Inverse consistency

In image registration, inverse consistency measures the consistency of mappings between images produced by a registration algorithm. The inverse consistency error, introduced by Christiansen and Johnson in 2001, quantifies the distance between the composition of the mappings from each image to the other, produced by the registration procedure, and the identity function, and is used as a regularisation constraint in the loss function of many registration algorithms to enforce consistent mappings. ^{[1] [2] [3] [4]} Inverse consistency is necessary for good image registration but it is not sufficient, since a mapping can be perfectly consistent but not register the images at all. ^[5]

Definition

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]

References

1. Christiansen and Johnson (2001)
2. Bhatia et al. (2004)
3. Johnson and Christensen (2002)
4. Arganda-Carreras et al. (2006)
5. Rohlfing (2012)
6. The transformation goes in this direction in order to allow resampling the source image in target space.
7. Bender and Tomé (2009)
8. Leow et al. (2005)
9. Beg and Khan (2007)

• 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.

External links

• Inverse consistency error