motionSimToolBox provides four different motion artifact simulation approaches in 3D MRI. This toolbox was used in "Accuracies of Four Simulation Approaches in Reproducing Motion Artifacts and Morphometric Parameter Biases". In tis study, the 3D FatNavs were used to monitor head movements. 

Motion effects were introduced in these simulations either (1) by modifying k-space coordinates (for rotational motion) and data phase (for translational motion), and then obtaining the final image by inverse non-uniform Fourier transform, or (2) by modifying k-space data value by non-uniform Fourier transform according to different head orientations and obtaining the final image by inverse Fourier transform. These two approaches will be referred to as MCP (modification of coordinate and phase) and MDV (modification of data value), respectively.

motionSimulation_method1 adopts MDV based on coil-combined magnitude image.
motionSimulation_method2 adopts MCP based on coil-combined magnitude image.
motionSimulation_method3 adopts MDV based on multi-channel raw k-space data.
motionSimulation_method4 adopts MCP based on multi-channel raw k-space data.

Example data were provided at [https://pan.baidu.com/s/1BjZaf36oMYqYKzB9-_D_Zw?pwd=ctvm]. Please run run_motionSim_demo.m.

Things to note:
1. motionPar
	(1) The motion parameters should be in AFNI 3dvolreg dfile convention, that is
              [n  roll  pitch  yaw  dS  dL  dP]
                    n: shotIndex [1,2,3,4...]. k-space lines are assigned to different shots using the acqInfo.shotIndex parameter.
                    roll  = rotation about the I-S axis }
                    pitch = rotation about the R-L axis } degrees CCW
                    yaw   = rotation about the A-P axis }
                    dS  = displacement in the Superior direction  }
                    dL  = displacement in the Left direction      } mm
                    dP  = displacement in the Posterior direction }
	(2) the displacement parameters should be recalculated if the center of rotation for motion parameter estimation is different from the FOV center by:
                shift=shift+((xform(1:3,1:3)-eye(3))*ds(:))';
	where ds is the center shift (=FOVcenter-Center_of_rotation), xform is the 3*3 rotation matrix obtained from rotational parameters.

2. weight: density compensation matrix for nufft is required when using motionSimulation_method3 and motionSimulation_method4.
   In the demo, it was generated using nufft_weight_MPRAGEmatrix.m