Analytical Analysis of Image Representation by Their Discrete Wavelet Transform

In this paper, we present an analytical analysis of the representation of images as the magnitudes of their transform with the discrete wavelets. Such a representation plays as a model for complex cells in the early stage of visual processing and of high technical usefulness for image understanding, because it makes the representation insensitive to small local shifts. We found that if the signals are band limited and of zero mean, then reconstruction from the magnitudes is unique up to the sign for almost all signals. We also present an iterative reconstruction algorithm which yields very good reconstruction up to the sign minor numerical errors in the very low frequencies.

Authors:

• R. M. Farouk

Keywords:

• Wavelets
• Image processing signal processing
• Image reconstruction

References:

[1] J. Jones and L. Palmer , An evaluation of two-dimensional Gabor filter
model of simple receptive fields in cat striate cortex, J. Neurophysiology
1987, pp.1233-1258.
[2] M. Lades, C. Vorbr├╝ggen, J. Buhmann J. Lange C. von. der MalsburgR
.P. W├╝rtz, Distortion Invariant Object Recognition in the Dynamic Link
Architecture, IEEE Transaction on computer, vol.42, number 3, 1993,
pp.300-310.
[3] G. Kaiser, Friendly Guide to Wavelets, Birkhäuser, 1994.
[4] R. M. Farouk , A system for Finding and segmenting a hand in Partially
cluttered scene, may 2007, Proceeding ICAS 29-31.
[5] J. G. Daugman Uncertainty relation for resolution in space, spatial
frequency, and orientation optimized by two-dimensional visual cortical
filters, Journal of the Optical Society of America vol. 2 number 7, 1985,
pp. 1362-1373.
[6] D. A. Pollen, S. F. Ronner, Phase relationships between adjacent
simple cells in the visual cortex, Science vol.212 , 1981, pp.1409-1411.
[7] C A. Daniel Steven F. Ronner, Visual cortical neurons as localized
spatial frequency filter, IEEE Trans. on systems vol.38, number 2, 1992,
pp. 587-607.
[8] R. M. Farouk, Reconstruction of objects from images with partial
occlusion, PhD thesis 2006.
[9] I. Fogel D. Sagi, Gabor filters as texture discriminator, Biological
Cybernetics, vol 16, 1989, pp.103-113.
[10] W. Xing B. Bhanu, Gabor Wavelet Representation for 3-D Object
Recognition, IEEE Transactions on Image Processing vol. 6 number 1,
1997, pp.47-64.
[11] L. Shen L. Bai M. Fairhurst, Gabor wavelets and General Discriminant
Analysis for face identification and verifications, J. Image vision
computing vol. 25, 2007, pp. 553-563.
[12] G. Xijin S. Iwata, Learning the parts of objects by Auto-association,
vol.15, 2002, pp.285-295.
[13] P.H. Gardenier , B.C. McClellan R. H. T. Bates, Fourier transform
magnitudes are unique pattern recognition templates, Biological
Cybernetics vol. 54 pp.385-391 1986.
[14] M. Nabti A. Bouridane, An effective and fast iris recognition system
based on a combined multiscale feature extraction technique, Pattern
Recognition vol. 41 2008.
[15] M.H. Hayes , The reconstruction of a multidimensional sequence from
the phase or magnitude of its Fourier transform, IEEE Trans. on
Acoustics, speech, and Signal Processing vol. 30 number 2 , pp.140-154
1982.
[16] M. H. Hayes H. J. McClellan, Reducible polynomials in more than one
variable, Proceeding of IEEE vol.70 number 2 pp. 197-198 1982.
[17] J. R. Fienup, Reconstruction of complex-valued object from the modulus
of its Fourier transform using a support constraint, J. of Optical Society
of America vol.4 number 1 pp. 118-123 1987.