Seiji Kumazawa^{1}, Takashi Yoshiura^{2}, Takumi Tanikawa^{1}, and Yuji Yaegashi^{1}

Our purpose was to develop an image-based method for undistorted image estimation from the distorted EPI image using T1 weighted image. Our basic idea to estimate the field inhomogeneity map is to reproduce the distorted EPI image, and estimates the undistorted image using the estimated field inhomogeneity map based on the signal equation in a single-shot EPI k-space trajectory. The value of the NRMSE between the measured EPI and synthesized EPI was 0.017, and both images were in good agreement. Results demonstrate that our proposed method was able to perform a reasonable estimation of the field inhomogeneity map and undistorted EPI image.

We propose the
iterative-method to estimate an undistorted image and its associated field inhomogeneity
map using conjugate gradient algorithm. Our basic idea to estimate the field
inhomogeneity map in EPI is to reproduce the distorted EPI image based on the
signal equation in a single-shot EPI k-space trajectory. To synthesize the
distorted EPI image, we make use of the image-based method^{3} which
can reduce the computation time compared to the k-space texture method^{4}.
The MR signal in synthesized EPI was calculated on a voxel-by-voxel basis using
the estimated T2 value and field inhomogeneity ΔB by the following
equation:

$$I(x,y,T2, \Delta B)=\sum_{u=0}^{M-1} \sum_{v=0}^{N-1} A(u,v)\cdot exp(-\frac{TEeff}{T2(u,v)})\cdot sinc (\pi (x-u-\frac{\Delta B(u,v)}{\Delta x \cdot Gx})) \cdot sinc (\pi (y-v-\frac{\Delta B(u,v)}{\Delta y \cdot Gy}))$$

where
Δx
and Δy
are pixel spacing in x and y direction, respectively. G_{x} is the gradient in the x
direction, and G_{y} = G_{b}τ /Δt_{y} (G_{b} is the average blip gradient in the y direction during the duration τ, and Δt_{y} is the time intervals between adjacent points in the phase-encoding
directions in k-space). To estimate T2 value and ΔB at
each point, we defined the cost function using the synthesized EPI image and
the measured EPI image with geometric distortion as follows.

$$J(T2, \Delta B) = \sum_{x=0}^{M-1}\sum_{y=0}^{N-1}|Y(x,y)-I(x,y,T2, \Delta B) |^2 +\beta_1 R_1(\Delta B)+\beta_2 R_2 (T2)$$

where Y(x,y)
is the measured EPI image, R_{1}(ΔB(x,y))
and R_{2}(T2(x,y)) are regularization terms. To result in a relatively smooth
field inhomogeneity map, R_{1}(ΔB(x,y))
penalizes the roughness of the estimated field inhomogeneity map. To obtain the
structural information in the distorted area in the measured EPI image, we used
the same slice of the T1WI as the measured EPI image. We generated the initial
estimated T2map by applying the gray-scale inversion and histogram
specification^{5} to the T1WI. We used Tikhonov regularization for R_{2}(T2(x,y)),
the initial estimated T2map was used as the reference image in Tikhonov
regularization. The estimation of T2 and ΔB maps was performed to minimize the cost function using
an iterative conjugate gradient
algorithm. The spin echo (SE) EPI and T1WI data of a healthy volunteer were
acquired using a 1.5-tesla clinical Siemens scanner. The SE EPI data was
obtained by a single-shot EPI pulse sequence (FOV: 230 mm, TR=8600ms, TE=119ms,
128×128 in-plane resolution, 3 mm thickness). Three
dimensional T1WI covering the same area in EPI was obtained by MPRAGE sequence
(FOV: 230 mm, TR=2090ms, TE=3.93ms, TI=1100ms, FA=15°, 256×256 in-plane resolution,
1 mm thickness). To evaluate the performance of our methods, we used the
normalized root mean square error (NRMSE) between the measured EPI and synthesized
EPI.

1. Jezzard P, Balaban RS. 1995. Correction for geometric distortion in echo planar images from B0 field variations. Magn Reson Med 34:65-73.

2. Sutton BP, Noll DC, Fessler JA, 2004. Dynamic field map estimation using a spiral-in/spiral-out acquisition. Magn Reson Med. 51:1194-1204.

3. Kumazawa S, Yoshiura T, Kikuchi A, et al. 2017. An improved image-based method for field inhomogeneity map in distorted brain EPI image. Proc ISMRM, 1526.

4. Kumazawa S, Yoshiura T, Honda H. 2015. Image-based estimation method for field inhomogeneity in brain echo-planar images with geometric distortion using k-space textures. Concept Magn Reson B. 45:142-152.

5. Coltuc D, Bolon P, Chassery JM. 2006. Exact histogram specification. IEEE Trans Image Process. 15:1143-1152.

Figure 1: The convergence of the cost
function over iteration in our estimation process.

Figure 2: (a) Measured EPI image from a
healthy volunteer, (b) the synthesized EPI image by proposed method, and (c)
the absolute difference image between them.

Figure 3: (a) The same slice of the T1 weighted
image as the measured EPI image. (b)
the initial estimated T2map by applying the gray-scale inversion and histogram
specification to the T1WI, (c) estimated undistorted EPI
image, and
(d) the estimated field inhomogeneity map by proposed method.