Abstract
This work presents new ideas in isotropic multi-dimensional phase based signal theory. The novel approach, called the conformal monogenic signal, is a rotational invariant quadrature filter for extracting local features of any curved signal without the use of any heuristics or steering techniques. The conformal monogenic signal contains the recently introduced monogenic signal as a special case and combines Poisson scale space, local amplitude, direction, phase and curvature in one unified algebraic framework. The conformal monogenic signal will be theoretically illustrated and motivated in detail by the relation between the Radon transform and the generalized Hilbert transform. The main idea of the conformal monogenic signal is to lift up n-dimensional signals by inverse stereographic projections to a n-dimensional sphere in ℝn+1 where the local signal features can be analyzed with more degrees of freedom compared to the flat n-dimensional space of the original signal domain. As result, it delivers a novel way of computing the isophote curvature of signals without partial derivatives. The philosophy of the conformal monogenic signal is based on the idea to use the direct relation between the original signal and geometric entities such as lines, circles, hyperplanes and hyperspheres. Furthermore, the 2D conformal monogenic signal can be extended to signals of any dimension. The main advantages of the conformal monogenic signal in practical applications are its compatibility with intrinsically one dimensional and special intrinsically two dimensional signals, the rotational invariance, the low computational time complexity, the easy implementation into existing software packages and the numerical robustness of calculating exact local curvature of signals without the need of any derivatives.
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References
Carneiro, G., Jepson, A.D.: Phase-based local features. In: 7th European Conference on Computer Vision-Part I. LNCS, vol. 2350, pp. 282–296. Springer, Berlin, Heidelberg, New York (2002)
Coope, I.D.: Circle fitting by linear and nonlinear least squares. J. Optim. Theory Appl. 76(2), 381–388 (1993)
do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall, New York (1976)
Felsberg, M., Sommer, G.: The monogenic signal. IEEE Trans. Signal Process. 49(12), 3136–3144 (2001)
Felsberg, M., Sommer, G.: The monogenic scale-space: a unifying approach to phase-based image processing in scale-space. J. Math. Imaging Vis. 21, 5–26 (2004)
Fleet, D.J., Jepson, A.D.: Stability of phase information. IEEE Trans. Pattern Anal. Mach. Intell. 15(12), 1253–1268 (1993)
Fleet, D.J., Jepson, A.D., Jenkin, M.R.M.: Phase-based disparity measurement. CVGIP, Image Underst. 53, 198–210 (1991)
Gabor, D.: Theory of communication. J. IEE (Lond.) 93, 429–457 (1946)
Gander, W., Golub, G.H., Strebel, R.: Least-squares fitting of circles and ellipses. BIT 34(4), 558–578 (1994)
Huang, T., Burnett, J., Deczky, A.: The importance of phase in image processing filters. IEEE Trans. Acoust. Speech Signal Process. 23(6), 529–542 (1975)
Krause, M., Sommer, G.: A 3D isotropic quadrature filter for motion estimation problems. In: Proc. Visual Communications and Image Processing, Beijing, China, vol. 5960, pp. 1295–1306. The International Society for Optical Engineering, Bellingham (2005)
Lichtenauer, J., Hendriks, E.A., Reinders, M.J.T.: Isophote properties as features for object detection. CVPR (2), 649–654 (2005)
Luo, Y., Al-Dossary, S., Marhoon, M., Alfaraj, M.: Generalized Hilbert transform and its applications in geophysics. Lead. Edge 22(3), 198–202 (2003)
Oppenheim, A.V., Lim, J.S.: The importance of phase in signals. Proc. IEEE 69(5), 529–541 (1981)
Romeny, B.M. (ed.): Geometry-Driven Diffusion in Computer Vision. Kluwer Academic, Dordrecht (1994)
Stein, E.M.: Singular Integrals and Differentiability Properties of Functions (PMS-30). Princeton University Press, Princeton (1971)
van de Weijer, J., van Vliet, L.J., Verbeek, P.W., van Ginkel, M.: Curvature estimation in oriented patterns using curvilinear models applied to gradient vector fields. In: IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 23, pp. 1035–1042 (2001)
van Ginkel, M., van de Weijer, J., van Vliet, L.J., Verbeek, P.W.: Curvature estimation from orientation fields. In: Ersboll, B.K. (ed.) 11th Scandinavian Conference on Image Analysis, pp. 545–551. Pattern Recognition Society of Denmark (1999)
Wietzke, L., Sommer, G.: The signal multi-vector. J. Math. Imaging Vis. 37, 132–150 (2010)
Xiaoxun, Z., Yunde, J.: Local Steerable Phase (LSP) feature for face representation and recognition. In: CVPR’06: Proceedings of the 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp. 1363–1368. IEEE Comput. Soc., Los Alamitos (2006)
Zang, D., Wietzke, L., Schmaltz, C., Sommer, G.: Dense optical flow estimation from the monogenic curvature tensor. In: Scale Space and Variational Methods. LNCS, vol. 4485, pp. 239–250. Springer, Berlin, Heidelberg, New York (2007)
Zetzsche, C., Barth, E.: Fundamental limits of linear filters in the visual processing of two-dimensional signals. Vis. Res. 30, 1111–1117 (1990)
Zhang, L., Qian, T., Zeng, Q.: Radon measure formulation for edge detection using rotational wavelets. Commun. Pure Appl. Anal. 6(3), 899–915 (2007)
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We acknowledge funding by the German Research Foundation (DFG) under the projects SO 320/4-2 and We 2602/5-1.
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Fleischmann, O., Wietzke, L. & Sommer, G. Image Analysis by Conformal Embedding. J Math Imaging Vis 40, 305–325 (2011). https://doi.org/10.1007/s10851-011-0263-5
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DOI: https://doi.org/10.1007/s10851-011-0263-5