# Image Analysis and Reconstruction using a Wavelet Transform Constructed from a Reducible Representation of the Euclidean Motion Group

- 210 Downloads
- 33 Citations

## Abstract

Inspired by the early visual system of many mammalians we consider the construction of-and reconstruction from- an orientation score \({\it U_f}:\mathbb{R}^2 \times S^{1} \to \mathbb{C}\) as a local orientation representation of an image, \(f:\mathbb{R}^2 \to \mathbb{R}\). The mapping \(f\mapsto {\it U_f}\) is a wavelet transform \(\mathcal{W}_{\psi}\) corresponding to a reducible representation of the Euclidean motion group onto \(\mathbb{L}_{2}(\mathbb{R}^2)\) and oriented wavelet \(\psi \in \mathbb{L}_{2}(\mathbb{R}^2)\). This wavelet transform is a special case of a recently developed generalization of the standard wavelet theory and has the practical advantage over the usual wavelet approaches in image analysis (constructed by irreducible representations of the similitude group) that it allows a stable reconstruction from one (single scale) orientation score. Since our wavelet transform is a unitary mapping with stable inverse, we directly relate operations on orientation scores to operations on images in a robust manner.

Furthermore, by geometrical examination of the Euclidean motion group \(G=\mathbb{R}^2 \mathbb{R}\times \mathbb{T}\), which is the domain of our orientation scores, we deduce that an operator Φ on orientation scores must be left invariant to ensure that the corresponding operator \(\mathcal{W}_{\psi}^{-1}\Phi \mathcal{W}_{\psi}\) on images is Euclidean invariant. As an example we consider all linear second order left invariant evolutions on orientation scores corresponding to stochastic processes on *G*. As an application we detect elongated structures in (medical) images and automatically close the gaps between them.

Finally, we consider robust orientation estimates by means of channel representations, where we combine robust orientation estimation and learning of wavelets resulting in an auto-associative processing of orientation features. Here linear averaging of the channel representation is equivalent to robust orientation estimation and an adaptation of the wavelet to the statistics of the considered image class leads to an auto-associative behavior of the system.

## Keywords

Orientation Score Reproduce Kernel Hilbert Space Fourier Domain Orientation Estimation Scale Space Theory## Preview

Unable to display preview. Download preview PDF.

## References

- Ali S.T., Antoine J.P. and Gazeau J.P. 1999.
*Coherent States, Wavelets and Their Generalizations*. Springer Verlag, New York, Berlin, Heidelberg.zbMATHGoogle Scholar - Antoine J.P. 1999. Directional wavelets revisited: Cauchy wavelets and symmetry detection in patterns.
*Applied and Computational Harmonic Analysis*, 6:314–345.zbMATHMathSciNetCrossRefGoogle Scholar - Aronszajn, N. 1950. Theory of reproducing kernels.
*Trans. A.M.S*., 68:337–404.zbMATHMathSciNetCrossRefGoogle Scholar - August J. and Zucker S.W. 2003. The curve indicator random field and markov processes.
*IEEE-PAMI, Pattern Recognition and Machine Intelligence*, 25.Google Scholar - Bosking W.H., Zhang Y., Schofield B. and Fitzpatrick D. 1997. Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex.
*The Journal of Neuroscience*, 17(6):2112–2127.Google Scholar - Duits M. 2004. A functional Hilbert space approach to frame transforms and wavelet transforms. Master thesis in Applied Analysis group at the department of Mathematics and Computer Science at the Eindhoven University of Technology.Google Scholar
- Duits, M. and Duits, R. 2004. A functional Hilbert space approach to the theory of wavelets. Technical report, TUE, Eindhoven, RANA/CASA Report RANA-7-2004, available on the web: ftp://ftp.win.tue.nl/pub/rana/rana04-07.pdf Department of Mathematics Eindhoven University of Technology.
- Duits, R. 2005.
*Perceptual Organization in Image Analysis*. PhD thesis, Eindhoven University of Technology, Department of Biomedical Engineering, The Netherlands. A digital version is available on the web: URL: http://www.bmi2.bmt.tue.nl/Image-Analysis/People/RDuits/THESISRDUITS.pdf. - Duits, R. and van Almsick, M. 2005. The explicit solutions of the left invariant evolution equations on the Euclidean motion group. Eindhoven University of Technology, Eindhoven, 5–43. Available on the web http://yp.bmt.tue.nl/pdfs/6321.pdf. An improved version of which is recently submitted to the Quarterly of Applied Mathematics (journal of American Mathetical Society.)
- Duits, R., Duits, M. and van Almsick, M. 2004. Invertible orientation scores as an application of generalized wavelet theory. Technical report, TUE, Eindhoven. Technical Report 04-04, Biomedical Image and Analysis, Department of Biomedical Engineering, Eindhoven University of Technology.Google Scholar
- Duits, R., Florack, L.M.J., de Graaf, J. and ter Haar Romeny, B. 2004. On the axioms of scale space theory.
*Journal of Mathematical Imaging and Vision*, 20:267–298.MathSciNetCrossRefGoogle Scholar - Duits, R., van Almsick, M., Duits, M., Franken, E. and Florack, L.M.J. 2004. Image processing via shift-twist invariant operations on orientation bundle functions. In Niemann Zhuralev et al. Geppener, Gurevich, editor,
*7th International Conference on Pattern Recognition and Image Analysis: New Information Technologies*, 193–196, St. Petersburg. Extended version is to appear in special issue of the International Journal for Pattern Recognition and Image Analysis MAIK.Google Scholar - Dungey, N., ter Elst, A.F.M. and Robinson, D.W. 2003.
*Analysis on Lie groups with polynomial growth*, volume 214. Birkhauser-Progress in Mathematics, Boston.Google Scholar - Eijndhoven, S.J.L. and de Graaf, J. 1982. Some results on hankel invariant distribution spaces.
*Proceedings of the Koninklijke Akademie van Wetenschapppen, Series A*, 86(1):77–87.Google Scholar - Faraut, J. and Harzallah, K. 1984.
*Deux cours d’analyse harmonique*. Birkhaeuser, Tunis.Google Scholar - Felsberg, M., Forssén, P.-E. and Scharr, H. 2004. Efficient robust smoothing of low-level signal features. Technical Report LiTH-ISY-R-2619, SE-581 83 Linkoping, Sweden.Google Scholar
- Felsberg, M., Forssén, P.-E. and Scharr, H. 2006. Channel smoothing: Efficient robust smoothing of low-level signal features.
*IEEE Transactions on Pattern Analysis and Machine Intelligence*, 28(2):209–222.CrossRefGoogle Scholar - Florack, L.M.J., 1997.
*Image Structure*. Kluwer Academic Publishers, Dordrecht, The Netherlands.Google Scholar - Forssén, P.-E. and Granlund, G. H. 2000. Sparse feature maps in a scale hierarchy. In G. Sommer and Y.Y. Zeevi, editors,
*Proc. Int. Workshop on Algebraic Frames for the Perception-Action Cycle*, volume 1888 of*Lecture Notes in Computer Science*, Kiel, Germany, Springer, Heidelberg.Google Scholar - Forssén, P.E. 2004.
*Low and Medium Level Vision using Channel Representations*. PhD thesis, Linkoping University, Dept. EE, Linkoping, Sweden.Google Scholar - van Ginkel, M. 2002.
*Image Analysis using Orientation Space based on Steerable Filters*. PhD thesis, Delft University of Technology, Delft, Netherlands.Google Scholar - Granlund, G.H. 2000. An associative perception-action structure using a localized space variant information representation. In
*Proceedings of Algebraic Frames for the Perception-Action Cycle (AFPAC)*, Kiel, Germany, Also as Technical Report LiTH-ISY-R-2255.Google Scholar - Grossmann, A., Morlet, J. and Paul, T. 1985. Integral transforms associated to square integrable representations.
*J.Math.Phys*., 26:2473–2479.zbMATHMathSciNetCrossRefGoogle Scholar - Isham, C.J. and Klauder J.R. 1991. Coherent states for
*n*-dimensional euclidean groups*e*(*n*) and their application.*Journal of Mathematical Physics*, 32(3):607–620.zbMATHMathSciNetCrossRefGoogle Scholar - Kalitzin, S.N., ter Haar Romeny, B.M. and Viergever, M.A. 1999. Invertible apertured orientation filters in image analysis.
*International Journal of Computer Vision*, 31(2/3):145–158.CrossRefGoogle Scholar - Lee T.S., 1996. Image representation using 2d gabor wavelets.
*IEEE-Transactions on Pattern Analysis and Machine Inteligence*, 18(10):959–971.CrossRefGoogle Scholar - Louis, A.K., Maass, P. and Rieder, A. 1997.
*Wavelets, Theory and Applications*. Wiley, New York.zbMATHGoogle Scholar - Martens, F.J.L. 2004.
*Spaces of analytic functions on inductive/projective limits of Hilbert Spaces*. PhD thesis, University of Technology Eindhoven, Department of Mathematics and Computing Science, Eindhoven, The Netherlands, 1988. This PHD thesis is available on the webpages of the Technische Universiteit Eindhoven. Webpage in: http://alexandria.tue.nl/extra3/proefschrift/PRF6A/8810117.pdf. - Mumford, D. 1994. Elastica and computer vision.
*Algebraic Geometry and Its Applications. Springer-Verlag*, 491–506.Google Scholar - van der Put, R.W. 2005. Methods for 3d orientation analysis and their application to the study of arterial remodelling. Master’s thesis, Department of Biomedical Engineering Eindhoven University of Technology, Technical Report BMIA-0502.Google Scholar
- Sugiura, M. 1990.
*Unitary representations and harmonic analysis*. North-Holland Mathematical Library, 44., Amsterdam, Kodansha, Tokyo, second edition.Google Scholar - Thornber, K.K. and Williams, L.R. 1996. Analytic solution of stochastic completion fields.
*Biological Cybernetics*, 75:141– 151.zbMATHCrossRefGoogle Scholar - Ts’0, D.Y., Frostig, R.D., Lieke, E.E. and Grinvald, A. 1990. Functional organization of primate visual cortex revealed by high resolution optical imaging.
*Science*, 249:417–20.CrossRefGoogle Scholar - Twareque Ali, S. 1998. A general theorem on square-integrability: Vector coherent states.
*Journal of Mathematical Physics*, 39.Google Scholar - van Almsick, M.A., Duits, R., Franken E. and ter Haar Romeny, B.M. 2005. From stochastic completion fields to tensor voting. In
*Proceedings DSSCC-workshop on Deep Structure Singularities and Computer Vision*, Maastricht the Netherlands, Springer-Verlag.Google Scholar - Williams, L.R. and Zweck, J.W. 2003. A rotation and translation invariant saliency network.
*Biological Cybernetics*, 88:2–10.zbMATHCrossRefGoogle Scholar