×

zbMATH — the first resource for mathematics

Multi-resolution image analysis using the quaternion wavelet transform. (English) Zbl 1068.65151
Summary: 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.

MSC:
65T60 Numerical methods for wavelets
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
65T50 Numerical methods for discrete and fast Fourier transforms
Software:
DT-CWT
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] E. Bayro-Corrochano, Geometric Computing for Perception Action Systems (Springer, Boston, 2001). · Zbl 1076.68500
[2] T. Bülow, Hypercomplex spectral signal representations for the processing and analysis of images, Ph.D. thesis, Christian Albrechts University of Kiel (1999).
[3] 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.
[4] D.J. Fleet and A.D. Jepson, Computation of component image velocity from local phase information, Internat. J. Comput. Vision 5 (1990) 77-104. · doi:10.1007/BF00056772
[5] W.R. Hamilton, Elements of Quaternions (Longmans Green/Chelsea, London/New York, 1866/1969).
[6] G. Kaiser, A Friendly Guide to Wavelets (Birkhäuser, Cambridge, MA, 1994). · Zbl 0839.42011
[7] N. Kingsbury, Image processing with complex wavelets, Phil. Trans. Roy. Soc. London Ser. A 357 (1999) 2543-2560. · Zbl 0976.68527 · doi:10.1098/rsta.1999.0447
[8] J.-M. Lina, Complex Daubechies wavelets: Filters design and applications, in: ISAAC Conference, University of Delaware (June 1997). · Zbl 0904.65143
[9] J.F.A. Magarey and N.G. Kingsbury, Motion estimation using a complex-valued wavelet transform, IEEE Trans. Image Process. 6 (1998) 549-565. · Zbl 1054.94500
[10] S. Mallat, A theory for multiresolution signal decomposition: the wavelet representation, IEEE Trans. Pattern Anal. Machine Intell. 11(7) (1989) 674-693. · Zbl 0709.94650 · doi:10.1109/34.192463
[11] S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. (Academic Press, San Diego, CA, 2001). · Zbl 0998.94510
[12] M. Mitrea, Clifford Waveletes, Singular Integrals and Hardy Spaces, Lecture Notes in Mathematics, Vol. 1575 (Springer, Berlin, 1994).
[13] H.-P. Pan, Uniform full information image matching complex conjugate wavelet pyramids, in: XVIII ISPRS Congress, Vol. XXXI, Viena (July 1996).
[14] 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.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.