T2 restoration and noise suppression of hybrid MR images using Wiener and linear prediction techniques

Hybrid imaging (HI) sequences used for magnetic resonance (MR) imaging and inherently degraded by T2 effects and additive measurement noise are enhanced. Wiener filter and linear prediction (LP) technique is used to process HI MR signals in the spatial frequency domain (K-space) and the hybrid domain respectively. Based on the average amplitude symmetry constraint of the spin echo signal, the amplitude frequency response function of the T2 distortion is estimated and used in the Wiener filter for a global T2 amplitude restoration. Then a linear prediction technique is utilized to obtain local signal amplitude and phase estimates around discontinuities of the frequency response function of the equivalent T2 distortion filter. These estimates are used to make local amplitude and phase corrections. The effectiveness of this combined technique in correcting T2 distortion and reducing the measurement noise is analyzed and demonstrated using experiments on both phantoms and humans.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
This invention relates generally to the field of magnetic resonance (MR) 
imaging (MRI) utilizing NMR phenomena. It is particularly related to 
enhancement of MR imaging data acquired using hybrid imaging (HI) MR data 
acquisition sequences which include T2 and additive noise degradation 
effects. 
2. Related Art 
Magnetic resonance (MR) imaging techniques are now extensively used for 
noninvasive investigations of living matter in medicine and biology. 
Researchers in different MR groups are trying to further improve the 
performance and the accessibility of MR image machines through lower cost, 
shorter scan time, higher resolution and contrast, higher signal/noise 
ratio and fewer artifacts. [L. Kaufman, L. E. Crooks and J. Carlson, 
"Technology Requirements for Magnetic Resonance Imaging System, in 
Proceedings of Technology Requirements for Biomedical Imaging, IEEE 
Computer Society Press, May 1991] Among these, reducing imaging time while 
maintaining image quality has been a very active topic. 
In 1978, Mansfield et al. demonstrated echo planar imaging (EPI) [P. 
Mansfield and P. G. Morris, "NMR Imaging in Biomedicine," in Advances in 
Magnetic Resonance, Edited by J. S. Waugh, Academic Press, New York, 1982] 
The basic concept behind EPI is that successive spin echoes can be used to 
encode position information using just a single shot (i.e., a single NMR 
RF excitation data acquisition sequence). Because of high requirements on 
gradient coils and power supplies for achieving rapid echo train 
generation and some other problems, various hybrid imaging (HI) 
approaches, incorporating aspects of both conventional two dimensional 
(2-D) FT imaging and EPI, have been proposed. [Hennig et al. J. Hennig, A. 
Nauerth and H. Friedberg, "RARE Imaging: A Fast Imaging Method for 
Clinical MR." Magne, Reson, Med., Vol. 3, pp. 823-33, 1986; Van Uijen et 
al. C. M. J. Van Uijen, J. H. Den Boer and F. J. J. Verschuren; Haacke et 
al. E. M. Haacke, F. H. Bearden, J. R. Clayton and N. R. Lingar, 
"Reduction of MR Imaging Time and Hybrid Fast Scan Technique," Radiology, 
Vol. 158, pp. 521-29, 1986; and others] These techniques use multiple (M) 
excitations and after each excitation, multiple (N) echoes are used to 
encode positional information. HI techniques are far less demanding on 
hardware and thus can be used to decrease imaging time without the cost 
and technical constraints of EPI. 
Since in EPI and HI, phase encoding measurements acquired at different echo 
times are used to form an image, there are inherently T2 distortions in 
the acquired data along the phase encoding direction. Depending upon the 
phase encoding schemes used and the object under the study, loss of 
spatial resolution and/or contrast may be introduced. [R. T. Constable and 
J. C. Gore, "The Loss of Small Objects in Variable TE Imaging: 
Implications for FSE, RARE, and EPI." Magne, Reson. Med., Vol. 28, pp. 
9-24, 1992; D. A. Ortendahl, L. Kaufman and D. M. Kramer, "analysis of 
Hybrid Imaging Techniques, Magne, Reson. Med., Vol. 26, pp. 155-73, 1992] 
Furthermore, there are discontinuities in the frequency response of the 
effective T2 distortion filter and these discontinuities generate ringing 
artifacts in the image. Techniques such as inverse filtering have been 
tried to reduce these T2 effects, based on some prior knowledge about the 
T2 values of the objects under study. The success of this approach is 
often limited by lack of knowledge about the T2 values and the existence 
of measurement noise. The problem of ringing artifacts caused by local 
discontinuities in the frequency response function of the T2 filter have 
not yet been successfully addressed. 
It is well known that a Wiener filter performs better for image restoration 
in a noisy environment. [R. C. Gonzalez and P. Wintz, Digital Image 
Processing,Addison-Wesly Publishing Company, 1988; A. K. Jain, 
Fundamentals of Digital Image Processing, Prentice Hall, Englewood Cliffs, 
N.J. 07632, 1989] 
The techniques of linear and nonlinear prediction have also been used to 
reduce truncation artifacts in MR imaging by several people. [M. R. Smith, 
S. T. Nichols, R. M. Henkelman and M. L. Wood, "Application of 
Autoregressive Moving Average Parametric Modeling in Magnetic Resonance 
Image Reconstruction," IEEE Trans. Med. Imag., Vol. 5, pp. 132-39, 1986; 
J. F. Martin and C. F. Tirendi, "Modified Linear Prediction Modeling in 
Magnetic Resonance Imaging," J. Magn. Reson., Vol. 82, pp. 392-99, 1989; 
E. M. Haacke, Z. Liang and S. H. Izen, "Super Resolution Reconstruction 
through Object Modeling and Parameter Estimation, IEEE Trans. ASSP., Vol. 
37, pp. 592-95, 1989; H. Yan and J. Mao, "Data Truncation Artifact 
Reduction in MR Imaging Using A Multilayer Neural Network," IEEE Trans. 
Med. Imag., Vol. 12, pp. 73-77, 1993 ] 
BRIEF SUMMARY OF THE INVENTION 
It has now been discovered that a combined use of a Wiener filter and 
linear prediction (LP) to process HI images better moderates T2 and noise 
effects. In the first stage, based on the average amplitude symmetry 
constraint, a global T2 value of the object is estimated from acquired 
data and thus the amplitude frequency response function of the effective 
T2 distortion filter is determined. The Wiener filter is then used to make 
global T2 amplitude restoration and noise suppression in K-space. In the 
second stage, linear prediction is utilized to obtain local signal 
amplitude and phase estimates. That is, Wiener filter processed K-space 
signals are Fourier-transformed in the read-out direction to obtain a 
hybrid domain signal and LP is used to provide estimates of local signal 
amplitude and phase. These estimates are used to make local amplitude and 
phase corrections in the hybrid domain and thus reduce the effects caused 
by discontinuities of the T2 distortion filter frequency response. As a 
result of this two-stage processing, T2 effects on the image data can be 
reduced and, at the same time, measurement noise can also be suppressed.

DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS 
FIG. 1 depicts a typical conventional MRI system that has been adapted so 
as to practice an exemplary embodiment of this invention. One example of 
such system is the Toshiba ACCESS.TM. MRI system. For example, it may 
comprise a rather large NMR polarizing magnet structure 10 which generates 
a substantially uniform homogeneous NMR polarizing magnetic field B.sub.o 
within a patient imaging volume 12. A suitable carriage 14 is used for 
inserting the desired portion of patient 16 anatomy within the image 
volume 12. Magnetic NMR gradients in B.sub.0 can be selectively created by 
electromagnet gradient coils, NMR RF mutation pulses can be transmitted 
into the patient tissue within the image volume and NMR RF responses can 
be received from the patient tissue via suitable RF coil structures as 
will be appreciated by those in the art. A particular MRI data acquisition 
sequence of such magnetic gradient pulses, RF nutation pulses and NMR RF 
responses is conventionally achieved by an MRI sequence controller 18 
controlling the usual array of gradient drivers 20, RF transmitter 
circuits 22 and RF receiver circuits 24, all suitably interfaced with 
electromagnetic and RF coils within the MRI system gantry. The received 
NMR RF responses are digitized and passed to an MRI image processor 26 
which typically includes an array processor 28 and suitable computer 
program storage media 30 (e.g., RAM in silicon or magnetic media) wherein 
programs are storm and selectively utilized so as to control the 
processing of acquired MR image dam to produce digitized image displays on 
the CRT terminal 32. The control terminal 32 may also include suitable 
keyboard switches and the like for exerting operator control over the MRI 
sequence controller 18 and the interconnected cooperating MR image 
processor 26. 
In conjunction with the usual MRI processor 26, an operator is typically 
presented with a menu of choices. In the exemplary embodiment of this 
invention, one of those choices available to the operator would be 
"enhancement" or filtering of a hybrid MR image. In the preferred 
exemplary embodiment, a default set of filter parameters is provided for 
general usage. However, if desired, different sets of filter parameters 
may be chosen (possibly from a predetermined array of filter parameter 
sets) so as to provide the operator with some filtering alternatives that 
may provide better results in the case of a particular type of medical 
image. 
T2 effects on the quality of conventional MR images have been analyzed in 
many papers. The effects of T2 amplitude distortions on the HI images have 
also been investigated and reported in R. T. Constable and J. C. Gore, 
"The Loss of Small Objects in Variable TE Imaging: Implications for FSE, 
RARE, and EPI," Magne. Reson. Med., Vol. 28, pp. 9-24, 1992; D. A. 
Ortendahl, L. Kaufman and D. M. Kramer, "Analysis of Hybrid Imaging 
Techniques," Magne. Reson. Med., Vol. 26, pp. 155-73, 1992. The frequency 
responses of the effective T2 distortion filters and their effects on HI 
image quality continue to be of concern. Ringing artifacts introduced by 
discontinuities in the T2 filter frequency response are also of concern. 
It is well known that MR images depend on multiple tissue parameters: the 
hydrogen density N(H), the longitudinal and transverse relaxation times, 
T1 and T2, and the pulse sequence parameters: repetition time TR and echo 
time TE. [D. A. Ortendahl and N. M. Hylton, "MRI Parameter Selection 
Techniques," in Magnetic Resonance Imaging, edited by C. L. Partain et 
al., W. B. Saunders Company, 1988]. For example, given a certain shaped 
object of uniform N(H), T1 and T2, the K-space MR signal 
S(k.sub.x,k.sub.y) acquired in a conventional spin-echo experiment can be 
written as 
##EQU1## 
where s(n.sub.x,n.sub.y) is the observable magnetization from the object 
and (k.sub.x,k.sub.y) are spatial frequencies. Using the tissue parameters 
and the sequence parameters, equation (1) can also be expressed as 
EQU S(k.sub.x,k.sub.y)=N(H)(1-e.sup.-TR/T1)e.sup.-TE/T2 S(k.sub.x,k.sub.y) (2) 
where the signal S(k.sub.x,k.sub.y) is determined only by the position, 
size and shape of the object. From equation (2), it can be seen that if TR 
and TE are fixed constants for different phase encode echoes (different 
k.sub.y values), there is no image distortion except for a fixed 
attenuation. In fact, along the frequency encoding (k.sub.x) direction, 
there is also a T2 distortion factor of .sub.e -TS(k.sub.x)/T2 such that 
S(k.sub.x,k.sub.y) should be written as 
EQU S(k.sub.x,k.sub.y)=N(H)(1-e.sup.-TR/T1)e.sup.-(TS(k.sub.x)+TE)/T2S(k.sub.x, 
k.sub.y) (3) 
with TS(k.sub.x) being the corresponding time for the sampling position 
k.sub.x. In practice, the durations of echoes are usually short compared 
to T2 times, so this effect is not visually apparent. [D. A. Ortendahl, L. 
Kaufman and D. M. Kramer, "Analysis of Hybrid Imaging Techniques," Magne, 
Reson. Med., Vol. 26, pp. 155-73, 1992] 
When hybrid MR imaging techniques are used, different echoes in the echo 
train have different echo times TE(k.sub.y). Then, the acquired signal 
becomes 
EQU S(k.sub.x,k.sub.y)=N(H)(1-e.sup.-TR/T1)e.sup.-(TS(k.sub.x)+TE(k.sub.y))/T2S 
(k.sub.x,k.sub.y). (4) 
We can rewrite equation (4) as 
EQU S(k.sub.x,k.sub.y)=H(k.sub.x,k.sub.y)S(k.sub.x,k.sub.y) (5) 
where S(k.sub.x,k.sub.y) is the K-space signal without T2 distortion 
EQU S(k.sub.x,k.sub.y)=N(H)(1-e.sup.-TR/T1)S(k.sub.x,k.sub.y) (6) 
and H(k.sub.x,k.sub.y) is the frequency response of the effective T2 
distortion filter defined as 
EQU H(k.sub.x,k.sub.y)=e.sup.-(TS(k.sbsp.x.sup.)+TE(k.sbsp.y.sup.))/T2 (7) 
Thus, the effective T2 distortion filter equation (7) is separable in 
k.sub.x and k.sub.y directions. [D. E. Dudgeon and R. M. Mersereau, 
Multidimensional Digital Signal Processing, Prentice Hall, Englewood 
Cliffs, N.J. 07632, 1981]. That is, it can be written as 
EQU H(k.sub.x,k.sub.y)=e.sup.-(TS(k.sbsp.x.sup.)/T2 e.sup.-TE(k.sbsp.y.sup.)/T2 
=H.sub.x (k.sub.x)H.sub.y (k.sub.y) (8) 
where 
EQU H.sub.x (k.sub.x)=e.sup.-TS(k.sbsp.x.sup.)/T2, H.sub.y 
(k.sub.y)=e.sup.-TE(k.sbsp.y.sup.)/T2 (9) 
This fact is useful for simplifying implementation of the T2 correction 
filter in practice. [D. E. Dudgeon and R. M. Merserean, Multidimensional 
Digital Signal Processing, Prentice Hall, Englewood Cliffs, N.J. 07632, 
1981]. In MR imaging systems, noise results from multiple sources but it 
essentially consists of two major components: noise from the receiver 
circuits and noise from the excited tissues. These two components are 
affected by the system resonance frequency but are independent of echo 
times. Therefore, the T2 filter equation (7) has an effect on the signal 
but not on noise. JR. T. Constable and J. C. Gore, "The Loss of Small 
Objects in Variable TE Imaging: Implications for FSE, RARE, and EPI," 
Magne, Reson. Med., Vol. 28, pp. 9-24, 1992; D. A. Ortendahl, L. Kaufman 
and D. M. Kramer, "Analysis of Hybrid Imaging Techniques," Magne., Reson. 
Med., Vol. 26, pp. 155-73, 1992]. 
In the above formula for the T2 filter, absolute times TS(k.sub.x) and 
TE(k.sub.y) are used for the data sample S(k.sub.x,k.sub.y). For the 
purpose of explanation in the following discussion, relative times will be 
used for equation (7). In addition, it will be assumed that TS(k.sub.x)=0 
for the first sample in the frequency encoding direction and TE(k.sub.y)=0 
for the first echo in the echo train. This will normalize the T2 filter 
such that H(k.sub.x,k.sub.y).sub.max =1.0 but will not change the shapes 
of the frequency responses of the T2 distortion filter and therefore will 
not affect the structure of the resulting T2 correction filter. 
Depending on the phase encoding schemes chosen, the T2 filter can have 
different frequency responses and thus have different effects on the 
resulting HI images. Consider, for instance, an M excitation HI sequence, 
each M containing N phase-encoded echoes to form an MN-line acquisition. 
If the earliest echo is assigned to the lowest spatial frequencies, with 
later echoes assigned to progressively higher spatial frequencies, the T2 
filter has a low-pass frequency response in the k.sub.y direction. In this 
case, the image spatial resolution will be reduced. This case is similar 
to the blurting problems in many other imaging systems. On the other hand, 
if the earliest echo is assigned to the highest spatial frequencies, with 
later echoes assigned to progressively lower spatial frequencies, T2 
filter has a high-pass frequency response in k.sub.y direction. Then, some 
edge enhancement may occur but the image contrast of large areas will be 
attenuated. In FIGS. 2 and 3 are shown the frequency responses H.sub.y 
(k.sub.y) of a low-pass T2 filter and a high-pass T2 filter, respectively, 
for two HI sequences with M=64 and N=4. The object is assumed to have a T2 
value of 70 (ms). The echo times TE(i)=20 .i ms are used for the echo 
number i=1,2,3,4, respectively, in the echo train. In these figures and 
the following discussions, the position of k.sub.y =129 corresponds to the 
zero-phase encode projection. 
The T2 filter can also have frequency responses of band-pass, ramp and 
other shapes by using different sequence specifications. [R. T. Constable 
and J. C. Gore, "The Loss of Small Objects in Variable TE Imaging: 
Implications for FSE, RARE, and EPI," Magne,Reson. Med., Vol. 28, pp. 
9-24, 1992; D. A. Ortendahl, L. Kaufman and D. M. Kramer, "Analysis of 
Hybrid Imaging Techniques," Magne.,Reson. Med., Vol. 26, pp. 155-73, 
1992]. In FIG. 4 is shown the frequency response of a band-pass T2 filter 
where the projections from the first echo are located in the middle 
frequency band. This encoding scheme has a better trade-off for image 
resolution and contrast than the low-pass and the high-pass encoding 
schemes. However, it usually generates more ringing artifacts because of 
the existence of bigger jumps in the frequency response shown in FIG. 4 
than those shown in FIGS. 2 and 3. Note that for different phase encoding 
schemes, the T2 filter H.sub.x (k.sub.x) in the k.sub.x direction always 
has an exponential ramp frequency response. 
In the above, only the amplitude frequency responses of T2 distortion 
filters is considered. It is known that for image representation, the 
phase components of an image Fourier transform often have a more important 
role than the amplitude components. [M. H. Hayes, "The Reconstruction of a 
Multidimensional Sequence from the Phase or Magnitude of Its Fourier 
Transform," IEEE Trans. ASSP., Vol. 30, No. 2, pp. 140-54, 1982]. In MR 
imaging, in addition to amplitude distortions introduced by T2 effects, 
due to the imperfection of the practical imaging system and the difficulty 
of exact phase control in the HI sequence, there are also phase 
distortions caused by different phase shifts to the signal components from 
different echoes. These phase distortions will also generate ringing 
artifacts (even if there were no T2 amplitude distortions). Furthermore, 
the amplitude and phase discontinuities introduced by the T2 effects are 
signal dependent and thus cannot be smoothed by simple windowing. In FIG. 
5A, the original image of two rectangular objects is shown, the image is 
shown with amplitude distortion only in FIG. 5B, the image with phase 
distortion only is shown in FIG. 5C, and the image with both amplitude and 
phase distortions is shown in FIG. 5D. The amplitude distortion is caused 
by an effective T2 distortion filter with a frequency response similar to 
that shown in FIG. 4. The phase distortion is caused by a 90.degree. phase 
shift to the signal components from the second echo. The periods of these 
tinging artifacts are determined by the positions of the discontinuities 
in the T2 distortion filter frequency response. For the example of FIGS. 
5A-5D, there is a big discontinuity in k.sub.y at position 96 which 
corresponds to the digital frequency f.apprxeq.0.13. Therefore, the period 
of the major tinging artifacts is about 7.7 pixels in this image of 256 
pixels. These tinging artifacts seriously disturb accurate diagnosis using 
HI MR images. To improve the HI image quality, both the amplitude and 
phase distortions should be reduced. 
In the previous discussion, one object with uniform tissue parameters has 
been assumed. In practice, an object under study usually has complicated 
distributions of these tissue parameters and the T2 values vary in 
different regions of the object. Signals from these different tissues are 
first distorted by the corresponding T2 distortion filters and then added 
together, further contaminated by measurement noise. Therefore, complete 
T2 correction is a challenging task. Nevertheless, as explained below, the 
use of a Wiener filter to make a global T2 amplitude correction (based on 
an estimated frequency response H(k.sub.x,k.sub.y) from the acquired data) 
provides a major improvement. 
As previously shown, the T2 effects on images can be modeled as an original 
image distorted by a T2 distortion filter. If the frequency response of 
the T2 distortion filter is known, inverse filtering can recover the 
original image from the acquired data. That is, 
EQU S(k.sub.x,k.sub.y)=H.sup.I (k.sub.x,k.sub.y)S(k.sub.x,k.sub.y) (10) 
where the frequency response H.sup.I (k.sub.x,k.sub.y) of the inverse 
filter is the reciprocal of the frequency response of the T2 filter and 
can be written using equation (7) as 
##EQU2## 
The main problem with the inverse filter is its sensitivity to measurement 
noise. When the acquired data S(k.sub.x,k.sub.y) is in the form of the T2 
distorted signal plus noise, which is the case in any practical imaging 
system, we have 
EQU S(k.sub.x,k.sub.y)=H(k.sub.x,k.sub.y)S(k.sub.x,k.sub.y)+N(k.sub.x,k.sub.y) 
(12) 
where N(k.sub.x,k.sub.y) is the measurement noise component. Then the 
inverse filter will give 
##EQU3## 
Since the noise component N(k.sub.x,k.sub.y) is not affected by the T2 
filter and H(k.sub.x,k.sub.y).ltoreq.1, we have 
N(k.sub.x,k.sub.y).gtoreq.N(k.sub.x,k.sub.y). The measurement noise will 
be amplified in this T2 correction process and the resulting images are 
often not acceptable. 
The Wiener algorithm provides a better solution to the T2 correction 
problem in a noisy environment. Let the signal s(n.sub.x,n.sub.y) and the 
noise n(n.sub.x,n.sub.y) be arbitrary, zero mean, random sequences, 
respectively. If the acquired sequence s(n.sub.x,n.sub.y) is modeled as 
EQU s(n.sub.x,n.sub.y)=h(n.sub.x,n.sub.y)**s(n.sub.x,n.sub.y)+n(n.sub.x,n.sub.y 
) (14) 
where h(n.sub.x,n.sub.y) is the distortion filter and ** denotes 2-D 
convolution, then the best linear estimate s(n.sub.x,n.sub.y) for the 
original signal s(n.sub.x,n.sub.y) can be obtained from the distorted data 
s(n.sub.x,n.sub.y) using a Wiener filter w(n.sub.x,n.sub.y) in the sense 
that the mean square error between the original signal s(n.sub.x,n.sub.y) 
and the estimated signal s(n.sub.x,n.sub.y) defined by 
EQU .sigma..sup.2.sub.w =.epsilon.[.vertline.e.sub.w 
(n.sub.x,n.sub.y).vertline.]=.epsilon.[.vertline.s(n.sub.x,n.sub.y)-s(n.su 
b.x,n.sub.y).vertline..sup.2 ] (15) 
is minimized where e represents the expectation operation. The Wiener 
restoration filter w(n.sub.x,n.sub.y) has a frequency response 
W(k.sub.x,k.sub.y) in the form 
##EQU4## 
where H(k.sub.x,k.sub.y) is the frequency response of the distortion 
filter h(x,y) and the superscript * represents conjugation. P.sub.ss 
(k.sub.x,k.sub.y) and P.sub.nn (k.sub.x,k.sub.y) are the power spectrums 
of the signal process s(x,y) and the noise process n(x,y), respectively. 
[R. C. Gonzalez and P. Wintz, Digital Image Processing,Addison-Wesley 
Publishing Company, 1988; A. K. Jain, Fundamentals of Digital Image 
Processing, Prentice Hall, Englewood Cliffs, N.J. 07632, 1989] 
The restoration process of the Wiener filter is adjusted adaptively by the 
local signal-to-noise ratio P.sub.ss (k.sub.x,k.sub.y)/P.sub.nn 
(k.sub.x,k.sub.y). In the small noise case, P.sub.nn 
(k.sub.x,k.sub.y).apprxeq.0.0, the Wiener filter reduces to the inverse 
filter 
##EQU5## 
and distortion effects can be removed. If the T2 effects are very weak 
such that H(k.sub.x,k.sub.y).apprxeq.1.0 as in conventional (i.e., 
non-hybrid) MR sequence cases, then the Wiener filter becomes 
##EQU6## 
which smooths the image data. When both T2 effects and noise exist, the 
Wiener filter reduces T2 effects and, at the same time, suppresses 
measurement noise. 
In MR imaging, the power spectrums of the signal and of the noise are not 
known and thus have to be estimated from the acquired data. Since 
S(k.sub.x,k.sub.y) is the inverse Fourier transform of the object, it 
usually has smaller values at higher spatial frequencies and thus the 
noise is often dominant in these frequencies. This will be especially true 
for hybrid fast MR imaging using a low-pass scheme since longer echo times 
further attenuate these high frequency signal components. Therefore, the 
average noise power P.sub.nn can be rather accurately estimated from the 
data S(k.sub.x,k.sub.y) on a support S.sub.n in the high spatial 
frequencies from the latest echo in the echo train using the standard 
formula 
##EQU7## 
where N.sub.p is the total data number used for the P.sub.nn estimation on 
the data support S.sub.n. Since the measurement noise N(k.sub.x,k.sub.y) 
is often assumed to be white, we have the same noise power P.sub.nn 
everywhere in K-space. In practice, some DC component may exist. In this 
case, any such DC should be removed before making the estimation by 
equation (19). 
Then the periodogram spectral estimation is used to obtain an estimate of 
the spectrum of the image data. [S. M. Kay, Modern Spectral Estimation: 
Theory & Application, Prentice Hall, Englewood Cliffs, N.J. 07632, 1988] 
That is, we use the local data power P.sub.dd (k.sub.x,k.sub.y) defined by 
EQU P.sub.dd (k.sub.x,k.sub.y)=.vertline.S(k.sub.x,k.sub.y).vertline..sup.2 ( 
20) 
to replace the signal power spectrum P.sub.ss (k.sub.x,k.sub.y) in equation 
(16) and thus obtain the frequency response of K-space Wiener T2 
correction filter is 
##EQU8## 
Since the frequency response H(k.sub.x,k.sub.y) of the T2 distortion 
filter is real in our discussion, the conjugate operation (*) and the 
absolute value operation (.vertline...vertline.) have been dropped in 
equation (21). The signal power can also be estimated by 
EQU P.sub.dd 
(k.sub.x,k.sub.y)=max{(.solthalfcircle.S(k.sub.x,k.sub.y).vertline..sup.2 
-P.sub.nn),0.0} (22) 
This can be used in equation (21)instead of P.sub.dd (k.sub.x,k.sub.y) and 
will, in general, give stronger noise suppression. Once the frequency 
response of the Wiener filter is determined, the restored K-space image 
data can be obtained as 
##EQU9## 
One approach to determine the frequency response function 
H(k.sub.x,k.sub.y) from the acquired data is described below and from this 
the Wiener filter equation (21) can be completely specified for the image 
data restoration. 
Since the frequency response H(k.sub.x,k.sub.y) of the T2 distortion filter 
depends on the spatial distribution and spin density of T2 values within 
the object under study, one cannot determine the exact frequency response 
of the T2 filter with just one data acquisition. Nevertheless, one 
approach for estimating H(k.sub.x,k.sub.y) from the acquired data based on 
the average amplitude symmetry constraint of the spin echo signal is set 
forth below. 
It is well known from MR spectroscopy that the time-domain echo signal is 
symmetric around its peak if the T2 value is much longer than the echo 
observation time. In MR imaging, when the same TE is used for all phase 
encode (k.sub.y) values, ideally, the K-space data 
S(129+k.sub.x,129+k.sub.y) and S(129-k.sub.x,129-k.sub.y) should be 
conjugate symmetric about the peak signal position (k.sub.x =129, k.sub.y 
=129) in a 256.times.256 data matrix and have the same amplitude values if 
the T2 values of the object are much longer than the sampling window width 
in the frequency encoding (read-out, k.sub.x) direction. [D. A. Feinberg, 
J. D. Hale, J. C. Watts, L. Kaufman and A. Mark, "Halving MR Imaging Time 
by Conjugation: Demonstration at 3.5 kG." Radiology 164, pp. 527-31, 1986] 
When the T2 distortion effect exists, data amplitude distributions are 
biased. Therefore, one can estimate a global T2 value from the acquired 
data using this symmetric constraint. 
FIG. 6 is shown a prone of the K-space data .vertline.S(k.sub.x, 
129).vertline. for a rectangular object. It is symmetric about and peaked 
at k.sub.x =129. The curve on the top part of FIG. 6 is the exponential 
function e.sup.-2 t/T2 with T2=100 ms and the sampling window width 
T.sub.x =10 ms. If there is no noise and there is only a global T2 effect, 
any two values of .vertline.S(k.sub.x,129).vertline. from the two points 
symmetric about the peak would provide an estimate of the global T2 value. 
But there are many factors, including measurement noise and phase shifts, 
which affect the amplitude symmetry property. Therefore, one may first 
calculate the sums of the acquired signal amplitude 
.vertline.S(k.sub.x,129).vertline. over tow equal time spans symmetric 
about the echo peak. For example, if S(k.sub.x, 129) has N.sub.x samples 
in the k.sub.x direction with the sampling window time T.sub.x, we 
calculate the average amplitude A1 and A2 over a fraction of T.sub.x as 
##EQU10## 
where N.sub.T2,x is the number of data samples in the T2 estimate window 
T.sub.w. An average T2 value can be calculated from these two amplitude 
values as 
EQU T2=(N.sub.x -N.sub.T2,x)T.sub.p /ln(A1/A2)) (25) 
where T.sub.p is the period of the data sampling in the k.sub.x direction 
and thus T.sub.x =N.sub.x T.sub.p. From equation (1), it is known that 
##EQU11## 
and thus the echo signal S(k.sub.x,129) is composed of contributions from 
all elements s(n.sub.x,n.sub.y) of the object. Therefore, (25) provides 
only a global T2 estimate of the object. 
The derivation of equation (25) is based on an assumption that the signal 
has a flat Fourier spectrum over the T2 estimate window T.sub.w. For real 
signal spectrums with arbitrary shapes, the estimated global T2 will 
deviate from the true global T2 value and the difference between the true 
T2 value and estimated T2 value is affected by the width T.sub.w and the 
shape of the signal spectrum S(k.sub.x,k.sub.y). When there is no noise 
and no phase error, a shorter estimate window generates a smaller estimate 
difference (bias) from the true T2 value. When there are noise and other 
disturbing factors, the estimate window cannot be too short. It is the 
summation (low-pass filtering) process that reduces the disturbing effects 
such as noise and phase shifts, and helps obtain a stable T2 estimation. 
This is a typical tradeoff between the bias and the variance in the 
estimation problem [L. L. Scharf, Statistical Signal Processing: 
Detection, Estimation, and Time Series Analysis, Addison-Wesley Publishing 
Company, 1990] In practice, the T2 estimate window size T.sub.w should be 
adjusted according to the MR imaging conditions. 
To reduce the effect of spectrum shape on the T2 estimate, one may use a 
root filtering technique to compress the dynamic range of 
S(k.sub.x,k.sub.y). That is, the two variables A1 and A2 are calculated by 
##EQU12## 
where the root factor .alpha.(&gt;0) determines the degree of compression and 
a smaller .alpha. generates a smoother spectrum. Since the estimated T2 
value is affected only by the ratio of A1 and A2, the normalizing factor 
1/N.sub.T2,x used in equation (24) has been omitted in equation (27) 
above. Then the average T2 value is determined as 
EQU T2=.alpha.(N.sub.x -N.sub.T2,x)T.sub.p /ln(A1/A2) (28) 
Some estimated T2 values using equations (27) and (28) are listed in Table 
1 below for different values of the T2 estimate window width T.sub.w and 
the root factor .alpha.. The time-domain signal has a Sinc shape as shown 
in FIG. 6 and a T2 value of 100 ms. A sampling window time T.sub.x =10 ms 
has been used. It can be seen that the shorter window T.sub.w and the 
smaller .alpha. provide better estimates for the T2 value. 
A1 and A2 could also be calculated over a wider window in k.sub.y direction 
such that 
##EQU13## 
where N.sub.T2,y determines the width (2N.sub.T2,y +1) of the T2 estimate 
window in the k.sub.y direction. When N.sub.T2,y =0, equation (29) reduces 
to equation (27). After a global T2 value has been estimated, the 
frequency response H(k.sub.x,k.sub.y) of the T2 distortion filter can be 
determined using equation (7), according to the MR sequence specification 
of echo times TE(k.sub.y) and the data sampling period T.sub.p. 
TABLE 1 
______________________________________ 
Estimated T2 Values Using Different Values of T.sub.w and .alpha. 
T2 = 100.00 ms, T.sub.x = 10.00 ms 
.alpha. 
T.sub.w 2.00 1.00 0.50 0.25 
______________________________________ 
T.sub.x /A 
109.65 ms 105.22 ms 102.75 ms 
104.44 ms 
T.sub.x /8 
102.85 ms 104.89 ms 101.15 ms 
100.64 ms 
T.sub.x /16 
104.76 ms 104.32 ms 100.84 ms 
100.48 ms 
______________________________________ 
The approach discussed above makes the estimation of H(k.sub.x,k.sub.y) 
from the acquired image data directly and thus does not require extra data 
acquisition. But this approach can only estimate the amplitude distortion 
function of the T2 filter and therefore the resulting Wiener filter based 
on these estimates can only reduce the amplitude distortion caused by the 
T2 effect. 
Linear prediction techniques have been successfully used for time-series 
analysis, high resolution spectrum estimation, speech and image signal 
coding, and many other applications [S. M. Kay, Modern Spectral 
Estimation: Theory & Application, Prentice Hall, Englewood Cliffs, N.J. 
07632, 1988] The technique has also been used for MR data extrapolation to 
reduce the truncation artifacts and improve spatial resolution [M. R. 
Smith, S. T. Nichols, R. M. Henkelman and M. L. Wood, "Application of 
Autoregressive Moving Average Parametric Modeling in Magnetic Resonance 
Image Reconstruction," IEEE Trans. Med. lmag., Vol. 5, pp. 132-39, 1986; 
J. E Martin and C. F. Tirendi, "Modified Linear Prediction Modeling in 
Magnetic Resonance Imaging," J. Magn. Reson., Vol. 82, pp. 392-99, 1989; 
E. M. Haacke, Z. Liang and S. H. Izen, "Super Resolution Reconstruction 
Through Object Modeling and Parameter Estimation," IEEE Trans. ASSP., Vol. 
37, pp. 592-95, 1989]. The application of linear prediction to local T2 
correction of both amplitude and phase distortions is discussed below. 
Given p observed data samples x(n),x(n+1) . . . x(n+p-1), the unobserved 
data sample x(n+p) can be predicted using one-step forward linear 
prediction as 
##EQU14## 
Similarly, the unobserved data sample x(n-1) can be predicted using 
one-step backward linear prediction as 
##EQU15## 
In both cases, the predicted data sample is a linear combination of the p 
observed samples. In accordance with the Wiener filter, forward prediction 
coefficients {.alpha..sub.1,.alpha..sub.2 . . . .alpha..sub.p } and 
backward prediction coefficients {.beta..sub.1,.beta..sub.2, . . . 
.beta..sub.p } are chosen to minimize the mean square error between the 
original signal x(n) and the predicted signal x(n) 
EQU .sigma..sup.2.sub.p =.epsilon.[.vertline.e.sub.p (n).vertline..sup.2 
]=.epsilon.[.vertline.x(n)-x(n).vertline..sup.2 ] (32) 
In fact, Wiener filter coefficients and linear predictor coefficients are 
both defined by the Wiener-Hopf equations. [S. Haykin, Adaptive Filter 
Theory, Second Edition, Prentice Hall, Englewood Cliffs, N.J. 07632, 1991] 
There are several methods that may be used for determination of the 
coefficients of a linear predictor. For example, the autocorrelation 
method may be desirable because the resulting predictor is guaranteed to 
be stable and the very efficient Levinson-Durbin algorithm can be used for 
determination of the predictor coefficients. [L. B. Jackson, Digital 
Filters and Signal Processing, Second Edition, Kluwer Academic Publishers, 
1989; S. Haykin, Adapative Filter Theory, Second Edition, Prentice Hall, 
Englewood Cliffs, N.J. 07632, 1991] 
Assume that the global T2 corrected MR data by the Wiener filter are 
represented by a 2-D function S(k.sub.x,k.sub.y) where k.sub.x is the 
index of sampling points in the frequency encode direction and k.sub.y is 
the index of sampling points in the phase encode direction. First, the 
inverse Fourier transform of the time-domain data S(k.sub.x,k.sub.y) is 
taken with respect to k.sub.x to obtain the hybrid-domain data set 
X(n.sub.x,k.sub.y) and then the data X(n.sub.x,k.sub.y) is considered for 
each n.sub.x value. Given the discontinuity positions in the frequency 
response of T2 distortion filters, the linear prediction is used to 
produce data samples across these positions. For example, if the T2 
distortion filter is as shown in FIG. 2, a total of 31 data samples from 
X(n.sub.x,130) to X(n.sub.x,160) can be used to predict data samples on 
the other side of the discontinuity position by forward linear prediction, 
starting from k.sub.y =161. The prediction is performed from data samples 
of lower frequency to data samples of higher frequency since low frequency 
data samples usually have higher signal-to-noise ratio and therefore the 
resulting predicted data have lower prediction errors. 
From the old data sample X(n.sub.x,161) and the first predicted sample 
X(n.sub.x,161), the local phase distortion factor can be estimated as 
EQU .PSI.(n.sub.x)=phase[X((n.sub.x,161)]-phase[X(n.sub.x,161)](33) 
This local phase distortion factor is then used to make a local zero-order 
phase correction by multiplying all the old data samples of the second 
echo from k.sub.y =161 to k.sub.y =192. The resulting phase corrected data 
are denoted as X(n.sub.x,k.sub.y). That is 
EQU X(n.sub.x,k.sub.y)=exp(+i.PSI.(n.sub.x)).X(n.sub.x,k.sub.y) for 
161.ltoreq.k.sub.y .ltoreq.192 (34) 
The estimated local phase distortion factor .PSI.(n.sub.x) from different 
column data X(n.sub.x,k.sub.y) can be first low-pass filtered and the 
resulting smoothed phase estimates are then used in equation (34) for 
phase correction. This enhances the stability of local phase correction. 
Since the prediction usually has quite precise results for data samples 
close to the starting position (k.sub.y =161) in the echo train boundary 
and becomes less accurate as k.sub.y moves away from the boundary, 
predicted data X(nx,k.sub.y) are combined with zero-order phase corrected 
data x(n.sub.x,k.sub.y) as a weighted average over a data merging band to 
obtain local amplitude and phase corrected data X(n.sub.x,k.sub.y). That 
is, 
EQU X(n.sub.x,k.sub.y)=w.sub.1 (k.sub.y)X(n.sub.x,k.sub.y)+w.sub.2 
(k.sub.y)X(n.sub.x,k.sub.y) for 161.ltoreq.k.sub.y .ltoreq.(161+N.sub.m) 
(35) 
where N.sub.m is the width of the data merging band. Two weighting factors 
w.sub.1 (k.sub.y) and w.sub.2 (k.sub.y) are some monotonously decreasing 
and increasing functions of k.sub.y, respectively. They are related as 
w.sub.1 (k.sub.y)+w.sub.2 (k.sub.y)=1.0 for all k.sub.y. The use of these 
weighting functions are intended to ensure a smooth transition from the 
predicted data to the originally observed data. In this study, w.sub.1 
(k.sub.y) and w.sub.2 (k.sub.y) are chosen as linear functions of k.sub.y 
and thus 
##EQU16## 
Although other weighting functions such as quadratic and exponential 
functions have been tried to make different weightings on the two data 
sets, it has been found that they do not make a significant performance 
difference. The determination of the data merging band N.sub.m is affected 
by several considerations. A wider merging band usually provides stronger 
ringing artifact reduction but it also results in bigger changes in the 
original signal spectrum. Also, its value is limited by the data number in 
each echo group. A band around ten data samples gives good results for 
four-echo HI images of 256 phase encoding projections. 
A similar local amplitude and phase correction is performed around other 
discontinuity positions in the data set X(n.sub.x,k.sub.y) except that for 
the discontinuity positions of k.sub.y &lt;129, a backward linear prediction 
is used to estimate data samples across these positions (based also on the 
data samples with lower frequencies). Finally, the inverse Fourier 
transform of the hybrid domain data X(n.sub.x,k.sub.y) is taken with 
respect to k.sub.y to reconstruct the HI image. As shown by the 
subsequently discussed examples, this processing effectively reduces 
effects from amplitude and phase discontinuities in the frequency response 
of the effective T2 distortion filter and therefore reduces ringing 
artifacts in the reconstructed HI image. 
First, a one-dimensional image consisting of two rectangular objects with 
T2 values of 100 ms and 200 ms respectively is considered. A sequence with 
the band-pass T2 filter response defined in FIG. 4 has been used for data 
generation. The echo times of four echoes are TE(i)=20.i ms for i=1,2,3,4. 
It has also assumed that signal components from the second echo have a 
90.degree. phase shift with respect to those from the other three echoes. 
The reconstructed image is shown in FIG. 7A. Comparing with the original 
image shown by dotted line also in FIG. 7A, it can also be seen that the 
simulated HI image has amplitude attenuation, resolution loss and serious 
ringing artifacts. Using a global T2 value of 500 ms in the Wiener 
restoration filter, amplitude distortions on both objects have been 
reduced as shown in FIG. 7B. Note that since the noiseless case is now 
being considered, the Wiener filter is just a simple inverse filter. 
Because the phase error has not been corrected, it is noted that image 
resolution loss and ringing artifacts still exist. If local amplitude and 
phase correction is applied to the distorted image data directly, a local 
corrected image is obtained as shown in FIG. 7C with improved resolution 
and reduced ringing artifacts. In this case, 30 data samples were used to 
calculate the autocorrelation function (ACF) with a biased ACF estimator 
[L. B. Jackson, Digital Filters and Signal Processing, Second Edition, 
Kluwer Academic Publishers, 1989] For the predictor order selection, there 
are many techniques available [S. M. Kay, Modern Spectral Estimation: 
Theory & Application, Prentice Hall, Englewood Cliffs, N.J. 07632, 1988] 
The model order selection rule N/3.ltoreq.p.ltoreq.N/2 by Ulrych and 
Bishop [T. J. Ulrych and T. N. Bishop, "Maximum Entropy Spectral Analysis 
and Autoregressive Decomposition," Rev. Geophys. Space Phys., Vol. 13, pp. 
183--200, 1975] where N is the number of data samples used for the ACF 
estimation was used here. Therefore, the order p of the linear predictor 
for this case has been chosen to be 12. The predictor coefficients are 
determined with the Levinson-Durbin algorithm. Then new data samples can 
be predicted using equations (30) or (31) based on known data samples and 
previously predicted data samples. The estimated local phase distortion 
using equation (33) from the forward predicted data X(161) and the old 
data sample X(161) is 97.15.degree. and the estimated local phase 
distortion is 92.54.degree. from the backward predicted data X(96) and the 
old data X(96). The phase estimate error is less than 10%. As described 
above, these phase estimates are used to make the zero-order phase 
correction on data of k.sub.y .gtoreq.161 and k.sub.y .ltoreq.96, 
respectively. Then the phase corrected data are merged with the estimated 
data to form a new data set for final image reconstruction. A data merging 
band of ten data samples has been used for all discontinuity positions in 
FIG. 4. 
Finally, the local amplitude and phase correction technique is employed on 
data processed by the global T2 amplitude correction. The resulting 
corrected HI image by this two-stage procedure is given in FIG. 7D. It is 
clear from these figures that the original HI image degradation caused by 
amplitude attenuation, resolution loss and ringing artifacts has been 
effectively improved by this combined global and local data processing 
technique (except near the sharp edges of the objects). 
The proposed method has been applied to an image of a phantom acquired in a 
low-field-strength (0.064 T) permanent magnet imager (ACCESS.TM. 
Toshiba-America MRI Inc.). The phantom object is a rectangular container 
full of mineral oil with a T2 value of about 100 ms. 
The asymmetric Fourier imaging (AFI) approach has been employed to further 
reduce the imaging time. A total of 144 phase encode projections has been 
acquired by 36 data acquisition shots using a four-echo band-pass sequence 
with echo times TE(i)=10.i ms for i=1,2,3,4. The data is then conjugated 
to a full image size of 256.times.256 pixels. The resulting T2 filter has 
a frequency response similar to that shown in FIG. 4 and phase encode is 
in the vertical direction. Because of amplitude and phase distortions 
introduced by the T2 distortion filter, the reconstructed image has 
serious tinging artifacts along the phase encoding direction as evidenced 
in FIG. 8A. For demonstration purposes, one of the worst cases of ringing 
artifacts is illustrated. 
With a global T2 estimate window of N.sub.T2,x =64, N.sub.T2,y =2 and the 
root factor .alpha.=0.5, the global T2 value estimated using equations 
(27) and (28) is 127 ms. This estimated T2 value is used in a Wiener 
filter for global amplitude correction on the originally acquired 144 
projections. Now, 21 data samples are used for the ACF estimate and the 
predictor order is ten. For the AFI data case, both the global and the 
local corrections should be performed before conjugation and this will 
save computations needed for the data correction. For the given AFI data 
of 144 phase projections, the saving in the global T2 amplitude correction 
is about 44% compared with the full size data of 256 phase projections. 
For local corrections, the saving is 50% since only three instead of six 
discontinuities need to be processed. T2 corrected data are conjugated to 
the full image size and the improved image as shown in FIG. 8B is obtained 
which has much less ringing artifact. The standard deviation of the noise 
has been reduced form 144 to 55 by the Wiener filtering. 
As a further example, one image from a sequence with both sin echo and 
gradient echo has been tested. The sequence consists of three echoes in 
the echo train, one spin echo and two gradient echoes. The spin echo has 
an echo time 25 ms and generates the signal components S(k.sub.x,k.sub.y) 
for 86.ltoreq.k.sub.y .ltoreq.170. The first graident ehco has an echo 
time 19 ms and generates signal components S(k.sub.x,k.sub.y) for 
1.ltoreq.k.sub.y .ltoreq.85. The second gradient echo has an echo time 32 
ms, generating the remaining signal components. The image was acquired in 
a high-field-strength (1.5 T) MR image (Toshiba MRT200/FXIII). Similar to 
the fast spin echo case, because of T2 and T2* effects, the tinging 
artifacts in the reconstructed image are obvious as shown in FIG. 9A, 
which seriously degrade the image quality. In this image, phase encoding 
is in the horizontal direction. 
Although the echo times given above seem to define a ramp T2 filter, the 
actual T2 filter has a low-pass shape since the dephasing effects (T2* 
effects) caused by field inhomogeneity results in faster decay for the 
gradient echo signal components than from the T2 effect. In this case, the 
T2 amplitude attenuation factors A.sub.g1 for the first gradient echo and 
A.sub.g2 for the second gradient echo are estimated using signals from 
three extra zero phase encoding projections in the end of the sequence. 
They are determined by 
##EQU17## 
where S.sub.s (k.sub.x,129) is the zero-phase encoding signal from the 
standard spin echo, S.sub.g1 (k.sub.x,129) and S.sub.g2 (k.sub.x,129) are 
the zero-phase encoding signals from the first and the second gradient 
echoes, respectively. For the given image, estimated attenuation factors 
are A.sub.g1 =0.84 and A.sub.g2 =0.72 with respect to the spin echo 
signal. These attenuation factors were used in the Wiener filter to make a 
global amplitude correction. This estimation method for global T2 
amplitude attenuation factors can also be used for the FSE case. Then, the 
Wiener filter and the linear prediction are applied to this image and the 
resulting image after the two-stage processing is shown in FIG. 9B (which 
has imroved resolution and fewer tinging artifacts). The standard 
deviation of noise in the jost homogeneous tissue region has also been 
reduced from 149 to 118. 
In the above experiments, the Wiener filter and the linear prediction 
technique have been used together to effectively improve HI image quality. 
For some applications, the Wiener filter can be used only to perform 
partial global T2 amplitude correction. For example, the sequence with a 
band-pass frequency response as shown in FIG. 4 is often used to emphasize 
signal components with middle range frequencies and obtain an edge enhance 
dimage. In this case, for preserving the desired objective, a global 
Wiener T2 amplitude correction can be employed only for signal components 
from the third and fourth echoes to further increase high frequency 
components (leaving the low frequency components from the second echo 
untouched). Then the linear prediction technique is used to reduce tinging 
artifacts. 
The combined use of a Wiener filter and a linear prediction technique for 
T2 restoration and noise suppression has been presented. The effectiveness 
of this method in improving HI image quality has been illustrated using 
some experimental results in which only one estimated global T2 value (and 
therefore a simple frequency response function H(k.sub.x,k.sub.y) of T2 
filter) is available for the T2 amplitude restoration. This simplified 
model may result in under and/or over compensations for certain objects in 
a given image. Methods for more accurate estimate of T2 values and 
distributions are required to obtain better T2 correction. For local 
amlpitude and phase correction, the simple linear prediction has been used 
for computation simplification. Other modeling techniques such as 
autoregressive moving average (ARMA) [M. R. Smith, S. T. Nichols, R. M. 
Henkelman and M. L. Wood, "Application of Autoregressive Moving Average 
Parametric Modeling in Magnetic Resonance Image Reconstruction," IEEE 
Trans. Med. Imag., Vol. 5, pp. 132-39, 1986; S. M. Kay, Modern Spectral 
Estimation: Theory & Application, Prentice Hall, Englewood Cliffs, N.J. 
07632, 1988; L. B. Jackson, Digital Filters and Signal Processing, Second 
Edition, Kluwer Academic Publishers, 1989] and nonlinear prediction with 
neural network [H. Yan and J. Mao, "Data Truncation Artifcat Reduction in 
MR Imaging Using a Multilayer Neural Network," IEEE Trans. Med. Imag., 
Vol. 12, pp. 73-77, 1993 ] can be employed to improve the results. 
As indicated above, this invention may be implemented by suitably 
programming the image data processor of a conventional MRI system so as to 
implement the above-described two-stage T2 correction (global, then local) 
process. The program may be provided as optional additional subroutines if 
desired or incorporated as permanently used non-optional added data 
processing routines. In any such case, the programs may be conventionally 
written by those skilled in the art to straightforwardly implement the 
above-described processes. Accordingly, no further detailed description of 
such computer programs is believed to be necessary. 
While only a few exemplary embodiments of this invention have been 
described in detail, those skilled in the art will appreciate that 
variations and modifications in these examples may be made while yet 
retaining some of the novel features and advantages of the invention. All 
such variations and modifications are intended to be included within the 
scope of the appended claims.