Abstract
In this work, edge sets are mapped one to the other by representing these zero area sets as diffuse images which have positive measure supports that can be registered elastically. The driving application for this work is to map a Purkinje fiber network in the endocardium of one heart to the endocardium of another heart. The approach is to register sufficiently accurate diffuse surface representations of two endocardia and then to apply the resulting transformation to the points of the Purkinje fiber network. To create a diffuse image from a given edge set, a region growing method is used to approximate diffusion of brightness from an edge set to a given point. To be minimized is the sum of squared differences of the transformed diffuse images along with a linear elastic penalty for the registration transformation. A Newton iteration is employed to solve the optimality system, and the degree of diffusion is larger in initial iterations while smaller in later iterations so that a desired local minimum is selected by means of vanishing diffusion. Favorable results are shown for registering highly detailed cardiac geometries.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ambrosio L, Tortorelli V (1990) Approximation of functionals depending on jumps by elliptic functionals via \(\gamma \)-convergence. Commun Pure Appl Math 43:999–1036
Ansari A, Ho SY, Anderson RH (1999) Distribution of the purkinje fibres in the sheep heart. Anat Rec 254(1):92–97
Aubert G, Kornprobst P (2006) Mathematical problems in image processing. In: Applied mathematical sciences, vol 147. 2nd edn. Springer, Berlin
Bishop MJ, Plank G, Burton RA, Schneider JE, Gavaghan DJ, Grau V, Kohl P (2010) Development of an anatomically detailed MRI-derived rabbit ventricular model and assessment of its impact on simulations of electrophysiological function. Am J Physiol Heart Circ Physiol 298(2):H699–718
Ciarlet P (1978) The finite element method for elliptic problems. North-Holland, Amsterdam
De Giorgi E (1979) Convergence problems for functionals and operators. In: Proceedings of the international meeting on recent methods in non-linear analysis, pp 131–188, Rome, 1978
Dennis J, Schnabel R (1983) Numerical methods for unconstrained optimization and nonlinear equations. Prentice-Hall, Englewood Cliffs
Droske M, Ring W (2006) A mumford-shah level-set approach for geometric image registration. SIAM J Appl Math 66(6):2127–2148
Fitzpatrick J, Hill D, Maurer C (2000) Image registration, medical image processing. In: Chapter 8: Handbook of Medical Imaging, vol 2. SPIE Press, Bellingham
Fuchs M, Jüttler B, Scherzer O, Yang H (2009) Shape metrics based on elastic deformations. J Math Imaging Vis 35(1):86–102
Fürtinger S, Keeling SL, Plank G, Prassl AJ (2010) Registration of edge sets for mapping a purkinje fiber network onto an endocardium. Technical report, Institute for Mathematics and Scientific Computing, Karl-Franzens University, Graz, Austria
Fürtinger S, Keeling SL, Plank G, Prassl AJ (2011) Elastic registration of edge sets by means of diffuse surfaces—with an application to embedding purkinje fiber networks. In: Mestetskiy L, Braz J (eds) VISAPP, SciTePress, pp 12–21
Hill W, Baldock RA (2006) The constrained distance transform: interactive atlas registration with large deformations through constrained distances. In: Workshop on image registration in deformable environments (DEFORM’06), Edinburgh
Holland RP, Brooks H (1976) The qrs complex during myocardial ischemia. an experimental analysis in the porcine heart. J Clin Invest 57(3):541–550 doi:10.1172/JCI108309
Huelsing DJ, Spitzer KW, Cordeiro JM, Pollard AE (1998) Conduction between isolated rabbit purkinje and ventricular myocytes coupled by a variable resistance. Am J Physiol 274(4 Pt 2):H1163–H1173
Keeling S, Ring W (2005) Medical image registration and interpolation by optical flow with maximal rigidity. J Math Imaging Vis 23(1):47–65
Knauer C, Kriegel K, Stehn F (2009) Minimizing the weighted directed hausdorff distance between colored point sets under translations and rigid motions. In: Proceedings of the 3d international workshop on frontiers in algorithmics (FAW ’09), Springer, Berlin, pp 108–119
Modersitzki J (2004) Numerical methods for image registration. Oxford Science Publications, New York
Mumford D, Shah J (1989) Optimal approximations by piecewise smooth functions and associated variational problems. Commun Pure Appl Math 42(5):577–685
Nocedal J, Wright S (2000) Numerical optimization. Springer, New York
Ortega JM (1968) The Newton–Kantorovich theorem. The Am Math Monthly 7(6):658–660
Paragios N, Ramesh MRV (2002) Matching distance functions: a shape-to-area variational approach for global-to-local registration. In: Proceedings of the 7th European conference on computer vision-part II
Peckar W, Schnörr C, Rohr K, Stiehl HS (1999) Parameter-free elastic deformation approach for 2d and 3d registration using prescribed displacements. J Math Imaging Vis 10(2):143–162
Plank G, Burton RA, Hales P, Bishop M, Mansoori T, Bernabeu MO, Garny A, Prassl AJ, Bollensdorff C, Mason F, Mahmood F, Rodriguez B, Grau V, Schneider JE, Gavaghan D, Kohl P (2009) Generation of histo-anatomically representative models of the individual heart: tools and application. Philos Trans A Math Phys Eng Sci 367(1896):2257–2292
Prassl AJ, Kickinger F, Ahammer H, Grau V, Schneider JE, Hofer E, Vigmond EJ, Trayanova NA, Plank G (2009) Automatically generated, anatomically accurate meshes for cardiac electrophysiology problems. IEEE Trans Biomed Eng 56(5):1318–1330
Toriwaki J, Yoshida H (2009) Fundamentals of three-dimensional digital image processing. Springer, New York
Tranum-Jensen J, Wilde AA, Vermeulen JT, Janse MJ (1991) Morphology of electrophysiologically identified junctions between purkinje fibers and ventricular muscle in rabbit and pig hearts. Circ Res 69(2):429–437
Vetter F, McCulloch A (1998) Three-dimensional analysis of regional cardiac function: a model of rabbit ventricular anatomy. Prog Biophys Mol Biol 69(2–3):157–183
Vigmond EJ, Clements C (2007) Construction of a computer model to investigate sawtooth effects in the purkinje system. IEEE Trans Biomed Eng 54(3):389–399
Yang S, Kohler D, Teller K, Cremer T, Le Baccon P, Heard E, Eils R, Rohr K (2008) Nonrigid registration of 3-d multichannel microscopy images of cell nuclei. IEEE Trans Image Process 17:493–499
Acknowledgments
All authors are supported by the Austrian Science Fund Fond zur Förderung der Wissenschaftlichen Forschung (FWF) under grant SFB F032 (“Mathematical Optimization and Applications in Biomedical Sciences” http://math.uni-graz.at/mobis).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Fürtinger, S., Keeling, S., Plank, G., Prassl, A.J. (2013). Elastic Registration of Edges Using Diffuse Surfaces. In: González Hidalgo, M., Mir Torres, A., Varona Gómez, J. (eds) Deformation Models. Lecture Notes in Computational Vision and Biomechanics, vol 7. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5446-1_11
Download citation
DOI: https://doi.org/10.1007/978-94-007-5446-1_11
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-5445-4
Online ISBN: 978-94-007-5446-1
eBook Packages: EngineeringEngineering (R0)