Gabor filters have attracted researchers since they extract information quanta in forms of time and frequency, two physically measurable quantities, combined in the most elegant way by the Heisenberg's uncertainty relation. For decades time-frequency analysis has played a central role in signal processing as it combines two fundamental domains and allows simultaneous representation of the signals in both. However, a signal must often be processed to mine important properties for further tasks, e.g., recognizing words from a speech or detecting objects from an image. The process that refines a signal to more meaningful pieces of information is often referred to as feature extraction. And here in this paper we are trying to explain this feature extraction using Gabor filter from its earlier track of the evolution from the very first paper by Dennis Gabor in 1946 to the present state of art.
Gabor research community lacks a good survey and the authors have thus found it necessary to gather all information into this work and enlighten the basic principles behind the theories. Different Gabor researchers have combined theories from many different contexts and as a result the Gabor function remains an alien concept, confusing researchers who are not aware of the context they should work with; Gabor filter, Gabor expansion, Gabor transform, Gabor jet, or Gabor wavelet?
[...] The major impact on the development and use of 2-d Gabor filters has been image processing and especially 2-d feature extraction. The development of the 2-d Gabor elementary functions began from Granlund in 1978, when he defined some fundamental properties and proposed the form of a general picture processing operator. The general picture processing operator had a form of the Gabor elementary function in two dimensions and it was derived directly from the needs of the image processing without a connection to Gabor's work [36]. [...]
[...] On the detection of Gabor signals and discrimination of Gabor textures. Vision Research (1985) J. Jones. (1991, May 10). Networks (2nd ed.) [Online]. Available: Carmona, R., Hwang, W., and Torrésani, B. Multiridge detection and timefrequency reconstruction. IEEE Transactions on Signal Processing (1999), Casasent, D., and Psaltis, D. New optical transforms for pattern recognition. Proceedings of the IEEE (1977) Chan, W., Coghill, G., and Sivaswamy, J. A simple mechanism for curvature detection. Pattern Recognition Letters 22, 6-7 (2001) Chen, J., Sato, Y., and Tamura, S. [...]
[...] This study's main focus is in feature extraction and its subsequent object recognition, and thus, the self-similarity of frequency wave forms, which would not be the case for example with a space spanned by polynomials of different orders, will be essential for further considerations. Frequency content of a Fourier spectrum is semantically the same regardless of signal shifts in the time domain due to the quadrature nature of complex frequencies. This would not be the case for example with splines or wavelets because they also have time dependent characteristics and non-continuous spectrum. [...]
[...] Now, the frequency constraint for the 2-d filter is λπ ( ) γπ (λπ ) + erf f 2 L ηπ ( ) ηπ > Pf (ηπ ) + erf f 2 L and the spatial constraint f L f L erf ( ( Pt 2 2 Finally, using the discrete forms of the Gabor filter in Eq or Eq or corresponding 2-d forms, and by satisfying the given constraints Eq and Eq or Eq and Eq a proper construction of the Gabor filters can be achieved and reliable results in applications can be expected. [...]
[...] Three main objectives of paper are: to cover the history of the fundamental theories and innovations involved in the discovery and development of Gabor filters in signal processing; to revisit existing and examine new filter properties useful in problems encountered in feature extraction; and (iii) to place the theory utilized thereby. Gabor research community lacks a good survey and the authors has thus found it necessary to gather all information into this work and enlighten the basic principles behind the theories. [...]
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