Abstract
This paper presents the theory and practicalities of the quaternion wavelet transform. The contribution of this work is to generalize the real and complex wavelet transforms and to derive for the first time a quaternionic wavelet pyramid for multi-resolution analysis using the quaternion phase concept. The three quaternion phase components of the detail wavelet filters together with a confidence mask are used for the computation of a denser image velocity field which is updated through various levels of a multi-resolution pyramid. Our local model computes the motion by the linear evaluation of the disparity equations involving the three phases of the quaternion detail high-pass filters. A confidence measure singles out those regions where horizontal and vertical displacement can reliably be estimated simultaneously. The paper is useful for researchers and practitioners interested in the theory and applications of the quaternion wavelet transform.
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References
E. Bayro-Corrochano, Geometric Computing for Perception Action Systems (Springer, Boston, 2001).
T. Bülow, Hypercomplex spectral signal representations for the processing and analysis of images, Ph.D. thesis, Christian Albrechts University of Kiel (1999).
V.M. Chernov, Discrete orthogonal transforms with the data representation in composition algebras, in: Scandinavian Conference on Image Analysis, Uppsala, Sweden (1995) pp. 357–364.
D.J. Fleet and A.D. Jepson, Computation of component image velocity from local phase information, Internat. J. Comput. Vision 5 (1990) 77–104.
W.R. Hamilton, Elements of Quaternions (Longmans Green/Chelsea, London/New York, 1866/1969).
G. Kaiser, A Friendly Guide to Wavelets (Birkhäuser, Cambridge, MA, 1994).
N. Kingsbury, Image processing with complex wavelets, Phil. Trans. Roy. Soc. London Ser. A 357 (1999) 2543–2560.
J.-M. Lina, Complex Daubechies wavelets: Filters design and applications, in: ISAAC Conference, University of Delaware (June 1997).
J.F.A. Magarey and N.G. Kingsbury, Motion estimation using a complex-valued wavelet transform, IEEE Trans. Image Process. 6 (1998) 549–565.
S. Mallat, A theory for multiresolution signal decomposition: the wavelet representation, IEEE Trans. Pattern Anal. Machine Intell. 11(7) (1989) 674–693.
S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. (Academic Press, San Diego, CA, 2001).
M. Mitrea, Clifford Waveletes, Singular Integrals and Hardy Spaces, Lecture Notes in Mathematics, Vol. 1575 (Springer, Berlin, 1994).
H.-P. Pan, Uniform full information image matching complex conjugate wavelet pyramids, in: XVIII ISPRS Congress, Vol. XXXI, Viena (July 1996).
L. Traversoni, Image analysis using quaternion wavelet, in: Geometric Algebra in Science and Engineering Book, eds. E. Bayro Corrochano and G. Sobczyk (Springer, Berlin, 2001) chapter 16.
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Bayro-Corrochano, E. Multi-resolution image analysis using the quaternion wavelet transform. Numer Algor 39, 35–55 (2005). https://doi.org/10.1007/s11075-004-3619-8
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DOI: https://doi.org/10.1007/s11075-004-3619-8