The Monogenic ScaleSpace: A Unifying Approach to PhaseBased Image Processing in ScaleSpace
 M. Felsberg,
 G. Sommer
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In this paper we address the topics of scalespace and phasebased image processing in a unifying framework. In contrast to the common opinion, the Gaussian kernel is not the unique choice for a linear scalespace. Instead, we chose the Poisson kernel since it is closely related to the monogenic signal, a 2D generalization of the analytic signal, where the Riesz transform replaces the Hilbert transform. The Riesz transform itself yields the flux of the Poisson scalespace and the combination of flux and scalespace, the monogenic scalespace, provides the local features phasevector and attenuation in scalespace. Under certain assumptions, the latter two again form a monogenic scalespace which gives deeper insight to lowlevel image processing. In particular, we discuss edge detection by a new approach to phase congruency and its relation to amplitude based methods, reconstruction from local amplitude and local phase, and the evaluation of the local frequency.
 J. Babaud, A.P.Witkin, M. Baudin, and R.O. Duda, “Uniqueness of the Gaussian kernel for scalespace filtering,” IEEE Transactions onPattern Analysis andMachine Intelligence,Vol. 8, No. 1, pp. 26–33, 1986.
 J. Behar, M. Porat, and Y.Y. Zeevi. “Image reconstruction from localized phase,” IEEE Transactions on Signal Processing, Vol. 40, No. 4, pp. 736–743, 1992.
 R.N. Bracewell, The Fourier Transform and its Applications, McGraw Hill, 1986.
 R.N. Bracewell, TwoDimensional Imaging, Prentice Hall Signal Processing Series. Prentice Hall, Englewood Cliffs, 1995.
 F. Brackx, R. Delanghe, and F. Sommen, Clifford Analysis, Pitman: Boston, 1982.
 I. Bronstein, K. Semendjajew, G. Musiol, and H. Mühlig, Taschenbuch der Mathematik, Verlag Harri Deutsch, Frankfurt, 1993.
 K. Burg, H. Haf, and F. Wille, Höhere Mathematik für Ingenieure, Band V Funktionalanalysis und Partielle Differentialgleichungen, Teubner Stuttgart, 1993.
 K. Burg, H. Haf, and F. Wille, Höhere Mathematik für Ingenieure, Band IV Vektoranalysis und Funktionentheorie, Teubner Stuttgart, 1994.
 P.J. Burt and E.H. Adelson, “The Laplacian pyramid as a compact image code,” IEEE Trans. Communications, Vol. 31, No. 4, pp. 532–540, 1983.
 F. Catté, P.L. Lions, J.M. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Analysis, Vol. 32, pp. 1895–1909, 1992.
 R. Duits, L.M.J. Florack, J. de Graaf, and B.M. ter Haar Romeny, “On the axioms of scale space theory,” Journal of Mathematical Imaging and Vision, 2002 (accepted).
 R. Duits, L.M.J. Florack, B. M. ter Haar Romeny, and J. de Graaf, “Scalespace axioms critically revisited,” in Signal and Image Processing, N. Younan (Ed.), IASTED, ACTA Press, Kauai, August 2002, pp. 304–309.
 M. Evans, N. Hastings, and J.B. Peacock, Statistical Distributions, 3rd. ed., WileyInterscience, 2000.
 M. Felsberg, “Disparity from monogenic phase,” in 24. DAGM Symposium Mustererkennung, Zürich, L.V. Gool (Ed.), Vol. 2449 of Lecture Notes in Computer Science, Springer, Heidelberg, 2002, pp. 248–256.
 M. Felsberg, LowLevel Image Processing with the Structure Multivector, Ph.D. thesis, Institute of Computer Science and Applied Mathematics, ChristianAlbrechtsUniversity of Kiel, 2002. TR no. 0203, available at http://www.informatik.unikiel. de/reports/2002/0203.html.
 M. Felsberg, R. Duits, and L. Florack, “The monogenic scale space on a bounded domain and its applications,” in Scale Space Conference, 2003 (accepted).
 M. Felsberg and G. Sommer, “A new extension of linear signal processing for estimating local properties and detecting features,” in 22. DAGMSymposium Mustererkennung, G. Sommer, N. Krüger, and C. Perwass (Eds.), Springer, Heidelberg, Kiel, 2000, pp. 195–202.
 M. Felsberg and G. Sommer, “The monogenic signal,” IEEE Transactions on Signal Processing, Vol. 49, No. 12, pp. 3136–3144, 2001.
 M. Felsberg and G. Sommer, “Scale adaptive filtering derived from the Laplace equation,” in 23. DAGM Symposium Mustererkennung, B. Radig and S. Florczyk (Eds.),Vol. 2191 of Lecture Notes in Computer Science, Springer, Heidelberg, München, 2001, pp. 124–131.
 M. Felsberg and G. Sommer, “The Poisson scalespace: A unified approach to phasebased image processing in scalespace,” Tech. Rep. LiTHISYR2453, Dept. EE, Linköping University, SE581 83 Linköping, Sweden, 2002.
 M. Felsberg and G. Sommer, “The structure multivector,” in Applied Geometrical Algebras in Computer Science and Engineering, Birkhäuser, Boston, 2002, pp. 437–448.
 L. Florack, Image Structure, Vol. 10 of Computational Imaging and Vision, Kluwer Academic Publishers, 1997.
 L. Florack and A. Kuijper, “The topological structure of scalespace images,” Journal of Mathematical Imaging and Vision, Vol. 12, No. 1, pp. 65–79, 2000.
 G.H. Granlund, “In search of a general picture processing operator,” Computer Graphics and Image Processing, Vol. 8, pp. 155–173, 1978.
 G.H. Granlund and H. Knutsson, Signal Processing for Computer Vision, Kluwer Academic Publishers: Dordrecht, 1995.
 S.L. Hahn, Hilbert Transforms in Signal Processing, Artech House: Boston, London, 1996.
 D. Hestenes, “Multivector calculus,” J. Math. Anal. and Appl., Vol. 24, No. 2, pp. 313–325, 1968.
 R.A. Hummel, “Representations based on zerocrossings in scale space,” in Proc. IEEE Comp. Soc. Conf. Computer Vision and Pattern Recognition, Miami Beach, 1986, pp. 204–209.
 T. Iijima, “Basic theory of pattern observation,” in Papers of Technical Group on Automata and Automatic Control, IECE, Japan, December 1959.
 T. Iijima, “Basic theory on normalization of pattern (In case of a typical onedimensional pattern),” Bulletin of the Electrotechnical Laboratory, Vol. 26, pp. 368–388, 1962.
 T. Iijima, “Observation theory of twodimensional visual patterns,” in Papers of Technical Group on Automata and Automatic Control, IECE, Japan, October 1962.
 B. Jähne, Digitale Bildverarbeitung, Springer: Berlin, 1997.
 J.J. Koenderink, “The structure of images,” Biological Cybernetics, Vol. 50, pp. 363–370, 1984.
 P. Kovesi, “Image features from phase information,” Videre: Journal of Computer Vision Research, Vol. 1, No. 3, 1999.
 S.G. Krantz, Handbook of Complex Variables, Birkhäuser: Boston, 1999.
 G. Krieger and C. Zetzsche, “Nonlinear image operators for the evaluation of local intrinsic dimensionality,” IEEE Transactions on Image Processing, Vol. 5, No. 6, pp. 1026–1041, 1996.
 T. Lindeberg, ScaleSpace Theory in Computer Vision, The Kluwer International Series in Engineering and Computer Science. Kluwer Academic Publishers: Boston, 1994.
 T. Lindeberg, “Linear spatiotemporal scalespace,” in ScaleSpace Theory in Computer Vision, Vol. 1252 of Lecture Notes in Computer Science, Springer: Utrecht, Netherlands, 1997.
 T. Lindeberg, On the Axiomatic Foundations of Linear ScaleSpace: Combining SemiGroup Structure with Causality vs. Scale Invariance, Ch. 6, Kluwer Academic, 1997.
 A. Papoulis, The Fourier Integral and its Applications, McGrawHill: New York, 1962.
 A. Papoulis, Probability, Random Variables and Stochastic Processes, McGrawHill, 1965.
 E.J. Pauwels, L.J. Van Gool, P. Fiddelaers, and T. Moons, “An extended class of scaleinvariant and recursive scale space filters,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 17, No. 7, pp. 691–701, 1995.
 P. Perona and J. Malik, “Scalespace and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. 12, No. 7, pp. 629–639, 1990.
 D. Reisfeld, “The constrained phase congruency feature detector: Simultaneous localization, classification and scale determination,” Pattern Recognition Letters, Vol. 17, pp. 1161–1169, 1996.
 J.L. Schiff, The Laplace Transform, Undergraduate Texts in Mathematics. Springer: New York, 1999.
 N. Sochen, R. Kimmel, and R. Malladi, “A geometrical framework for low level vision,” IEEE Trans. on Image Processing, Special Issue on PDE Based Image Processing, Vol. 7, No. 3, pp. 310–318, 1998.
 E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press: New Jersey, 1971.
 J. Weickert, “Anisotropic siffusion in image processing,” Ph.D. thesis, Faculty of Mathematics, University of Kaiserslautern, 1996.
 J. Weickert, “A review of nonlinear diffusion filtering,” in ScaleSpace Theory in Computer Vision, B. ter Haar Romeny, L. Florack, J. Koenderink, and M. Viergever (Eds.), Vol. 1252 of LNCS, Springer: Berlin, 1997, pp. 260–271.
 J. Weickert, S. Ishikawa, and A. Imiya, “Scalespace has first been proposed in Japan,” Mathematical Imaging and Vision, Vol. 10, pp. 237–252, 1999.
 A.P. Witkin, “Scalespace filtering,” in Proc. 8th Int. Joint Conf. Art. Intell., 1983, pp. 1019–1022.
 A.L. Yuille and T. Poggio, “Scaling theorems for zerocrossings,” IEEE Trans. Pattern Analysis and Machine Intell., Vol. 8, pp. 15–25, 1986.
 Title
 The Monogenic ScaleSpace: A Unifying Approach to PhaseBased Image Processing in ScaleSpace
 Journal

Journal of Mathematical Imaging and Vision
Volume 21, Issue 12 , pp 526
 Cover Date
 20040701
 DOI
 10.1023/B:JMIV.0000026554.79537.35
 Print ISSN
 09249907
 Online ISSN
 15737683
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 Poisson kernel
 scalespace
 local phase
 analytic signal
 Riesz transform
 monogenic signal
 Industry Sectors
 Authors

 M. Felsberg ^{(1)}
 G. Sommer ^{(2)}
 Author Affiliations

 1. Department of Electrical Engineering, Linköping University, Sweden
 2. Institute of Computer Science and Applied Mathematics, ChristianAlbrechtsUniversity of Kiel, Germany