Inverse consistency

Image registration is the process of establishing a common coordinate system between two images, and given two images

${\displaystyle {\begin{aligned}I_{1}:\Omega _{1}\to \mathbb {R} \\I_{2}:\Omega _{2}\to \mathbb {R} \end{aligned}}}$

registering a source image ${\displaystyle I_{1}}$ to a target image ${\displaystyle I_{2}}$ consists of determining a transformation ${\displaystyle f_{1}:\Omega _{2}\to \Omega _{1}}$ that maps points from the target space to the source space.[6] An ideal registration algorithm should not be sensitive to which image in the pair is used as source or target, and the registration operator should be antisymmetric such that the mappings

${\displaystyle {\begin{aligned}f_{1}:\Omega _{2}\to \Omega _{1}\\f_{2}:\Omega _{1}\to \Omega _{2}\end{aligned}}}$

produced when registering ${\displaystyle I_{1}}$ to ${\displaystyle I_{2}}$ and ${\displaystyle I_{2}}$ to ${\displaystyle I_{1}}$ respectively should be the inverse of each other, i.e. ${\displaystyle f_{2}=f_{1}^{-1}}$ and ${\displaystyle f_{1}=f_{2}^{-1}}$ or, equivalently, ${\displaystyle f_{2}\circ f_{1}=\operatorname {id} _{\Omega _{2}}}$ and ${\displaystyle f_{1}\circ f_{2}=\operatorname {id} _{\Omega _{1}}}$, where ${\displaystyle \circ }$ denotes the function composition operator.[7][8]

Real algorithms are not perfect, and when swapping the role of source and target image in a registration problem the so obtained transformations are not the inverse of each other. Inverse consistency can be enforced by adding to the loss function of the registration a symmetric regularisation term that penalises inconsistent transformations[1]

${\displaystyle \int _{\Omega _{2}}\left\Vert f_{2}(f_{1}(x))-x\right\Vert ^{2}\mathrm {d} x+\int _{\Omega _{1}}\left\Vert f_{1}(f_{2}(x))-x\right\Vert ^{2}\mathrm {d} x.}$

Inverse consistency can be used as a quality metric to evaluate image registration results. The inverse consistency error (${\displaystyle ICE}$) measures the distance between the composition of the two transforms and the identity function, and it can be formulated in terms of both average (${\displaystyle ICE_{a}}$) or maximum (${\displaystyle ICE_{m}}$) over a region of interest ${\displaystyle \Omega }$ of the image:[9]

${\displaystyle {\begin{aligned}ICE_{a}&={\frac {1}{\int _{\Omega }\mathrm {d} x}}\int _{\Omega }\left\Vert f_{2}(f_{1}(x))-x\right\Vert \mathrm {d} x\\ICE_{m}&=\max _{x\in \Omega }\left\Vert f_{2}(f_{1}(x))-x\right\Vert .\end{aligned}}}$

While inverse consistency is a necessary property of good registration algorithms, inverse consistency error alone is not a sufficient metric to evaluate the quality of image registration results, since a perfectly consistent mapping, with no other constraint, may be not even close to correctly register a pair of images.[5]

• Arganda-Carreras, Ignacio; Sorzano, Carlos OS; Marabini, Roberto; Carazo, José María; Ortiz-de-Solorzano, Carlos; Kybic, Jan (2006). Consistent and elastic registration of histological sections using vector-spline regularization. International Workshop on Computer Vision Approaches to Medical Image Analysis. Springer. pp. 85–95.
• Bhatia, Kanwal K.; Hajnal, Joseph V.; Puri, Basant K.; Edwards, A. David; Rueckert, Daniel (2004). Consistent groupwise non-rigid registration for atlas construction. IEEE International Symposium on Biomedical Imaging. IEEE. pp. 908–911.
• Christensen, Gary E; Johnson, Hans J (2001). "Consistent image registration". IEEE Transactions on Medical Imaging. IEEE. 20 (7): 568–582. doi:10.1109/42.932742. PMID 11465464.
• Bender, Edward T.; Tomé, Wolfgang A. (2009). "The utilization of consistency metrics for error analysis in deformable image registration". Physics in Medicine & Biology. 54 (18): 5561–77. Bibcode:2009PMB....54.5561B. doi:10.1088/0031-9155/54/18/014. PMC 2798737. PMID 19717890.
• Leow, Alex; Huang, Sung-Cheng; Geng, Alex; Becker, James; Davis, Simon; Toga, Arthur; Thompson, Paul (2005). "Inverse consistent mapping in 3D deformable image registration: its construction and statistical properties". Biennial International Conference on Information Processing in Medical Imaging.
• Beg, Mirza Faisal; Khan, Ali (2007). "Symmetric data attachment terms for large deformation image registration". IEEE Transactions on Medical Imaging. 26 (9): 1179–1189. doi:10.1109/TMI.2007.898813. PMID 17896591.
• Johnson, Hans J.; Christensen, Gary E. (2002). "Consistent landmark and intensity-based image registration". IEEE Transactions on Medical Imaging. 21 (5): 450–461. doi:10.1109/TMI.2002.1009381. PMID 12071616.
• Rohlfing, Torsten (2012). "Image similarity and tissue overlaps as surrogates for image registration accuracy: widely used but unreliable". IEEE Transactions on Medical Imaging. IEEE. 31 (2): 153–163. doi:10.1109/TMI.2011.2163944. PMC 3274625. PMID 21827972.

• Inverse consistency error