## 4 Respiratory Motion Correction Strategies
### Higher Order Correction  
While translational correction is adequate in the majority of cases if the CMRA scan is gated to the most quiescent respiratory phase, rotation and non-rigid motion correction such as affine correction has to be considered if no respiratory gating is employed [7], which is desirable to minimize scan time. Affine models describe a combination of linear transformations including rotation, shear and scale, as well as translation. A 3D affine transformation \(A\) transforms the point r = [_x_, \(y\), \(z\), _1_]^T to r' = [_x_, _y_', \(z\), _1_]^T_ according to:  
\[\begin{bmatrix}x^{\prime}\\ y^{\prime}\\ z^{\prime}\\ 1\end{bmatrix}=\begin{bmatrix}M_{00}&M_{01}&M_{02}&T_{x}\\ M_{10}&M_{11}&M_{12}&T_{y}\\ M_{20}&M_{21}&M_{22}&T_{z}\\ 0&0&0&1\end{bmatrix}\begin{bmatrix}x\\ y\\ z\\ 1\end{bmatrix} \tag{3}\]  
where \(t=[T_{\infty}\ T_{\infty}\ T_{z}]^T\) denotes the 3D translational components and the sub-matrix \(\mathbf{M}=\mathbf{A}\)(4;4) defines the rotation, scale and shear. Prospective 3D affine correction has been implemented for CMRA in which the affine motion model is calculated from a calibration scan and a d1D NAV used to steer the model during the CMRA scan, which allows for data acquisition throughout almost the entire respiratory cycle [50]. Previously described approaches using respiratory binning have been used to retrospectively correct for affine and non-linear motion, either in image space [11; 67] or in _k_-space [68; 69], with reported navigator efficiencies approaching 100%. Affine correction in _k_-space can be performed based on the following relationship:  
\[F^{\prime}(\mathbf{k^{\prime}})=\frac{e^{\frac{z\pi t\mathbf{k^{\prime}}}{\mathbf{M^{ \prime}}}}{\mathbf{M^{\prime}}}}{\left|\det(\mathbf{M})\right|}F(\mathbf{M^{-T}}\mathbf{k}) \tag{4}\]  
where _F_(\(\mathbf{k}\)) denotes the uncorrected and _F_(\(\mathbf{k^{\prime}}\)) the corrected _k_-space. Non-linear correction can be performed by generalizing the signal equation to also incorporate the different motion states [86]. In this way, the motion corrupted _k_-space _F_(\(\mathbf{k}\)) can be written as:  
\[F(\mathbf{k})=\sum_{s}\mathbf{W}_{s}\mathbf{E}\mathbf{P}_{s}\mathbf{o} \tag{5}\]  
where \(\mathbf{o}\) represents the motion-free imaged object, \(\mathbf{E}\) the data encoding matrix, \(\mathbf{W}_{s}\) and \(\mathbf{P}_{s}\) the non-linear warping and data sampling operators, respectively, at motion state \(s\). The motion corrected image can be obtained by performing matrix inversion on Equation (5). A generalized schematic of motion correction using respiratory binning is shown in Figure 6, where images reconstructed for different bins can be used to calculate and thereafter apply affine or non-linear transformations between each bin, to allow for 100% navigator efficiency.  
A drawback of higher order motion correction is that the data post-processing becomes increasingly computationally expensive and time-consuming as respiratory motion estimation and correction is performed with more degrees of freedom. Excessive post-processing durations may impede the clinical utility of advanced motion correction strategies, although in many cases the post-processing can be expedited by the use of parallel computing [87].