Figure 1.

Implementation of iterative model‐based reconstruction (step 3 of proposed framework) using data from both blip‐up and blip‐down EPI scans. (a) Given the input k‐space data $\tilde{\mathbf{Y}}={[{\tilde{\mathbf{Y}}}_1{\tilde{\mathbf{Y}}}_2]}^{\text{T}}$ that have been phase corrected in step 2, in each iteration of the conjugate gradient (CG) algorithm, we iterate back and forth between the k‐space and image x by encoding operator E = [E ₁ E ₂ ] ^T and its adjoint E^H that include the center frequency offset corrected B₀ field. The convergence is achieved when the residual $r=\frac{{\mathbf{E}}^{\mathbf{H}}\mathbf{Ex}-{\mathbf{E}}^{\mathbf{H}}{\tilde{\mathbf{Y}}}_2}{{\mathbf{E}}^{\mathbf{H}}{\tilde{\mathbf{Y}}}_2}$ in the current iteration becomes smaller than ϵ (ϵ being a small number) giving the final distortion corrected output image $\dot{\mathbf{x}}$. (b) Details of the forward encoding operator E _i that takes the input single image x to the multicoil output k‐space Y _i for phase‐encoding direction i, i = 1 corresponds to the blip‐up and i = 2 corresponds to the blip‐down scans, respectively. The image x is first multiplied by the coil sensitivity C ^j (j = 1, 2,..,J). This is followed by a modified Fourier transform operator G that maps the product of image and coil sensitivity from image to Fourier space, taking into account the susceptibility effects determined by B₀ field and k‐space sampling times t_i, resulting in k‐space data ${\mathbf{Y}}_{\text{i}}^j$ (j = 1, 2,..,J). (c) Details of the adjoint encoding operator ${\mathbf{E}}_{\text{i}}^H$ that takes the multicoil k‐space data ${\mathbf{Y}}_{\text{i}}^j$ (j = 1, 2,..,J) to single image x for phase‐encoding direction i, i = 1 corresponds to the blip‐up and i = 2 corresponds to the blip‐down scans, respectively. Each individual coil k‐space data ${\mathbf{Y}}_{\text{i}}^j$ is transformed to image space by the adjoint of a modified Fourier transform operator (G^H). The resulting images are individually multiplied by complex conjugated coil sensitivities ${({\mathbf{C}}_i^j)}^{\ast }$ (j = 1, 2,..,J) and summed to give the final image x