A variational approach to cardiac motion estimation based on covariant derivatives and multi-scale Helmholtz decomposition. (English) Zbl 1261.49008
Summary: The investigation and quantification of cardiac motion is important for assessment of cardiac abnormalities and treatment effectiveness. Therefore we consider a new method to track cardiac motion from Magnetic Resonance (MR) tagged images. Tracking is achieved by following the spatial maxima in scale-space of the MR images over time. Reconstruction of the velocity field is then carried out by minimizing an energy functional which is a Sobolev norm expressed in covariant derivatives. These covariant derivatives are used to express prior knowledge about the velocity field in the variational framework employed. Furthermore, we propose a multi-scale Helmholtz decomposition algorithm that combines diffusion and Helmholtz decomposition in one nonsingular analytic kernel operator in order to decompose the optic flow vector field in a divergence-free and a rotation-free part. Finally, we combine both the multi-scale Helmholtz decomposition and our vector field reconstruction (based on covariant derivatives) in a single algorithm and show the practical benefit of this approach by an experiment on real cardiac images.
MSC:
| 49M25 | Discrete approximations in optimal control |
| 49M27 | Decomposition methods |
| 47A05 | General (adjoints, conjugates, products, inverses, domains, ranges, etc.) |
| 47A10 | Spectrum, resolvent |
| 68U10 | Computing methodologies for image processing |
| 92C55 | Biomedical imaging and signal processing |
Keywords:
covariant derivatives; fiber bundles; Helmholtz decomposition; inverse problems; optical flow methods in image analysis; scale spaceReferences:
| [1] | George B. Arfken and Hans J. Weber, Mathematical methods for physicists, 4th ed., Academic Press, Inc., San Diego, CA, 1995. · Zbl 0970.00005 |
| [2] | L. Axel and L. Dougherty, MR imaging of motion with spatial modulation of magnetization, Radiology 171 (1989), no. 3, 841-845. |
| [3] | J.L. Barron, D.J. Fleet, and S. Beauchemin, Performance of optical flow techniques, IJCV 12 (1994), no. 1, 43-77. |
| [4] | A. Becciu, Feature based estimation of myocardial motion from tagged MR images, Ph.D. thesis, Eindhoven University of Technology, Dept. of Biomedical Engineering, 2010. |
| [5] | A. Becciu, B.J. Janssen, H.C. van Assen, L.M.J. Florack, V. Roode, and B.M. ter Haar Romeny, Extraction of cardiac motion using scale-space features points and gauged reconstruction, CAIP ’09: Proceedings of the 13th International Conference on Computer Analysis of Images and Patterns (Berlin, Heidelberg), Springer-Verlag, 2009, pp. 598-605. |
| [6] | A. Bruhn, J. Weickert, T. Kohlber, and C. Schnoerr, A multigrid platform for real-time motion computation with discontinuity-preserving variational methods, IJCV 70 (2006), no. 3, 257-277. |
| [7] | T. Corpetti, E. Mémin, and P. Pérez, Dense estimation of fluid flows, IEEE PAMI 24 (2002), no. 3, 365-380. |
| [8] | A. Cuzol, P. Hellier, and E. Mémin, A low dimensional fluid motion estimator, International Journal of Computer Vision 75 (2007), no. 3, 329-349. |
| [9] | James Damon, Local Morse theory for solutions to the heat equation and Gaussian blurring, J. Differential Equations 115 (1995), no. 2, 368 – 401. · Zbl 0847.35056 · doi:10.1006/jdeq.1995.1019 |
| [10] | T. Delhaas, J. Kotte, and A. van der Toorn, Increase in Left Ventricular Torsion-to-Shortening Ratio in Children With Valvular Aorta Stenosis., Magnetic Resonance in Medicine 51 (2004), 135-139. |
| [11] | R. Duits, Perceptual organization in image analysis, Ph.D. thesis, Eindhoven University of Technology, Department of Biomedical Engineering, The Netherlands, 2005. |
| [12] | R. Duits, A. Becciu, B.J. Janssen, H.C. van Assen, L.M.J. Florack, and B.M. ter Haar Romeny, Cardiac motion estimation using covariant derivatives and Helmholtz decomposition, CASA-report 31, Eindhoven University of Technology, Department of Mathematics and Computer Science, 2010, http://www.win.tue.nl/casa/research/casareports/2010.html. |
| [13] | R. Duits, H. Fuehr, B.J. Janssen, and L.C.M. Bruurmijn, Left-invariant reassignment and diffusion on Gabor transforms, (2011), Submitted. |
| [14] | R. Duits, B.J. Janssen, F.M.W. Kanters, and L.M.J. Florack, Linear image reconstruction from a sparse set of alpha scale space features by means of inner products of Sobolev type, Lecture Notes in Computer Science, Springer-Verlag 3753 (2005), 96-111. · Zbl 1119.68481 |
| [15] | Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. · Zbl 0902.35002 |
| [16] | L.M.J. Florack, Image structure, Computational Imaging and Vision, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997. |
| [17] | -, Scale space representations locally adapted to the geometry of base and target manifold, Mathematical Methods for Signal and Image Analysis and Representation , Springer-Verlag, 2011, pp. 175-189. |
| [18] | Luc Florack and Arjan Kuijper, The topological structure of scale-space images, J. Math. Imaging Vision 12 (2000), no. 1, 65 – 79. · Zbl 0979.68585 · doi:10.1023/A:1008304909717 |
| [19] | L.M.J. Florack, W. Niessen, and M. Nielsen, The intrinsic structure of optic flow incorporating measurements of duality, International Journal of Computer Vision 27 (1998), no. 3, 263-286. |
| [20] | L.M.J. Florack and H.C. van Assen, A New Methodology for Multiscale Myocardial Deformation and Strain Analysis Based on Tagging MRI, International Journal of Biomedical Imaging (2010), 1-8, Published online http://downloads.hindawi.com/journals/ijbi/2010/341242.pdf. |
| [21] | P.E. Forssen, Low and medium level vision using channel representations, Ph.D. thesis, Linkoping University, Dept. EE, Linkoping, Sweden, March 2004. |
| [22] | D. Gabor, Theory of communication, J. IEE 93 (1946), no. 26, 429-457. |
| [23] | T. Georgiev, Relighting, retinex theory, and perceived gradients, Proceedings of Mirage 2005, INRIA Rocquencourt, 2005. |
| [24] | S. Gerschgorin, Ueber die abgrenzung der eigenwerte einer matrix, Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk. 7 (1931), 550-559. |
| [25] | Charles W. Groetsch, Elements of applicable functional analysis, Monographs and Textbooks in Pure and Applied Mathematics, vol. 55, Marcel Dekker, Inc., New York, 1980. · Zbl 0428.46016 |
| [26] | S. Gupta, E.N. Gupta, and J.L. Prince, Stochastic models for div-curl optical flow methods, IEEE Signal Processing Letters 3 (1996), 32-35. |
| [27] | Per Christian Hansen, James G. Nagy, and Dianne P. O’Leary, Deblurring images, Fundamentals of Algorithms, vol. 3, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2006. Matrices, spectra, and filtering. · Zbl 1112.68127 |
| [28] | C. O. Horgan and S. Nemat-Nasser, Bounds on eigenvalues of Sturm-Liouville problems with discontinuous coefficients, Z. Angew. Math. Phys. 30 (1979), no. 1, 77 – 86 (English, with French summary). · Zbl 0408.34025 · doi:10.1007/BF01597482 |
| [29] | B.K.P. Horn and B.G. Schunck, Determining optical flow, AI 17 (1981), 185-203. |
| [30] | T. Iijima, Basic theory on normalization of a pattern (in case of typical one-dimensional pattern), Bulletin of Electrical Laboratory 26 (1962), 368-388. |
| [31] | B.J. Janssen, Representation and manipulation of images based on linear functionals, Ph.D. thesis, Eindhoven University of Technology, Eindhoven, The Netherlands, 2009. |
| [32] | B.J. Janssen and R. Duits, Linear image reconstruction by Sobolev norms on the bounded domain, International Journal of Computer Vision. 84 (2009), no. 2, 205-219. |
| [33] | B.J. Janssen, R. Duits, and L.M.J. Florack, Coarse-to-fine image reconstruction based on weighted differential features and background gauge fields, Lecture Notes in Computer Science 5567 (2009), 377-388. |
| [34] | B.J. Janssen, L.M.J. Florack, R. Duits, and B.M. ter Haar Romeny, Optic flow from multi-scale dynamic anchor point attributes, Image Analysis and Recognition, Third International Conference, ICIAR 2006 (Berlin) , Lecture Notes in Computer Science, vol. 4141, Springer-Verlag, September 2006, pp. 767-779. |
| [35] | B.J. Janssen, F.M.W. Kanters, R. Duits, L.M.J. Florack, and B.M. ter Haar Romeny, A linear image reconstruction framework based on Sobolev type inner products., International Journal of Computer Vision 70 (2006), no. 3, 231-240. · Zbl 1119.68481 |
| [36] | F.M.W. Kanters, T. Denton, A. Shokoufandeh, L.M.J. Florack, and B.M. ter Haar Romeny, Combining different types of scale space interest points using canonical sets, Scale Space and Variational Methods (Berlin) , Lecture Notes in Computer Science, vol. 4485, Springer-Verlag, 2007, pp. 374-385. |
| [37] | F.M.W. Kanters, L.M.J. Florack, R. Duits, B. Platel, and B.M. ter Haar Romeny, Scalespacevis: \( \alpha \)-scale spaces in practice, Pattern Recognition and Image Analysis 17 (2007), no. 1. |
| [38] | F.M.W. Kanters, B. Platel, L.M.J. Florack and B.M. ter Haar Romeny, Image reconstruction from multiscale critical points, Scale Space Methods in Computer Vision, 4th International Conference, Scale Space 2003 (Isle of Skye, UK) , Springer, June 2003, pp. 464-478. · Zbl 1067.68746 |
| [39] | Jan J. Koenderink, The structure of images, Biol. Cybernet. 50 (1984), no. 5, 363 – 370. · Zbl 0537.92011 · doi:10.1007/BF00336961 |
| [40] | T. Kohlberger, E. Mémin, and P. Pérez, Variational dense motion estimation using the Helmholtz decomposition, Proc. Scale Space Conference , Lecture Notes in Computer Science, vol. 2695, Springer-Verlag, 2003, pp. 432-448. · Zbl 1067.68748 |
| [41] | M. Lillholm, M. Nielsen, and L.D. Griffin, Feature-based image analysis, International Journal of Computer Vision 52 (2003), no. 2/3, 73-95. |
| [42] | T. Lindeberg, Scale-space for discrete signals, PAMI 12 (1990), no. 3, 234-245. |
| [43] | -, Scale-space theory in computer vision, The Kluwer International Series in Engineering and Computer Science, Kluwer Academic Publishers, 1994. |
| [44] | B. Lucas and T. Kanade, An iterative image registration technique with application to stereo vision, DARPA, Image Process, vol. 21, 1981, pp. 85-117. |
| [45] | P. D. Lax and A. N. Milgram, Parabolic equations, Contributions to the theory of partial differential equations, Annals of Mathematics Studies, no. 33, Princeton University Press, Princeton, N. J., 1954, pp. 167 – 190. · Zbl 0058.08703 |
| [46] | I. Mirsky, J.M. Pfeffer, and M.A. Pfeffer, The contractile state as the major determinant in the evolution of left ventricular dysfunction in the spontaneously hypertensive rat. Circulation Research, 53 (1983), 767-778. |
| [47] | M. Nielsen and M. Lillholm, What features tell about images?, Lecture Notes in Computer Science 2106 (2001), 39-50. · Zbl 0991.68582 |
| [48] | T. Nir, A.M. Bruckstein, and R. Kimmel, Over-Parameterized variational optical flow, International Journal of Computer Vision 76 (2008), no. 2, 205-216. |
| [49] | N.F. Osman, W.S. McVeigh, and J.L. Prince, Cardiac motion tracking using sine harmonic phase (harp) magnetic resonance imaging, Magnetic Resonance in Medicine 42 (1999), no. 6, 1048-1060. |
| [50] | Walter Rudin, Functional analysis, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. · Zbl 0253.46001 |
| [51] | Yousef Saad, Iterative methods for sparse linear systems, 2nd ed., Society for Industrial and Applied Mathematics, Philadelphia, PA, 2003. · Zbl 1031.65046 |
| [52] | F. Sheehan, D. Stewart, H. Dodge, S. Mitten, E. Bolson and G. Brown, Variability in the measurement of regional left ventricular wall motion, Circulation 68 (1983), 550-559. |
| [53] | J. Staal, S. Kalitzin, B.M. ter Haar Romeny, and M. Viergever, Detection of critical structures in scale space, Lecture Notes in Computer Science, vol. 1682, 1999, pp. 105-116. |
| [54] | B.M. ter Haar Romeny, Front-end vision and multi-scale image analysis: Multiscale computer vision theory and applications, written in mathematica, Computational Imaging and Vision, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003. |
| [55] | J.P. Thirion, Image matching as a diffusion process: an analogy with Maxwell’s demons, Medical Image Analysis 2 (1998), no. 3, 243-260. |
| [56] | Y. Tong, S. Lombeyda, A.N. Hivani, and M. Desbrun, Discrete multiscale vector field decomposition, ACM Transactions on Graphics (TOG) 22 (2003), 445-452. |
| [57] | M. Unser, Splines: A perfect fit for signal and image processing, IEEE Signal Processing Magazine 16 (1999), no. 6, 22-38. |
| [58] | H.C. van Assen, L.M.J. Florack, F.J.J. Simonis, and J.J.M. Westenberg, Cardiac strain and rotation analysis using multi-scale optical flow, MICCAI workshop on Computational Biomechanics for Medicine 5, Springer-Verlag, 2011, pp. 91-103. |
| [59] | H.C. van Assen, L.M.J. Florack, A. Suinesiaputra, J.J.M. Westenberg, and B.M. ter Haar Romeny, Purely evidence based multiscale cardiac tracking using optic flow, Proc. MICCAI 2007 workshop on Computational Biomechanics for Medicine II , 2007, pp. 84-93. |
| [60] | H. von Helmholtz, Ueber integrale der hydrodynamischen gleichungen, welche den wirbelbewegungen entsprechen, Crelles J. 55 (1858), no. 25. · ERAM 055.1448cj |
| [61] | J. Weickert, S. Ishikawa, and A. Imiya, On the history of Gaussian scale-space axiomatics, Gaussian Scale-Space Theory, Computational Imaging and Vision Series, Kluwer Academic Publisher, 1997, pp. 45-59. |
| [62] | E.A. Zerhouni, D.M. Parish, W.J. Rogers, A. Yang, and E.P. Sapiro, Human heart: Tagging with MR imaging a method for noninvasive assessment of myocardial motion, Radiology 169 (1988), no. 1, 59-63. |
| [63] | H. Zimmer, A. Bruhn, J. Weickert, L. Valgaerts, A. Salgado, B. Rosenhahn, and H.-P. Seidel, Complementary optic flow, Proceedings of Energy Minimization Methods in Computer Vision and Pattern Recognition (EMMCVPR), Lecture Notes in Computer Science, Berlin, Springer-Verlag, vol. 5681, 2009, pp. 207-220. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
