assumed boundary values for unknown �rs, �xs and �ys were also modi�ed. Then, the values of theunknown parameters obtained in the local searches were used as the initial values of the corresponding parametersin the global search. By experience, it was found that after I iterations, the unknown parameters converged toa small boundary of p pixels and � degrees, respectively, for translational motions and rotation angle. Hence, toprevent saturation of the search algorithm, the initial values of unknown parameters and boundaries were replacedby the newly obtained values. In this method, each k-space segment with unknown parameters adds its own costto each iteration. Nevertheless, the total number of iterations depends on the number of unknown variables. Themethod described above for �nding unknown rigid motion parameters with M abrupt changes (M < N) duringMR data acquisition, can be simpli�ed in the following algorithm:

� Step 0: Acquire unknown kys, by using the paired t test [19] or spectrum shift algorithm (see explanationsof Fig. 2 in Section V).

� Step 1: Estimate the ROI as follows y:

{ Copy one of the M k-space segments to a zero image.

{ Compute the inverse 2DFT of the result.

{ Extract the ROI by thresholding the result.

� Step 2: Estimate the boundary and initial values of unknown parameters of each segment as follows:

{ Set the initial values of unknown �r, �x and �y to zero.

{ Set preliminary boundaries of unknown �r, �x and �yz: j�rj < �m degrees, j�xj < �1 pixels and

j�yj < �2 pixels.

{ Apply the direct Complex search algorithm to estimate unknown values in a three-dimensional space,iteratively, n times.

{ Use the obtained values of the unknown parameters as the initial values of the corresponding pa-rameters in the global search. Use the obtained values of �r, �x and �y to modify the correspondingboundary values (see explanations of Fig. 5 in Section V).

� Step 3: Apply the direct Complex search algorithm to estimate unknown values in a 3M -dimensional space,iteratively, I times.

� Step 4:

{ Replace the initial values of the non-linear search with the corresponding estimated values.

{ Renew the boundary values to p pixels and � degrees apart from the estimated values obtained inStep 3.

{ Set the loop-counter to zero.

� Step 5: Apply the direct Complex search algorithm to estimate unknown values in a 3M dimensional space,iteratively, I times.

� Step 6: Replace the initial values of the non-linear search with the estimated value obtained in the previousstep. If the loop-counter < L, go to Step 5, else end.

Remarks:y In practice, a physician with rough knowledge about the boundary of the imaging object (the brain image, forexample) interactively determines the most suitable segment as well as the appropriate value of the threshold.z In the case of an actual MR scan, as patient is recommended to avoid movement during the scan time, assumingboundary for rigid motion parameters is reasonable (for example: �m = 30 degrees, �1 = 30 pixels and �2 = 30pixels).

6

V Experimental ResultsIn this section, we evaluate experimentally the model and algorithms developed in the previous sections. Fig.

1 shows the results of an experiment to reconstruct an MRI artifact due to a known planar motion employingthe algorithm introduced in Section III. In the previous works, the reconstruction method was evaluated forseveral types of motion such as sinusoidal, single-shift [20], and four abrupt changes of the head [19] during dataacquisition. In the following, we con�rmed the reliability of the reconstruction algorithm for another type ofplanar rigid motion. The simulated image was a Shepp and Logan phantom [25] (for digitization process see [13]),which is shown in Fig. 1(a). Fig. 1(b) shows a typical rotational angle. The amplitude and width (numbers of thephase-encoding steps) of the motion were produced by random generators. The minimum k-space segment witha �xed rotation angle contained 10 phase-encoding steps. Figs. 1(c) and (d) respectively demonstrate randomX-directional and Y-directional translational motions. Fig. 1(e) shows the simulated MRI artifact obtained byapplying the above mentioned rigid motion. Fig. 1(f) shows the reconstructed image using superposition bilinearreconstruction. Fig. 1(g) is the MR image produced by adding zero mean Gaussian noise (S=N) = 16dB, to Fig.1(e). Fig. 1(h) shows the recovered noisy image. Comparison of Fig. 1(f) with 1(e) and 1(h) with 1(g) revealsthe e�ectiveness of the algorithm introduced in Section III. It is necessary to remember that as the interpolationmethod is used both to create the simulated artifact and to recover the original MR image, in all experimentsusing simulation the resultant interpolation errors are twice the actual values.

Figs. 2 and 3 show the results of applying the phase correction method (see Section IV.A) to improve thequality of an MRI artifact when the rotation angles and translational motions, respectively, are known andunknown during MR data acquisition. As it was shown in [21], the Y-directional Fourier transform of the MRoutput image is equal to the X-directional inverse Fourier transform of the MRI signal. We computed the Y-directional Fourier transform of Fig. 1(a), and then changed the result to a binary image by using a thresholdvalue of .005 (Fig. 1(a) was noiseless). The binary image is shown in Fig. 2(a). Figs. 2(b), (c), and (d)respectively demonstrate the shapes of the rotational angle, and the X-directional and Y-directional translationalmotions. Fig. 2(e) shows the artifacted image resulting from planar translational motion using Figs. 2(c) and(d). Fig. 2(f) illustrates the Y-directional Fourier transform of Fig. 2(e) and was obtained in the same manneras 2(a). We previously showed [21] that extracting X-directional motion (Fig. 2(c)) from the edges of Fig. 2(f)was possible. Fig. 2(g) shows the artifacted image arising from planar rotational motion using Fig. 2(b) and atypical rotation center of (128,200). Fig. 2(h) shows the Y-directional Fourier transform of Fig. 2(g) and wasobtained in the same manner as 2(a). Abrupt changes of the rotation angle are observable in 2(h). Hence, inaddition to the procedure of Wood et al [19] in which the paired t test was shown to be e�ective in identifyingky values acquired during subject movement, the spectrum shift algorithm is an alternative means of �ndingunknown phase-encoding steps, kys, where new rotations occur (see Section IV.B). Fig. 2(i) shows the artifactedimage resulting from planar rotational motion using Figs. 2(b), (c), (d), and a typical rotation center of (128,200).Fig. 2(j) shows the Y-directional Fourier transform of Fig. 2(i) and was obtained in the same manner as 2(a).A combination of the edge displacements of Figs. 2(f) and (h) is seen in Fig. 2(j). Our goal was to reduce theMRI artifact of Fig. 2(i). As noted earlier, for Figs. 2 and 3 the rotational motion parameters were known. Fig.3(a) shows the corrected image after reducing the rotational artifact of Fig. 2(i). In our previous work [20], weemployed the phase retrieval method [13] to remove the remaining artifact of Fig. 3(a). The result of using thismethod is shown in Fig. 3(b). The image is seen not to be improved su�ciently. As we described previously[21], for a large X-directional (read-out) translational motion, the rate of phase wrapping grows and the phaseretrieval algorithm fails to converge. To solve this problem, we used the following procedures. First, the edgesof the Y-directional Fourier transform of Fig. 2(i) ( Fig. 2(j)) were extracted. The detected displacement isdemonstrated in Fig. 3(c). Second, using the known places of abrupt changes in the k-space, assuming that theunknown X-directional motion of Fig. 2(c) was smooth, and Fig. 3(c), we estimated the unknown X-directionalmotion (Fig. 3(d)). Third, we corrected the artifact of Fig. 2(i) using the motions of Figs. 2(b) and 3(d). Theresult in shown in Fig. 3(e). Comparison of Fig. 3(e) with 3(a) shows that the image is improved. To designatethe ROI for the phase retrieval method, Fig. 3(e) was changed to a binary image with a threshold value of 30(Fig. 3(f)). Fig. 3(g) is the �nal result of our proposed method, which was obtained by reducing the remainingartifact of 3(e) with the phase retrieval algorithm (50 iterations). This experiment revealed the robustness ofthe phase correction method (see Section IV.A) in comparison with the previous technique [20] (Fig. 3(b)). Theremaining artifacts on Fig. 3(g) can be explained as follows: for a severe rotation angle (such as Fig. 2(b)) there-sampling technique discussed in Section III could not perfectly remove the e�ect of non-uniform sampling ofthe K-space. It meant that the phase information of the re-sampled K-space was a�ected by the reconstructionmethod. As the non-actual value of the phase was consequently employed in the phase retrieval method [13], thetechnique could not improve the quality of the artifact image as much as the previous work [21].

Figs. 4 and 5 evaluate the algorithm developed in Section IV.B to �nd unknown planar rigid motion param-eters. Previously [20] (using the linear search method), we con�rmed the capability of the algorithm to �nd theparameters of an unknown single-shift rotational motion. In the experimental results of Fig. 4, using a non-linearminimization method [32], [33], we attempted to �nd the unknown parameters of planar rotational motion (see(5)). In this experiment, it was assumed that during MR data acquisition, the imaging object was abruptly rotatedseveral times, so that the center of rotation (Figs. 4(a) and (b)) and the rotation angle (including the number

7

of abrupt changes), Fig. 4(c), respectively, were known and unknown planar rotational motion parameters. Theartifacted image resulting from the above motions is shown in Fig. 4(d). To estimate the unknown parameters,�rst, the number of abrupt changes (10 times) in k-space was found by the spectrum shift method (in the samemanner as in Fig. 2(h)). Then, the ROI (Fig. 4(e)) was obtained as follows: 2-D FFT of 4(d) was computed, andthe �rst upper part of the k-space with a constant rotation angle was copied to a zero image. Then, the inverse2-D FFT of the obtained image was made binary with some threshold value. Finally, the algorithm of SectionIV.B was used to estimate the 10 unknown rotation angles in an iterative scheme. each segment with unknownparameters needed 5 seconds. Hence, each iteration took 50 (5 x 10) seconds. After 300 iterations which took lessthan 4 12 hours, the algorithm succeed in �nding all 10 unknown rotation angles with less than one degree error.The corrected image is shown in Fig. 4(f). Fig. 5 shows the results of applying the minimum energy method (thealgorithm of Section IV.B) when all parameters of planar rigid motion are unknown (Figs. 5(a), (b), and (c)).The artifacted result after adding zero-mean Gaussian noise ((S=N) = 16dB)2 is demonstrated in Fig. 5(d). Thenumber of abrupt changes in k-space value (4) and the ROI were obtained in a similar to that used in Fig. 4. Thealgorithm of Section IV.B was then employed to �nd the 12 (4 x 3) unknown parameters of unknown planar rigidmotion. Again, each segment with unknown parameters needed 5 seconds. Hence, each iteration took 20 (5 x 4)seconds. After 300 iterations, which took less than 2 hours, the algorithm succeeded in �nding all 12 unknownrotation angles with errors of less than one degree and one pixel, respectively, for planar rotational and planartranslational motions. The corrected image is shown in Fig. 5(e). The same experiment as of Fig. 5 was repeatedwith higher values of noise. It was concluded that the algorithm was e�ective as long as (S=N) was greater than14 dB. When the (S=N) becomes very poor medical and diagnostic information of the MR image (non-movingtarget) will be lost. Hence, in such a noisy condition dealing with artifact-correction of a non-stationary target isnot clinically helpful.

The experimental results shown in Figs. 4 and 5 reveal the promising behavior of the minimum energy methodfor estimating unknown parameters of planar rigid motion.

Figs. 6, 7, and 8 show the results of experiments to eliminate MRI artifacts arising from unknown rigid motionoccurring during actual scans. The materials for the imaging object of Fig. 6(a) were selected as follows. Markerswere a water solution of Gadolinium-DTPA (a contrast medium for MRI), the gray region was a water-absorbedpolymer of sodium acrylate, and the white regions (inside the gray region) were obtained by a water solution ofnickel sulfate. The MR scan was taken by a General Electric Signa 1.5T System at Osaka University Hospital(TR = 1000 msec., TE = 20 msec., EC = 1/1 16kHz, FOV = 32cm x 32cm, 256 x 256/1 NEX). Fig. 6(b) is theartifact image resulting from imposing a controlled single-shift planar rotational motion about the center point(128,128) during the MR scan. The ROI (Fig. 6(c)) was obtained by thresholding 6(b) with a value of 50. Theminimum energy method (see Section IV.B) was employed to estimate two unknown motion parameters, ky and�r. Adding two high density markers, one to each side of the imaging object, to amplify the magnitudes of theedges of the X-directional Fourier transform of the MRI signal was suggested in the previous papers [14], [21]. Inthis experiment, the markers were used to amplify the change in the energy outside the boundary of the imagingobject (to speed up the convergence in the minimum energy method). Fig. 6(d) shows the corrected image afterusing the estimated parameters ky = 105 and �r = 4:5 degrees and applying the algorithm given in Section III.Comparison of Fig. 6(d) with 6(b) con�rms e�ectiveness of the methods discussed in this paper.

In the previous work, Wood et al. [19] reported on using a head-coil and pillow to control the head-noddingof a volunteer within the sagittal plane. Here, three of the present authors (R. A. Z., S. T., and Y. S.) actedas volunteers for the actual MR experiments. It should be emphasized that each volunteer did the experimentonly once without any preliminary trials. Head movement was not restricted by a head-coil and/or pillow. Thetrans-axial brain images were acquired with a standard spin-echo sequence (Figs. 7(a) and 8(a)). The volunteersmoved the head once during data acquisition. Two of the results are shown in Figs. 7(b) (R. A. Z.) and 8(b)(S. T.). The second volunteer (S. T.) had a larger movement. The unknown motion parameters (ky, �r, xc, andyc) were obtained in the same manner as in Fig. 5. The estimated values obtained for Fig. 7 were as follows:ky = 129, �r = 2:7 degrees, xc = 132, and yc = 210, while those for Fig. 8 were obtained as follows: ky = 129,�r = 3:8 degrees, xc = 128, and yc = 205. The corrected images are shown in Figs. 7(f) and 8(f). Theseexperiments proved the feasibility of using the developed techniques in a clinical setting. The third volunteer (Y.S.) moved his head too much so that we failed to recover the desired image; the result obtained after correctionwas the brain image of another slice. This revealed the need to increase the capability of the algorithm, to providek-space data of multiple slices in the neighborhood of the desired imaging plane (multiple 2-D slice acquisition).However, the promise shown by the basic method indicates that, the technique can be improved to correct MRIartifacts due to 3-D rigid motion [19].

2(S=N) = 10log[

PM

i=1

PM

j=1S(i;j)P

M

i=1

PM

i=1N(i;j)

], where the size of the image is (M x M) pixels; S(i; j) andN(i; j) respectively,

are the signal and noise-energy at (i,j) [31].

8

VI ConclusionThe e�ect of patient planar rigid motion during MR data acquisition is to impose a phase error and non-uniform

sampling on the corresponding k-space data. With known rigid motion parameters, the method developed here us-ing the super-position property and bilinear interpolation was e�ective in reconstructing an actual MRI artifactedimage. The result of the reconstruction algorithm can be expanded to correct other non-uniform sampling distor-tions in conventional MR Fourier imaging which satisfy the assumption of the inter-view e�ect. By experiments,we con�rmed the capability of the phase correction method (spectrum shift and phase retrieval algorithms) toreduce the artifact of an MR image when the rotation angles and translational motions were, respectively, knownand unknown during MR data acquisition. For planar rigid motion with all parameters unknown, we developeda method to estimate the motion parameters using the MR artifacted image and minimum energy method. Wecon�rmed the robustness of the method in estimating unknown motion parameters of simulated and real MR data.Finally, we showed the feasibility of utilizing the method clinically by decreasing the MR artifact of brain imagesarising from the rigid motion of volunteers. The next step in this research is to expand the current techniques toreduce the MRI artifact due to unknown 3-D rigid motion.

AcknowledgementsWe are grateful to the editors of the IEEE TMI for their valuable advice. We also wish to thank the sta� of

the Division of Functional Diagnostic Imaging, Osaka University Medical School for their technical assistance inusing software tools and hardware facilities.

References[1] image Z. Cho, J. P. Jones, and M. Singh, Foundations of Medical Imaging. New York: John Wiley, 1993, pp.

237-267.

[2] W. A. Edelstein, J. M. S. Hutchison, G. Johnson, and T. Redpath, \Spin warp NMR imaging and applicationsto human whole body imaging," Phys. Med. Biol., vol. 25, pp. 751-756, 1980.

[3] A. Kumar, D. Welti, and R. R. Ernst, \NMR Fourier zeugmatography," J. Magn. Reson., vol. 18, pp. 69-83,1975.

[4] E. L. Hahn, \Spin echoes,"Phys. Rev. 80: pp. 580-585, 1950.

[5] H. Y. Carr and E. M. Purcell, \E�ects of di�usion on free precession in nuclear magnetic resonance experi-ments,"Phys. Rev. 94: pp. 630-635, 1954.

[6] D. D. Stark and W. G. Bradley,Magnetic Resonance Imaging. St. Louis: MosbyYear Book, 1992, pp. 145-164.

[7] M. L. Wood and R. M. Henkelman, \Suppression of respiratory motion artifacts in magnetic resonanceimaging," Med. Phys., vol. 13, no. 6, pp. 794-805, 1986.

[8] P. M. Pattany, J. J. Phillips, L. C. Chiu, J. D. Lipcamon, J. L. Duerk, J. M. McNally, and S.N. Mohapatra,\Motion artifact suppression techniques (MAST) for MR imaging," J Comput Assist Tomogr 11(3), pp.369-377, May/June 1987.

[9] E. M. Haacke and G. W. Lenz, \Improving MR image quality in the presence of motion by using rephasinggradients," AJR 148: 1251-1258, Jun. 1987.

[10] H. W. Korin, F. Farzaneh, R. C. Wright, and S. J. Riederer, \Compensation for e�ects of linear motion inMR imaging," Magn Reson in Med 12, pp. 99-113, 1989.

[11] T. Mitsa, K. J. Parker, W. E. Smith, and A. M. Tekalp, \Correction of periodic motion artifacts along theslice selection axis in MRI," IEEE Trans. Med. Imag., vol. 9, no. 2, pp. 310-317, 1990.

[12] R. L. Ehman and J. P. Felmlee, \Adaptive Technique for high-de�nition MR imaging of moving structures,"Radiology 173: pp. 255-263, 1989.

[13] M. Hedley, H. Yan, and D. Rosenfeld, \An improved algorithm for 2-D translational motion artifact correc-tion," IEEE Trans. Medical Imaging, vol. 10, no. 4, pp. 548-553, Dec. 1991.

[14] J. P. Felmlee, R. L. Ehman, S. J. Riederer, and H. W. Korin, \Adaptive motion compensation in MR imagingwithout use of navigator echoes," Radiology 179: pp. 139-142, 1991.

9

[15] M. Hedley and H. Yan, \Suppression of slice selection axis motion artifact in MRI," IEEE Trans. Med.Imag., vol. 11, no. 2, pp. 233-237, Jun. 1992.

[16] M. Hedley, H. Yan, and D. Rosenfeld, \Motion artifact correction in MRI using generalized projections,"IEEE Trans. Med. Imag., vol. 10, no. 1, pp. 40-46, Mar. 1991.

[17] M. Hedley and H. Yan, \Iterative restoration of MR images corrupted with translational motion," J. ofVisual Comm. and Image Representation, vol. 3, no. 4, pp. 325-337, Dec. 1992.

[18] L. Tang, M. Ohya, Y. Sato, S. Tamura, H. Naito, K. Harada, and T. Kozuka, \MRI artifact cancellation fortranslational motion in the image plane," IEEE Nuclear Sc. Symp. and Med. Imag. Conf., pp. 1489-1493,1993.

[19] M. L. Wood, M. J. Shivji, and P. L. Stanchev, \Planar motion correction with use of k-space data acquiredin Fourier MR imaging," J. of MRI, vol. 5, no. 1, pp. 57-64, Jan./Feb. 1995.

[20] R. A. Zoroo�, Y. Sato, S. Tamura, and H. Naito, \ Reduction of MRI artifact due to rotational motion inthe imaging plane", Proceedings of Japanese association of medical imaging technology (JAMIT) Frontier95, pp. 37-44, Jan. 1995, Nagoya Univ., Japan.

[21] R. A. Zoroo�, Y. Sato, S. Tamura, H. Naito and L. Tang, \ An improved method for MRI artifact correctiondue to translational motion in the imaging plane," IEEE Trans. Med. Imag., vol. 14, no. 3, pp. 471-479, Sep.1995.

[22] W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy. New York: AcademicPress, 1978.

[23] R. W. Gerchberg and W. O. Saxton, \ A practical algorithm for the determination of phase image anddi�raction plane pictures," Optik, vol. 35, no. 2, pp. 237-248, 1972.

[24] R. W. Gerchberg, \Super resolution through error energy reduction," Optica Acta, vol. 21, no. 9, pp. 709-720,1974.

[25] L. A. Shepp and B. F. Logan, \Reconstructing interior head tissue from X-ray transmissions," IEEE Trans.Nuclear Science, vol. NS-21, pp. 228-236, 1974.

[26] J. L. Yen, \On nonuniform sampling of bandwidth-limited signals," IRE Trans. Circ. Theory, vol. CT-3, no.1, pp. 251-257, Dec. 1956.

[27] H. J. Trussell, L. L. Arnder, P. R. Moran, and R. C. Williams \Correction of nonuniform sampling distortionsin magnetic resonance imagery," IEEE Trans. Med. Imag., vol. 7, no. 1, pp. 32-44, March 1988.

[28] E. Atalar and L. Onural, \A respiratory motion artifact reduction method in magnetic resonance imagingof the chest," IEEE Trans. Med. Imag., vol. 10, no. 1, pp. 11-24, March 1991.

[29] R. N. Bracewell, The Fourier transform and its applications. 2nd ed. New York: MacGraw-Hill, 1978, p. 247.

[30] C. E. Floyd, R. J. Jaszczak, and R. E. Coleman, \Image resampling on a cylindrical sector grid," IEEETrans. Med. Imag., vol. MI-5, no. 3, pp. 128-131, Sep. 1986.

[31] R. C. Gonzales and P. Wints, Digital Image Processing. Massachusetts: Addison-Wesley, 1987, pp. 38-40,pp. 79-80, p.257.

[32] P. E. Gill, W. Murray and M. H. Wright, Practical Optimization. New York: Academic Press , 1981.

[33] J. A. Nedler and R. Mead, \A simplex method for function minimization," Computer Journal: 7, pp. 308-313,1965.

[34] D. J. Nishimura, Introductin to Magnetic Resonance Imaging. Text-book No. EE 369B, Stanford University,pp. 156-167, April 1994.

10

(a) (b)

(c) (d)

Fig. 1.

11

(e) (f)

(g) (h)

Fig. 1: (Cont.) Reconstruction of MRI artifact due to object planar rigid motion. (a) Shepp and Logan Phantom.(b) Typical rotation angle. (c) Random X-directional motion. (d) Random Y-directional motion. (e) Artifact imageresulting from rigid motion (using (b), (c), and (d)). (f) Reconstructed image using super-position bilinear methodand known motion parameters. (g) Artifact image containing additive noise (S=N = 16dB). (h) Reconstructednoisy image using super-position bilinear method and motion parameters (using (b), (c), and (d)). The super-position bilinear method is seen to be e�ective in reconstructing the MRI artifact due to planar rigid motion.

12

(a) (b)

(c) (d)

Fig. 2.

13

(e) (f)

(g) (h)

(i) (j)

Fig. 2: (cont.) E�ectiveness of spectrum shift algorithm in detecting parameters of unknown rigid motion. (a)Binary Y-directional FFT of Fig. 1(a). (b) Typical rotation angle. (c) X-directional motion. (d) Y-directionalmotion. (e) Artifact image after imposing translational motion (using (c) and (d)). (f) Binary Y-directional FFTof (e). In comparison with (a), the edges of the spectrum are displaced. The edge displacements are equal to theX-directional translational motion (see text). (g) Artifact image after imposing rotational motion (using (b) anda typical rotation center of (128,200)). (h) Binary Y-directional FFT of (g). The abrupt changes of the rotationangle ((b)) are visible in (h). Hence, it is possible to �nd the unknown phase-encoding steps (ky) in which thesubject rotation are occurred. (i) Artifact image resulting from rigid motion (using (b), (c) and (d)). (j) BinaryY-directional FFT of (i). A combination of the motions in (f) and (h) is observable in (j). Therefore, if thelocations of the abrupt changes in rotation angle ((b)) are given (see text), it is practical to detect unknown (butsmooth) X-directional translational motion from the binary Y-directional FFT of the artifact image.

14

(a) (b)

(c) (d)

Fig. 3.

15

(e)

(f) (g)

Fig. 3: (cont.) Experimental results for reducing MRI artifact (Fig. 2(i)) when planar rigid motion parameters(Figs. 2(b), 2(c), and 2(d)) are partly given such that (I) the 2-D rotational motion (Fig. 2(b)) is knownand 2-D translational motions (Figs. 2(c) and 2(d)), are unknown. (a) Corrected image after applying super-position bilinear reconstruction (and known rotation parameters) to remove rotational artifact of Fig. 2(i). (b)Corrected image using the phase retrieval algorithm to remove the remaining artifact of (a) (after 30 iterations).In comparison with (a), the artifact is not improved su�ciently. (c) Detected motion from the edges of thespectrum in Fig. 2(j). (d) Estimated X-directional motion using (c) and known locations of abrupt changes inrotation angle (Fig. 2(b)). (e) Corrected image using (d) to remove the X-directional translational artifact of(a). (f) ROI detected by thresholding (e) with a value of 30. (g) Corrected image by using (e) (rotational andX-corrected image) and the phase retrieval algorithm after 50 iterations. This experiment con�rms the ability ofour technique ((g)) to correct a MRI 2-D rigid motion artifact ((a)) more satisfactorily than the previous method((b)).

16

(a) (b)

(c) (d)

(e) (f)

Fig. 4: Experimental results for reducing MRI artifact with planar rotational motion when motion parametersare partly given such that (II) the angle of rotation �r is unknown and the center of rotation (xc; yc) is known. Inthis experiment, the k-space segment where the rotational motion parameters did not change contained at least10 phase-encoding steps. (a) Typical motion for xc. (b) Typical motion for yc. (c) Unknown rotation angle. Itis seen that the rotation angle contains 10 abrupt changes. (d) Artifact image resulting from rotational motion(using (a), (b), and (c)). (e) ROI. To create the ROI, the 2-D FFT of (d) was computed, and the �rst uppermostpart of the k-space with a constant rotation angle was copied to a zero image. Then, the inverse 2-D of the imageobtained was made binary with some threshold value. (f) Corrected image after estimating 10 motion parameters(rotation angles) using the minimum energy method. Before applying the method, spectrum shift algorithm wasused to �nd abrupt changes along the ky direction (to �nd segments with a �xed rotation angle). The algorithmwas reliable in estimating all 10 unknown rotation angles with less than 1 degree error.

17

(a) (b) (c)

(d) (e)

Fig. 5: (cont.) Experimental results for reducing MRI artifact due to completely unknown planar rigid motion.(a) X-directional motion. (b) Y-directional motion. (c) Rotation about (128,128). It is seen that the rigid motion((a), (b), and (c)) contains 4 abrupt changes. Hence, excluding unknown kys (that were estimated by the spectrumshift algorithm) for each of the four segments, we had to �nd 3 unknown parameters. In other words, the goalof this experiment was to estimate 12 unknown parameters of planar rigid motion. (d) Artifact image resultingfrom rigid motion (using (a), (b), and (c)) and additive zero mean Gaussian noise (S=N = 16dB) (e) Correctedimage after estimating 12 motion parameters using the minimum energy method. In the presence of noise, thealgorithm was reliable in estimating unknown translational and rotational parameters with errors of less than 1pixel and 1 degrees, respectively.

18

(a) (b)

(c) (d)

Fig. 6: Experimental results for real MR scan. (a) Actual MRI phantom without motion. (b) Artifacted imageafter imposing a controlled (planar) single-shift rotational motion about the rotation center (128,128) during dataacquisition time. In this experiment, the angle of rotation and phase-encoding step in which the rotation occurredwere two unknown motion parameters. (c) ROI detected by thresholding (b) with a value of 50. (d) Correctedimage after estimating 2 motion parameters by the minimum energy method. The markers were used to intensifythe change of the energy outside the ROI arising from phantom planar rigid motion.

19

(a) (b)

(c) (d)

(e) (f)

Fig. 7: Experimental results for MR brain image # 1. (a) MR image taken from a trans-axial cross-section withoutmotion. (b) Y-directional FFT of (a). (c) Artifacted image when a volunteer was requested to rotate his headonly once during the data acquisition time. In this experiment, the angle of rotation, the phase-encoding step inwhich the rotation occurred, and the center of rotation (xc; yc) were 4 unknown motion parameters. The headmotion was not restricted (by a using head coil and/or pillow, etc.). (d) Y-directional FFT of (c). In comparisonwith (b), due to the head movement, the edges of the spectrum are displaced. Using (d) and the spectrum shiftalgorithm, the unknown phase-encoding step was estimated. (e) ROI obtained in the same manner as in Fig. 3(e).(f) Corrected image after estimating the remaining 3 motion parameters by the minimum energy method. Thisexperiment con�rmed the feasibility of using the algorithm to reduce MRI artifacts caused by the patient rigidmotion in a clinical setting.

20

(a) (b)

(c) (d)

(e) (f)

Fig. 8: Experimental results for MR brain image # 2. In comparison with Fig. 7, the second volunteer can beobserved to have rotated his head with a larger rotation angle. Explanations are the same as in Fig. 7.

21