Evolution equations on Gabor transforms and their applications. (English) Zbl 1296.35204
Summary: We introduce a systematic approach to the design, implementation and analysis of left-invariant evolution schemes acting on Gabor transform, primarily for applications in signal and image analysis. Within this approach we relate operators on signals to operators on Gabor transforms. In order to obtain a translation and modulation invariant operator on the space of signals, the corresponding operator on the reproducing kernel space of Gabor transforms must be left-invariant, i.e. it should commute with the left-regular action of the reduced Heisenberg group \(H_r\). By using the left-invariant vector fields on \(H_r\) in the generators of our evolution equations on Gabor transforms, we naturally employ the essential group structure on the domain of a Gabor transform. Here we distinguish between two tasks. Firstly, we consider non-linear adaptive left-invariant convection (reassignment) to sharpen Gabor transforms, while maintaining the original signal. Secondly, we consider signal enhancement via left-invariant diffusion on the corresponding Gabor transform. We provide numerical experiments and analytical evidence for our methods and we consider an explicit medical imaging application.
MSC:
| 35R03 | PDEs on Heisenberg groups, Lie groups, Carnot groups, etc. |
| 68U10 | Computing methodologies for image processing |
| 94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |
| 94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |
| 92C55 | Biomedical imaging and signal processing |
Keywords:
evolution equations; Heisenberg group; differential reassignment; left-invariant vector fields; diffusion on Lie groups; Gabor transforms; medical imagingReferences:
| [1] | Akian, M.; Quadrat, J.; Viot, M., Bellman processes, (Lecture Notes in Control and Inform. Sci., vol. 199 (1994), Springer-Verlag), 302-311 · Zbl 0826.49021 |
| [2] | Arts, T.; Prinzen, F. W.; Delhaas, T.; Milles, J. R.; Rossi, A. C.; Clarysse, P., Mapping displacement and deformation of the heart with local sine-wave modeling, IEEE Trans. Med. Imag., 29, 5, 1114-1123 (May 2010) |
| [3] | Auger, F.; Flandrin, P., Improving the readability of time-frequency and time-scale representations by the reassignment method, IEEE Trans. Signal Process., 43, 5, 1068-1089 (May 1995) |
| [5] | Becciu, A.; Duits, R.; Janssen, B. J.; Florack, L. M.J.; ter Haar Romeny, B. M.; van Assen, H. C., Cardiac motion estimation using covariant derivatives and Helmholtz decomposition, (Statistical Atlases and Computational Models of the Heart, STACOM 2011 Held in Conjunction with MICCAI 2011. Statistical Atlases and Computational Models of the Heart, STACOM 2011 Held in Conjunction with MICCAI 2011, LNCS, vol. 7085 (2011)), 263-273 |
| [6] | Becciu, A.; Janssen, B. J.; van Assen, H. C.; Florack, L. M.J.; Roode, V.; ter Haar Romeny, B. M., Extraction of cardiac motion using scale-space features points and gauged reconstruction, (CAIPʼ09: Proceedings of the 13th International Conference on Computer Analysis of Images and Patterns (2009), Springer-Verlag: Springer-Verlag Berlin, Heidelberg), 598-605 |
| [7] | Bellaïche, B., The Tangent Space in Sub-Riemannian Geometry, Progr. Math. (1996), Birkhäuser: Birkhäuser New York · Zbl 0862.53031 |
| [8] | Bölcskei, H.; Hlawatsch, F., Discrete Zak transforms, polyphase transforms, and applications, IEEE Trans. Signal Process., 45, 4, 851-866 (April 1997) |
| [9] | Bruhn, A.; Weickert, J.; Kohlber, T.; Schnoerr, C., A multigrid platform for real-time motion computation with discontinuity-preserving variational methods, Int. J. Comput. Vis., 70, 3, 257-277 (2006) · Zbl 1477.68336 |
| [11] | Bryant, R. L.; Chern, S. S.; Gardner, R. B.; Goldschmidt, H. L.; Griffiths, P. A., Exterior Differential Systems, Math. Sci. Res. Inst. Publ., vol. 18 (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0726.58002 |
| [12] | Bukhvalov, A. V.; Arendt, W., Integral representation of resolvent and semigroups, Pacific J. Math., 10, 419-437 (1960) |
| [13] | Burgeth, B.; Weickert, J., An explanation for the logarithmic connection between linear and morphological systems, (Proc. 4th Int. Conference Scale Space. Proc. 4th Int. Conference Scale Space, Lecture Notes in Comput. Sci. (2003), Springer-Verlag), 325-339 · Zbl 1067.68723 |
| [14] | Chassande-Mottin, E.; Daubechies, I.; Auger, F.; Flandrin, P., Differential reassignment, IEEE Signal Proc. Lett., 4, 10, 293-294 (October 1997) |
| [15] | Chirikjian, G. S., Stochastic Models, Information Theory, and Lie Groups, Appl. Numer. Harmon. Anal., vols. 1-2 (2011), Birkhäuser: Birkhäuser New York |
| [16] | Cottet, G.-H.; Germaine, L., Image processing through reaction combined with nonlinear diffusion, Math. Comp., 61, 659-667 (1990) · Zbl 0799.35117 |
| [17] | Crandall, Michael G.; Lions, Pierre-Louis, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277, 1, 1-42 (1983) · Zbl 0599.35024 |
| [18] | Daubechies, I.; Maes, S., A nonlinear squeezing of the continuous wavelet transform based on auditory nerve models, (Wavelets in Medicine and Biology (1996), CRC Press), 527-546 · Zbl 0848.92003 |
| [19] | Daudet, L.; Morvidone, M.; Torrésani, B., Time-frequency and time-scale vector fields for deforming time-frequency and time-scale representations, (Proceedings of the SPIE Conference on Wavelet Applications in Signal and Image Processing (1999), SPIE), 2-15 |
| [20] | Dieudonné, J., Treatise on Analysis, vol. 5, Pure Appl. Math., vol. 10-V (1977), Academic Press: Academic Press New York, London, a subsidiary of Harcourt Brace Jovanovich Publishers, translated by I.G. Macdonald, Chapter XXI · Zbl 0418.22007 |
| [21] | Dorst, P. A.G.; Janssen, B. J.; Florack, L. M.J.; ter Haar Romeny, B. M., Optic flow using multi-scale anchor points, (Proceedings of the 13th International Conference on Computer Analysis of Images and Patterns (CAIP09). Proceedings of the 13th International Conference on Computer Analysis of Images and Patterns (CAIP09), LNCS (2009), Springer-Verlag: Springer-Verlag Berlin) |
| [23] | Duits, R.; Burgeth, B., Scale spaces on Lie groups, (Proceedings of SSVM 2007, 1st International Conference on Scale Space and Variational Methods in Computer Vision. Proceedings of SSVM 2007, 1st International Conference on Scale Space and Variational Methods in Computer Vision, Lecture Notes in Comput. Sci. (2007), Springer-Verlag), 300-312 |
| [24] | Duits, R.; Felsberg, M.; Granlund, G.; ter Haar Romeny, B. M., Image analysis and reconstruction using a wavelet transform constructed from a reducible representation of the Euclidean motion group, Int. J. Comput. Vis., 72, 79-102 (2007) |
| [25] | Duits, R.; Franken, E. M., Left invariant parabolic evolution equations on \(SE(2)\) and contour enhancement via invertible orientation scores, part I: Linear left-invariant diffusion equations on \(SE(2)\), Quart. Appl. Math., 68, 255-292 (June 2010) · Zbl 1202.35334 |
| [26] | Duits, R.; Franken, E. M., Left invariant parabolic evolution equations on \(SE(2)\) and contour enhancement via invertible orientation scores, part II: Nonlinear left-invariant diffusion equations on invertible orientation scores, Quart. Appl. Math., 68, 293-331 (June 2010) · Zbl 1205.35326 |
| [27] | Duits, R.; Franken, E. M., Left-invariant diffusions on the space of positions and orientations and their application to crossing-preserving smoothing of HARDI images, Int. J. Comput. Vis., 92, 231-264 (March 2011), published digitally online in 2010: · Zbl 1235.92032 |
| [30] | Duits, R.; Janssen, B. J.; Becciu, A.; van Assen, H., A variational approach to cardiac motion estimation based on covariant derivatives and multi-scale Helmholtz decomposition, Quart. Appl. Math., 71, 1, 1-3 (March 2013) · Zbl 1261.49008 |
| [31] | Duits, R.; van Almsick, M., The explicit solutions of linear left-invariant second order stochastic evolution equations on the 2D Euclidean motion group, Quart. Appl. Math., 66, 1, 27-67 (2008) · Zbl 1153.60040 |
| [32] | Evans, L. C., Partial Differential Equations, Grad. Stud. Math., vol. 19 (2002), American Mathematical Society |
| [33] | Feichtinger, H. G.; Gröchenig, K., Non-orthogonal wavelet and Gabor expansions, and group representations, (Wavelets and Their Applications (1992), Jones and Bartlett: Jones and Bartlett Boston, MA), 353-376 · Zbl 0832.42022 |
| [34] | Feichtinger, H. G.; Kozek, W.; Luef, F., Gabor analysis over finite abelian groups, Appl. Comput. Harmon. Anal., 26, 2, 230-248 (2009) · Zbl 1162.43002 |
| [35] | Florack, L. M.J.; Niessen, W. J.; Nielsen, M., The intrinsic structure of optic flow incorporating measurement duality, Int. J. Comput. Vis., 27, 3, 263-286 (May 1998) |
| [37] | Florack, L. M.J.; van Assen, H. C., A new methodology for multiscale myocardial deformation and strain analysis based on tagging MRI, Int. J. Biomed. Imaging, 1-8 (2010), published online: |
| [39] | Franken, E. M.; Duits, R., Crossing preserving coherence-enhancing diffusion on invertible orientation scores, Int. J. Comput. Vis., 85, 3, 253-278 (2009) |
| [40] | Gabor, D., Theory of communication, J. IEE (London), 93, 429-457 (1946) |
| [41] | Gaveau, B., Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents, Acta Math., 139, 1-2, 95-153 (1977) · Zbl 0366.22010 |
| [42] | Gröchenig, K., Foundations of Time-Frequency Analysis, Appl. Numer. Harmon. Anal. (2001), Birkhäuser Boston Inc.: Birkhäuser Boston Inc. Boston, MA · Zbl 0966.42020 |
| [43] | Helstrom, C. W., An expansion of a signal in Gaussian elementary signals (corresp.), IEEE Trans. Inform. Theory, 12, 1, 81-82 (Jan 1966) |
| [44] | Hörmander, L., Hypoelliptic second order differential equations, Acta Math., 119, 147-171 (1967) · Zbl 0156.10701 |
| [45] | Horn, B. K.P.; Shunck, B. G., Determining optical flow, Artificial Intelligence, 17, 185-203 (1981) · Zbl 1497.68488 |
| [46] | Janssen, A. J.E. M., The Zak transform: A signal transform for sampled time-continuous signals, Philips J. Res., 43, 1, 23-69 (1988) · Zbl 0653.94002 |
| [48] | Janssen, B. J.; Florack, L. M.J.; Duits, R.; ter Haar Romeny, B. M., Optic flow from multi-scale dynamic anchor point attributes, (Campilho, A.; Kamel, M., Third International Conference on Image Analysis and Recognition, ICIAR 2006. Third International Conference on Image Analysis and Recognition, ICIAR 2006, LNCS, vol. 4141 (September 2006), Springer-Verlag: Springer-Verlag Berlin), 767-779 |
| [49] | Kodera, K.; de Villedary, C.; Gendrin, R., A new method for the numerical analysis of non-stationary signals, Phys. Earth Planet. Inter., 12, 142-150 (1976) |
| [50] | Lions, P. L.; Magenes, E., Non-Homogeneous Boundary Value Problems and Applications, vols. 1-3 (1973), Springer-Verlag: Springer-Verlag Berlin · Zbl 0251.35001 |
| [51] | Loog, M.; Duistermaat, J. J.; Florack, L. M.J., On the behavior of spatial critical points under Gaussian blurring: A folklore theorem and scale-space constraints, (Kerckhove; etal., Proceedings of Scale Space Conf. 2001. Proceedings of Scale Space Conf. 2001, LNCS (2001)), 182-192 · Zbl 0991.68100 |
| [53] | Manfredi, J. J.; Stroffolini, B., A version of the Hopf-Lax formula in the Heisenberg group, Comm. Partial Differential Equations, 27, 5, 1139-1159 (2002) · Zbl 1080.49023 |
| [54] | Nitzberg, M.; Shiota, T., Nonlinear image filtering with edge and corner enhancement, IEEE Trans. Pattern Anal. Mach. Intell., 14, 826-833 (1992) |
| [55] | Osman, N. F.; Kerwin, W. S.; McVeigh, E. R.; Prince, J. L., Cardiac motion tracking using CINE harmonic phase (HARP) magnetic resonance imaging, Magn. Reson. Med., 42, 6, 1048-1060 (1999) |
| [56] | Prckovska, V.; Rodrigues, P.; Duits, R.; Vilanova, A.; ter Haar Romeny, B. M., Extrapolating fiber crossings from DTI data. Can we infer similar fiber crossings as in HARDI?, (CDMRIʼ10 MICCAI 2010 Workshop on Computational Diffusion MRI, vol. 1. CDMRIʼ10 MICCAI 2010 Workshop on Computational Diffusion MRI, vol. 1, Beijing, China, August 2010 (2010), Springer-Verlag), 26-37 |
| [57] | Rodrigues, P.; Duits, R.; Vilanova, A.; ter Haar Romeny, B. M., Accelerated diffusion operators for enhancing DW-MRI, (Eurographics Workshop on Visual Computing for Biology and Medicine. Eurographics Workshop on Visual Computing for Biology and Medicine, Leipzig, Germany, 2010 (2010), Springer-Verlag), 49-56 |
| [58] | Rund, H., The Hamilton-Jacobi Theory in the Calculus of Variations: Its Role in Mathematics and Physics (1966), D. Van Nostrand Co., Ltd.: D. Van Nostrand Co., Ltd. London/Toronto, ON, New York · Zbl 0141.10602 |
| [59] | Taylor, T., A parametrix for step-two hypoelliptic diffusion equations, Trans. Amer. Math. Soc., 296, 1, 191-215 (1986) · Zbl 0602.35021 |
| [60] | ter Elst, A. F.M.; Robinson, D. W., Weighted subcoercive operators on Lie groups, J. Funct. Anal., 157, 1, 88-163 (1998) · Zbl 0910.22005 |
| [61] | van Assen, H. C.; Florack, L. M.J.; Simonis, F. J.J.; Westenberg, J. J.M.; Strijkers, G. J., Cardiac strain and rotation analysis using multi-scale optical flow, (MICCAI Workshop on Computational Biomechanics for Medicine 5 (2011), Springer-Verlag), 91-103 |
| [63] | Weickert, J., Coherence-enhancing diffusion filtering, Int. J. Comput. Vis., 31, 2/3, 111-127 (1999) · Zbl 1505.94010 |
| [64] | Weisz, F., Inversion of the short-time Fourier transform using Riemann sums, J. Fourier Anal. Appl., 13, 3, 357-368 (2007) · Zbl 1138.42015 |
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