G-Lets: A New Signal Processing Algorithm

International Journal of Computer Applications
© 2012 by IJCA Journal
Volume 37 - Number 6
Year of Publication: 2012
Murali Rangarajan

B.Rajathilagam, Murali Rangarajan and K.P.Soman. Article: G-Lets: A New Signal Processing Algorithm. International Journal of Computer Applications 37(6):1-7, January 2012. Full text available. BibTeX

	author = {B.Rajathilagam and Murali Rangarajan and K.P.Soman},
	title = {Article: G-Lets: A New Signal Processing Algorithm},
	journal = {International Journal of Computer Applications},
	year = {2012},
	volume = {37},
	number = {6},
	pages = {1-7},
	month = {January},
	note = {Full text available}


Different signal processing transforms provide us with unique decomposition capabilities. Instead of using specific transformation for every type of signal, we propose in this paper a novel way of signal processing using a group of transformations within the limits of Group theory. For different types of signal different transformation combinations can be chosen. It is found that it is possible to process a signal at multiresolution and extend it to perform edge detection, denoising, face recognition, etc by filtering the local features. For a finite signal there should be a natural existence of basis in it’s vector space. Without any approximation using Group theory it is seen that one can get close to this finite basis from different viewpoints. Dihedral groups have been demonstrated for this purpose.


  • Rajathilagam, B., Murali Rangarajan, and Soman, K.P., 2011 G-lets: Signal Processing Algorithm Using Transformation Groups, IEEE Trans. Imag. Proc., submitted
  • Riley, K. F., Hobson, M. P., and Bence, S. J., 2002 Mathematical Methods for Physics and Engineering, Cambridge University Press.
  • Knapp, A. W., 1996 Group Representations and Harmonic Analysis, Part II, Amer. Math. Soc., 43, 537-549.
  • Duzhin, S.V., and Chebotarevsky, B.D., 2004 Transformation Groups for Beginners, Student Mathematical Library, 25.
  • Hamermesh, M., 1989 Group Theory and Its Application to Physical Problems, New York, Dover Publications.
  • Mallat, S., 1998 A Wavelet Tour of Signal Processing, Academic Press.
  • Fokas, A.S., 1997 A unified transform method for solving linear and certain nonlinear PDEs, Proc. R. Soc. London A., 453, 1411-1443.
  • Daubechies, I., 1988 Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math., 41, 909-996, October.
  • Daubechies, I., 1992 Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics.
  • Candes, E.J., 1998 Ridgelets: theory and applications, PhD thesis, Stanford University.
  • Candes, E.J., and Donoho, D.L., 1999 Ridgelets: the key to high dimensional intermittency?, Phil. Trans. R. Soc. London A, 357, 2495-2509.
  • Matus, F., and Flusser, J., 1993 Image representations via a finite Radon transform, IEEE Transactions on Pattern Analysis and Machine Intelligence, 15, 996-1006.
  • Candes, E.J., and Donoho, D.L., 2000 Curvelets and curvilinear integrals, J. Approx. Theory, 113, 59-90.
  • Do, M. N., and Vetterli, M., 2003 Contourlets, Beyond Wavelets, Stoeckler, J, Welland, G. V., Ed., Academic Press.
  • Donoho, D.L., 1999 Wedgelets: nearly-minimax estimation of edges, Ann. Statist, 27, 859-897.
  • Mallat, S., 2009 Geometrical grouplets, Appl. Comp. Harmonic Analysis, 26, 161-180.
  • Lenz,R., 1990 Group Theoretical Methods in Image Processing, Springer-Verlag New York, Inc.
  • Lenz,R., and Latorre Carmona, P., 2009 Octahedral transforms for 3-D image processing, IEEE Trans. Image Processing, 18, 2618-2628, August.
  • Vale, R., and Waldron, S., 2008 Tight frames generated by finite nonabelian groups, Numerical Algorithms, 48, 11-27, July.
  • Starck, J.L., Elad, M., and Donoho, D., 2005 Image decomposition via the combination of sparse representation and a variational approach, IEEE Trans. Image Processing, 14, 1570-1582.
  • Aharon, M., Elad, M., and Bruckstein, A.M., 2006 The K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation, IEEE Trans. Signal Processing, 54, 4311-4322, November.
  • Viana, M., and Lakshminarayanan, V., 2010 Dihedral Fourier analysis, Lecture Notes in Statistics, Springer, New York, NY.
  • Dresselhaus, M. S., 2008 Group Theory, Springer.
  • Hamermesh, M., 1965 Group theory and its applications to physical problems, Addison Wesley Publishing Company, Inc.
  • Assefa, D., Mansinha, L., Tiampo, K. F., Rasmussen, H., and Abdella, K., 2010 Local quaternion Fourier transform and color image texture analysis, Signal Processing, 90,1825-1835, June.
  • Stankovic, R. S., Moraga, C., and Astola, J., 1999 Readings in Fourier Analysis on Finite Non-Abelian Groups, TICSP Series,Sharp5, September.