Image Representation by the Magnitude of the Discrete Gabor Wavelet Transform

We present an analytical analysis of the representation of images as the magnitudes of their transform with complex-valued Gabor wavelets. Although reconstruction of the image is difficult such a representation is very useful for image understanding purposes. We show that if the sampling of the linear wavelet transform is appropriate then the representation by the nonlinearity introduced by the magnitude is unique up to the sign for almost all images. Finally, numerical experiments with a phase retrieval algorithm show that recognizable versions of the original image can be reconstructed from the magnitudes. Keywords— Gabor wavelets, feature extraction, phase retrieval, Gabor magnitudes, DFT, interpolation, image coding, visual cortex.

[1]  A.V. Oppenheim,et al.  The importance of phase in signals , 1980, Proceedings of the IEEE.

[2]  Gerald Kaiser,et al.  A Friendly Guide to Wavelets , 1994 .

[3]  Daniel A. Pollen,et al.  Visual cortical neurons as localized spatial frequency filters , 1983, IEEE Transactions on Systems, Man, and Cybernetics.

[4]  Dennis Gabor,et al.  Theory of communication , 1946 .

[5]  Joachim M. Buhmann,et al.  Distortion Invariant Object Recognition in the Dynamic Link Architecture , 1993, IEEE Trans. Computers.

[6]  Rolf P. Würtz,et al.  Object Recognition Robust Under Translations, Deformations, and Changes in Background , 1997, IEEE Trans. Pattern Anal. Mach. Intell..

[7]  Stéphane Mallat,et al.  A Theory for Multiresolution Signal Decomposition: The Wavelet Representation , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[8]  Jochen Triesch,et al.  Robust classification of hand postures against complex backgrounds , 1996, Proceedings of the Second International Conference on Automatic Face and Gesture Recognition.

[9]  J. P. Jones,et al.  An evaluation of the two-dimensional Gabor filter model of simple receptive fields in cat striate cortex. , 1987, Journal of neurophysiology.

[10]  D. Sagi,et al.  Gabor filters as texture discriminator , 1989, Biological Cybernetics.

[11]  J. Daugman.  Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters. , 1985, Journal of the Optical Society of America. A, Optics and image science.

[12]  W. Kilmer.  A Friendly Guide To Wavelets , 1998, Proceedings of the IEEE.

[13]  Edward H. Adelson,et al.  Shiftable multiscale transforms , 1992, IEEE Trans. Inf. Theory.

[14]  R. Duffin,et al.  A class of nonharmonic Fourier series , 1952 .

[15]  R. Murenzi.  Wavelet Transforms Associated to the n-Dimensional Euclidean Group with Dilations: Signal in More Than One Dimension , 1990 .

[16]  B. C. McCallum,et al.  Fourier transform magnitudes are unique pattern recognition templates , 1986, Biological Cybernetics.

[17]  Rolf P. Würtz,et al.  Corner detection in color images through a multiscale combination of end-stopped cortical cells , 2000, Image Vis. Comput..

[18]  Tai Sing Lee,et al.  Image Representation Using 2D Gabor Wavelets , 1996, IEEE Trans. Pattern Anal. Mach. Intell..

[19]  M. Hayes,et al.  Reducible polynomials in more than one variable , 1982, Proceedings of the IEEE.

[20]  M. Hayes.  The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform , 1982 .

[21]  A. Grossmann,et al.  DECOMPOSITION OF HARDY FUNCTIONS INTO SQUARE INTEGRABLE WAVELETS OF CONSTANT SHAPE , 1984 .

[22]  James R. Fienup,et al.  Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint , 1987 .