For the non-invasive imaging of moving organs, in this chapter, we investigate the generalisation of optical flow in three-dimensional Euclidean space. In computer vision, optical flow is dealt with as a local motion of pixels in a pair of successive images in a sequence of images. In a space, optical flow is defined as the local motion of the voxel of spatial distributions, such as x-ray intensity and proton distributions in living organs. Optical flow is used in motion analysis of beating hearts measured by dynamic cone beam x-ray CT and gated MRI tomography. This generalisation of optical flow defines a class of new constraints for optical-flow computation. We first develop a numerically stable optical-flow computation algorithm. The accuracy of the solution of this algorithm is guaranteed by Lax equivalence theorem which is the basis of the numerical computation of the solution for partial differential equations. Secondly, we examine numerically the effects of the divergence-free condition, which is required from linear approximation of infinitesimal deformation, for the computation of cardiac optical flow from images measured by gated MRI. Furthermore, we investigate the relation between the vector-spline constraint and the thin plate constraint. Moreover, we theoretically examine the validity of the error measure for the evaluation of computed optical flow.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Harvey, W., Exceritation anatomica de motu cordis et sanguinis animalibus Francofurti, Sumptibus G. Fitzel 1628, (Japanese Translation, Iwanami 1961).
Sachse, B. F., Computational Cardiology, Modelling of Anatomy, Electrophysi-ology, and Mechanics, LNCS 2966 Tutorial, 2004.
Ayache, N., ed. Handbook of Numerical Analysis Vol XII, Special Volume: Com-putational Models for the Human Body, Elsevier, 2004.
Morel, J.-M., Solimini, S., Variational Methods in Image Segmentation, Rirkhaäuser, 1995.
Aubert, G., Kornprobst, P., Mathematical Problems in Image Processing:Partial Differential Equations and the Calculus of Variations, Springer, 2002.
Sapiro, G., Geometric Partial Differential Equations and Image Analysis, Cambridge University Press, 2001.
Osher, S., Paragios, N., eds., Geometric Level Set Methods in Imaging, Vision, and Graphics, Springer, 2003.
Horn, B. K. P., Schunck, B. G., Determining optical flow, Artificial Intelligence, 17,185-204, 1981.
Nagel, H.-H., On the estimation of optical flow: Relations between different approaches and some new results. Artificial Intelligence, 33, 299-324, 1987.
Barron, J. L., Fleet, D. J., Beauchemin, S. S., Performance of optical flow tech-niques, International Journal of Computer Vision, 12, 43-77, 1994.
Beauchemin, S. S., Barron, J. L., The computation of optical flow, ACM Com-puter Surveys 26, 433-467, 1995.
Moore, J., Drangova, M., Wierzbicki, M., Barron, J., Peters, T., A High reso-lution dynamic heart model based on averaged MRI data, LNCS, 2878, 15-18, 2003.
Zhou, Z., Synolakis, C. E., Leahy, R. M., Song, S. M., Calculation of 3D internal displacement fields from 3D X-ray computer tomographic images, in Proceedings of Royal Society: Mathematical and Physical Sciences, 449, 537-554, 1995.
Song, S. M., Leahy, R. M., Computation of 3-D velocity fields from 3-D cine images of a human heart, IEEE Transactions on Medical Imaging, 10, 295-306, 1991.
Weickert, J., Schnörr, Ch., Variational optic flow computation with a spatio-temporal smoothness constraint, Journal of Mathematical Imaging and Vision 14,245-255, 2001.
Demmel, J.W., Applied Numerical Linear Algebra, SIAM, 1997.
Grossmann, Ch., Roos, H.-G., Numerik partieller Differentialgleichungen, Trubner, 1994.
Varga, R.S., Matrix Iteration Analysis, 2nd Edn., Springer, 2000.
Selig, J. M., Geometrical Method in Robotics, Springer, 1996.
Suter, D., Motion estimation and vector spline, Proceedings of CVPR’94, 939-942, 1994.
Suter, D., Chen F., Left ventricular motion reconstruction based on elastic vec-tor splines, IEEE Trans. Medical Imaging, 295-305, 2000.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer
About this chapter
Cite this chapter
Kameda, Y., Imiya, A. (2008). The William Harvey Code: Mathematical Analysis of Optical Flow Computation for Cardiac Motion. In: Rosenhahn, B., Klette, R., Metaxas, D. (eds) Human Motion. Computational Imaging and Vision, vol 36. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-6693-1_4
Download citation
DOI: https://doi.org/10.1007/978-1-4020-6693-1_4
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-6692-4
Online ISBN: 978-1-4020-6693-1
eBook Packages: Computer ScienceComputer Science (R0)