Gabor wavelet
Gabor wavelets are wavelets invented by Dennis Gabor using complex functions constructed to serve as a basis for Fourier transforms in information theory applications. They are very similar to Morlet wavelets. They are also closely related to Gabor filters. The important property of the wavelet is that it minimizes the product of its standard deviations in the time and frequency domain. Put another way, the uncertainty in information carried by this wavelet is minimized. However they have the downside of being non-orthogonal, so efficient decomposition into the basis is difficult. Since their inception, various applications have appeared , from image processing to analyzing neurons in the human visual system. Minimal uncertainty property. The motivation for Gabor wavelets comes from finding some function formula_1 which minimizes its standard deviation in the time and frequency domains. More formally, the variance in the position domain is: where formula_3 is the complex conjugate of formula_1 and formula_5 is the arithmetic mean, defined as: The variance in the wave number domain is: Where formula_8 is the arithmetic mean of the Fourier Transform of formula_1, formula_10: With these defined, the uncertainty is written as: This quantity has been shown to have a lower bound  of formula_13. The quantum mechanics view is to interpret formula_14 as the uncertainty in position and formula_15 as uncertainty in momentum. A function formula_1 that has the lowest theoretically possible uncertainty bound is the Gabor Wavelet. Equation. The equation of a 1-D Gabor wavelet is a Gaussian modulated by a complex exponential, described as follows: As opposed to other functions commonly used as bases in Fourier Transforms such as formula_18 and formula_19, Gabor wavelets have locality properties, meaning that as the distance from the center formula_20 increases, the value of the function becomes exponentially suppressed. formula_21 controls the rate  of this exponential drop-off and formula_8 controls the rate of modulation. It is also worth noting the Fourier transform of a Gabor wavelet, which is also a Gabor wavelet: An example wavelet is given here: Improved Gabor wavelet. propose an improved gabor wavelet transform which is invertible. The spectrum formula_24 using an improved wavlet formula_25 of signal formula_26 with frequency formula_27 at the time formula_28 is defined as formula_29 The signal can be reconstructed using formula_30 Note that formula_31 refers to the Convolution. The wavelet is defined as formula_32 formula_33