Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-22T07:11:42.850Z Has data issue: false hasContentIssue false
  • English
  • Français

CONFORMAL IMAGE REGISTRATION BASED ON CONSTRAINED OPTIMIZATION

Published online by Cambridge University Press:  12 January 2021

S. MARSLAND Open the ORCID record for S. MARSLAND [Opens in a new window] ,
R. I. MCLACHLAN Open the ORCID record for R. I. MCLACHLAN [Opens in a new window]  and
M. Y. TUFAIL Open the ORCID record for M. Y. TUFAIL [Opens in a new window]
S. MARSLAND
Affiliation:
School of Mathematics and Statistics, Victoria University of Wellington, Wellington, New Zealand; e-mail: [email protected].
R. I. MCLACHLAN
Affiliation:
School of Fundamental Sciences, Massey University, Palmerston North, New Zealand; e-mail: [email protected].
M. Y. TUFAIL*
Affiliation:
Department of Mathematics, NED University of Engineering and Technology, Karachi, Pakistan.
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Image registration is the process of finding an alignment between two or more images so that their appearances match. It has been widely studied and applied to several fields, including medical imaging and biology, where it is related to morphometrics. In this paper, we present a construction of conformal diffeomorphisms which is based on constrained optimization. We consider a set of different penalty terms that aim to enforce conformality, based on discretizations of the Cauchy–Riemann equations and geometric principles, and demonstrate them experimentally on a variety of images.

Keywords

conformal diffeomorphismsimage registrationconstrained optimizationnumerical computation

MSC classification

Primary: 68U10: Image processing
Type
Research Article
Information
The ANZIAM Journal , Volume 62 , Issue 3 , July 2020 , pp. 235 - 255
Check for updates
Copyright
© Australian Mathematical Society 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arsigny, V., Commowick, O., Ayache, N. and Pennec, X., “A fast and log-Euclidean polyaffine framework for locally linear registration”, J. Math. Imaging Vis. 33 (2009) 222–238; doi:10.1007/s10851-008-0135-9.CrossRefGoogle Scholar
Ashburner, J. and Friston, K. J., “Rigid body registration”, in: Statistical parametric mapping: the analysis of functional brain images (eds. Penny, W., Friston, K., Ashburner, J., Kiebel, S. and Nichols, T.), (Elsevier, Amsterdam, 2007) 49–62; doi:10.1016/b978-012372560-8/50004-8.CrossRefGoogle Scholar
Beg, M. F., Miller, M. I., Trouvé, A. and Younes, L., “Computing large deformation metric mappings via geodesic flows of diffeomorphisms”, Int. J. Comput. Vis. 61 (2005) 139–157; doi:10.1023/b:visi.0000043755.93987.aa.CrossRefGoogle Scholar
Bobenko, A. I., Mercat, C. and Suris, Y. B., “Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green’s function”, J Reine Angew. Math. 583 (2005) 117–161; doi:10.1515/crll.2005.2005.583.117.CrossRefGoogle Scholar
Brown, L. G., “A survey of image registration techniques”, ACM Comput. Surv. 24 (1992) 325–376; doi:10.1145/146370.146374.CrossRefGoogle Scholar
Calvetti, D., Morigi, S., Reichel, L. and Sgallari, F., “Tikhonov regularization and the l-curve for large discrete ill-posed problems”, J. Comput. Appl. Math. 123 (2000) 423–446; doi:10.1016/S0377-0427(00)00414-3.CrossRefGoogle Scholar
Elmore, K. L. and Richman, M. B., “Euclidean distance as a similarity metric for principal component analysis”, Mon. Weather Rev. 129 (2001) 540–549; doi:10.1175/1520-0493(2001)129<0540:EDAASM>2.0.CO;2.2.0.CO;2>CrossRefGoogle Scholar
Fletcher, P. T., “Geodesic regression and the theory of least squares on Riemannian manifolds”, Int. J. Comput. Vis. 105 (2013) 171–185; doi:10.1007/s11263-012-0591-y.CrossRefGoogle Scholar
Glasbey, C. A. and Mardia, K. V., “A review of image-warping methods”, J. Appl. Stat. 25 (1998) 155–171; doi:10.1080/02664769823151.CrossRefGoogle Scholar
Goshtasby, A. A., Image registration principles, tools and methods (Springer, London, 2012); doi:10.1007/978-1-4471-2458-0. CrossRefGoogle Scholar
Greenfield, S. J., “Cauchy–Riemann equations in several variables”, Ann. Sc. Norm. Super. Pisa Cl. Sci. 22 (1968) 275–314, available at https://eudml.org/doc/83459.Google Scholar
Grenander, U. and Miller, M. I., “Computational anatomy: an emerging discipline”, Q. Appl. Math. 56 (1998) 617–694; doi:10.1090/qam/1668732.CrossRefGoogle Scholar
Groetsch, C. W., The theory of Tikhonov regularization for Fredholm equations of the first kind, (Pitman, London, 1984), available at https://www.researchgate.net/publication/233814596_The_theory_of_Tikhonov_regularization_for_Fredholm_equations_of_the_first_kind.Google Scholar
Hsiao, H., Hsieh, C., Chen, X., Gong, Y., Luo, X. and Liao, G., “New development of nonrigid registration”, ANZIAM J. 55 (2014) 289–297; doi:10.1017/S1446181114000091.CrossRefGoogle Scholar
Joshi, S. C. and Miller, M. I., “Landmark matching via large deformation diffeomorphisms”, IEEE Trans. Image Process. 9 (2000) 1357–1370; doi:10.1109/83.855431.CrossRefGoogle ScholarPubMed
Keller, H. B., “Lectures on numerical methods in bifurcation problems”, Appl. Math. 217 (1987) 50, available at https://www.math.tifr.res.in/~publ/ln/tifr79.pdf.Google Scholar
Marsland, S. and McLachlan, R., “A Hamiltonian particle method for diffeomorphic image registration”, in: Biennial international conference on information processing in medical imaging, (Springer, Berlin, 2007) 396–407; doi:10.1007/978-3-540-73273-0_33. CrossRefGoogle Scholar
Marsland, S. and Twining, C. J., “Clamped-plate splines and the optimal flow of bounded diffeomorphisms”, in: Statistics of large datasets, proceedings of Leeds annual statistical research workshop, (University of Leeds, Leeds, 2002) 91–95, available at https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.18.5291rep=rep1type=pdf.Google Scholar
McLachlan, R. and Marsland, S., “Discrete mechanics and optimal control for image registration”, ANZIAM J. 48 (2007) 1–16; doi:10.21914/anziamj.v48i0.82.CrossRefGoogle Scholar
Miller, M. I., Trouvé, A. and Younes, L., “Hamiltonian systems and optimal control in computational anatomy: 100 years since D’Arcy Thompson”, Annu. Rev. Biomed. Eng. 17 (2015) 447–509; doi:10.1146/annurev-bioeng-071114-040601.CrossRefGoogle ScholarPubMed
Modersitzki, J., Numerical methods for image registration, (Oxford University Press, Oxford, 2004); doi:10.1093/acprof:oso/9780198528418.001.0001. Google Scholar
Pennec, X., “Intrinsic statistics on Riemannian manifolds: Basic tools for geometric measurements”, J. Math Imaging Vis. 25 (2006) 127; doi:10.1007/s10851-006-6228-4.CrossRefGoogle Scholar
Petukhov, S. V., “Non-Euclidean geometries and algorithms of living bodies”, Comput. Math. Appl. 17 (1989) 505–534; doi:10.1016/0898-1221(89)90248-4.CrossRefGoogle Scholar
Saxena, S. and Singh, R. K., “A survey of recent and classical image registration methods”, Int. J. Signal Process. Image Process. Pattern Recognit. 7 (2014) 167–176; doi:10.14257/ijsip.2014.7.4.16.Google Scholar
Sommer, S., Lauze, F., Hauberg, S. and Nielsen, M., “Manifold valued statistics, exact principal geodesic analysis and the effect of linear approximations”, in: European conference on computer vision, (Springer, Berlin, 2010) 43–56; doi:10.1007/978-3-642-15567-3_4. Google Scholar
Thompson, D. W., On growth and form, (Cambridge University Press, Cambridge, 1942); doi:10.5962/bhl.title.6462. Google Scholar
Tufail, M. Y., “Image registration under conformal diffeomorphisms”, Ph. D. Thesis, Massey University, 2017, available at https://mro.massey.ac.nz/handle/10179/12459.Google Scholar
Wells, J. R., “The Cauchy–Riemann equations and differential geometry”, Bull. Am. Math. Soc. 6 (1982) 187–199; doi:10.1090/s0273-0979-1982-14976-x. CrossRefGoogle Scholar
Younes, L., Shapes and diffeomorphisms, (Springer, Berlin, 2010); doi:10.1007/978-3-642-12055-8. CrossRefGoogle Scholar