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 (given by the variances defined below). 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 f ( x ) {\displaystyle f(x)} which minimizes its standard deviation in the time and frequency domains. More formally, the variance in the position domain is:
( Δ x ) 2 = ∫ − ∞ ∞ ( x − μ ) 2 f ( x ) f ∗ ( x ) d x ∫ − ∞ ∞ f ( x ) f ∗ ( x ) d x {\displaystyle (\Delta x)^{2}={\frac {\int _{-\infty }^{\infty }(x-\mu )^{2}f(x)f^{*}(x)\,dx}{\int _{-\infty }^{\infty }f(x)f^{*}(x)\,dx}}}
where f ∗ ( x ) {\displaystyle f^{*}(x)} is the complex conjugate of f ( x ) {\displaystyle f(x)} and μ {\displaystyle \mu } is the arithmetic mean, defined as:
μ = ∫ − ∞ ∞ x f ( x ) f ∗ ( x ) d x ∫ − ∞ ∞ f ( x ) f ∗ ( x ) d x {\displaystyle \mu ={\frac {\int _{-\infty }^{\infty }xf(x)f^{*}(x)\,dx}{\int _{-\infty }^{\infty }f(x)f^{*}(x)\,dx}}}
The variance in the wave number domain is:
( Δ k ) 2 = ∫ − ∞ ∞ ( k − k 0 ) 2 F ( k ) F ∗ ( k ) d k ∫ − ∞ ∞ F ( k ) F ∗ ( k ) d k {\displaystyle (\Delta k)^{2}={\frac {\int _{-\infty }^{\infty }(k-k_{0})^{2}F(k)F^{*}(k)\,dk}{\int _{-\infty }^{\infty }F(k)F^{*}(k)\,dk}}}
Where k 0 {\displaystyle k_{0}} is the arithmetic mean of the Fourier Transform of f ( x ) {\displaystyle f(x)} , F ( x ) {\displaystyle F(x)} :
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