U. Glaeser

FIGURE 28.17 The real (top) and imaginary (bottom) parts of a representative 2-D Gabor function.
where ∆ x, ∆ y, ∆ξ x, and ∆ξ y are the effective widths of the functions in the spatial and spatial-frequency
domains. By this measure, then, these functions are optimally local. Their real and imaginary parts also
agree reasonably well with measured receptive field profiles. The basis is not orthogonal, however.
Specifically, the Gabor transform is not equivalent to the finite-support Fourier transform with a Gaussian
window. For a cross-section of the state of the art in Gabor transform-based analysis, see [12].
The Derivative of Gaussian Transform
In 1987, Young [13] proposed a receptive field model based on the Gaussian and its derivatives. These
functions, like the Gabor functions, are spatially and spectrally local and consist of alternating regions
of excitation and inhibition in a decaying envelope. Young showed that Gaussian derivative functions
more accurately model the measured receptive field data than do the Gabor functions [14].
In [15], a spatial/spatial-frequency representation based on shifted versions of the Gaussian and its
derivatives was introduced (the derivative of Gaussian transform (DGT)). As with the Gabor transform,
although this transform is nonorthogonal, with a suitably chosen basis it is invertible. The DGT has
significant practical advantage over the Gabor transform in that both the basis functions and coefficients
of expansion are real-valued.
The family of 2-D separable Gaussian derivatives centered at the origin can be defined as
© 2002 by CRC Press LLC
g0,0( x, y)
= g0( x)g0(
y)
=
e (x2 +y 2 – )
/2σ 2
gm,n ( x, y)
= gm( x)gn(
y)
d m ( )
dx m ( )
( n)
d
dy n ( )
= -----------g 0( x)
----------g 0( y)
(28.35)
(28.36)
This set can then be shifted to any desired location. The variance σ defines the extent of the functions
in the spatial domain. There is an inverse relationship between the spatial and spectral extents, and the
value of this variable may be constant or may vary with context.
The 1-D Gaussian derivative function spectra are bimodal (except for that of the original Gaussian,
which is itself a Gaussian) with modes centered at ±Ω m rad/pixel:
m
Ωm =
-------σ
(28.37)