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Short introduction
HySCO is an academic software for the correction of susceptibility artifacts in diffusion weighted images based on a reversed gradient based acquisition scheme. It is developed by Lars Ruthotto and Jan Modersitzki as an addon to the registration toolbox FAIR.
HySCO requires the acquisition of a pair of images with reversed phaseencoding gradients that are oppositely affected by distortions. From the socalled "blipup" and "blipdown" image data, HySCO estimates the fieldinhomogeneity by solving a tailored image registration problem that incorporates a physical model of inhomogeneity artifacts in spinecho MRI.
HySCO's namegiving component is a special nonlinear regularization functional, which is inspired by hyperelasticity. It ensures smoothness of the field inhomogeneity and invertibility of the geometrical transformations regardless of the actual choice of regularization parameters.
Preprocessing
Use EC and Motion Correction to minimise eddy current and motion artefacts for each diffusion dataset separately (i.e. for blipup and blipdown). The EC and motion corrected images will have a prefix 'r'.
Overview over HySCO

Reference blipup image: Select one reference image volume acquired with blipup phase encoding direction (b=0 image is recommended, because it has higher SNR). Note that the ''true'' blipup and down directions can be unknown! Just define here one DTI dataset as ''blipup'' and the other as 'blipdown'. The field inhomogeneity is estimated by minimizing the sumofsquared difference between this image and the blipdown image chosen below and regularization.

Reference blipdown image: Select one reference image volume acquired with blipdown phase encoding direction (see above for details).

Other blipup images: (optional) Choose ''other image volumes'' acquired with blipup phase encoding direction. The data is corrected by applying the transformation estimated by the reference blipup/down data. If an equal number of blipup and blipdown data is provided as ''other image volumes'', you may also want to disable ''Apply to other images'' (see 7).

Other blipdown images: (optional) Choose ''other image volumes'' acquired with blipdown phase encoding direction.

Dimension of phaseencoding: Specify the phaseencoding direction of your data (i.e. the direction in which the susceptibility distortions will be greatest). Default is ydirection.

Maximal data resolution: Choose the finest discretization level for field inhomogeneity estimation. If set to ''full'' a multilevel strategy with three discretization levels is used, where the resolution on the finest level equals the data resolution. To save computation time, choose ''half''. The multilevel scheme will be stopped after the second level (i.e. half of data resolution) and the inhomogeneity estimate will be interpolated to the data resolution.

Apply to other images: If set to ''no'' and if the same number of diffusionweighted images is provided for blipup and blipdown, the field inhomogeneities are estimated for each set of blipup/down images separately (This might be useful to correct in addition to susceptibilityinduced distortion for the distortions due to nonlinear eddy current fields). To this end, the fieldinhomogeneity estimated from the nondiffusion weighted images is used as a starting guess for minimization of the distance between the respective diffusionweighted image pairs. Optimization is only carried out on the finest discretization level to save computation time.

Smoothing of splineinterpolation: Choose parameter theta that balances between the data fit and the smoothness of the cubic Bspline approximation of the image data. For theta equal to zero a standard cubic Bspline interpolation is used. For positive theta data is only approximated, but the image representation is smoother. Thus, theta can be used to adjust for noise level of the data. Note that this scheme is only used for the optimization and in particular that the corrected image data is obtained by resampling using a standard trilinear interpolation. For details see Section 3.4 in: Modersitzki, J. FAIR: Flexible Algorithms for Image Registration. Society for Industrial and Applied Mathematics (SIAM); 2009.

Weight for "diffusion" regularizer: For larger values of alpha, the computed solution will in general be smoother, but the image distance between the corrected blipup and blipdown image will be larger.

Weight for "Jacobian" regularizer: By design the value of this regularization functional grows to infinity when the Jacobian determinant of either of the geometrical transformations approaches zero. Thus, for larger/smaller values of beta, the range of the Jacobian determinants, which translates to the maximum compression/expansion of volume, becomes smaller/larger. However, for any beta that is greater than zero both transformations will be invertible.
Examples and Tutorials
References and further literature
Ruthotto, L, Kugel, H, Olesch, J, Fischer, B, Modersitzki, J, Burger, M, and Wolters, C H. Diffeomorphic Susceptibility Artefact Correction of DiffusionWeighted Magnetic Resonance Images. Physics in Medicine and Biology, 57(18), 57155731; 2012.
Ruthotto, L, Mohammadi, S, Heck, C, Modersitzki, J, and Weiskopf, N. HySCO  Hyperelastic Susceptibility Artifact Correction of DTI in SPM. Presented at the Bildverarbeitung fuer die Medizin 2013.
J. Modersitzki: FAIR: Flexible Algorithms for Image Registration. SIAM, Philadelphia, 2009.
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