Abstract
We define a colour monogenic wavelet transform. This is based on recent greyscale monogenic wavelet transforms and an extension to colour signals aimed at defining non-marginal tools. Wavelet based colour image processing schemes have mostly been made by separately using a greyscale tool on every colour channel. This may have some unexpected effects on colours because those marginal schemes are not necessarily justified. Here we propose a definition that considers a colour (vector) image right at the beginning of the mathematical definition so that we can expect to create an actual colour wavelet transform – which has not been done so far to our knowledge. This provides a promising multiresolution colour geometric analysis of images. We show an application of this transform through the definition of a full denoising scheme based on statistical modelling of coefficients.
Mathematics Subject Classification (2010). Primary 68U10; secondary 15A66, 42C40.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
References
W.L. Chan, H.H. Choi, and R.G. Baraniuk. Coherent multiscale image processing using dual-tree quaternion wavelets. IEEE Transactions on Image Processing, 17(7):1069–1082, July 2008.
I. Daubechies. Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics, 1992.
G. Demarcq, L. Mascarilla, and P. Courtellemont. The color monogenic signal: A new framework for color image processing. application to color optical flow. In 16th IEEE International Conference on Image Processing (ICIP), pages 481–484, 2009.
D.L. Donoho. De-noising by soft-thresholding. IEEE Transactions on Information Theory, 41(3):613–627, 1995.
M. Felsberg. Low-Level Image Processing with the Structure Multivector. PhD thesis, Christian-Albrechts-Universität, Institut für Informatik und Praktische Mathematik, Kiel, 2002.
P. Kovesi. Image features from phase congruency. VIDERE: Journal of Computer Vision Research, 1(3):2–26, 1999.
S. Mallat. A Wavelet Tour of Signal Processing. Academic Press, third edition, 2008. First edition published 1998.
I.W. Selesnick, R.G. Baraniuk, and N.G. Kingsbury. The dual-tree complex wavelet transform. IEEE Signal Processing Magazine, 22(6):123–151, Nov. 2005.
M. Unser, D. Sage, and D. Van De Ville. Multiresolution monogenic signal analysis using the Riesz-Laplace wavelet transform. IEEE Transactions on Image Processing, 18(11):2402–2418, Nov. 2009.
M. Unser and D. Van De Ville. The pairing of a wavelet basis with a mildly redundant analysis via subband regression. IEEE Transactions on Image Processing, 17(11):2040–2052, Nov. 2008.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Basel
About this chapter
Cite this chapter
Soulard, R., Carré, P. (2013). Colour Extension of Monogenic Wavelets with Geometric Algebra: Application to Color Image Denoising. In: Hitzer, E., Sangwine, S. (eds) Quaternion and Clifford Fourier Transforms and Wavelets. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0603-9_12
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0603-9_12
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0602-2
Online ISBN: 978-3-0348-0603-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)