  The receptive fields of simple cells in the visual cortex can be understood
as linear filters. These filters can be modelled by Gabor functions, or by
Gaussian derivatives. Gabor functions can also be combined in an `energy model'
of the complex cell response. This paper proposes an alternative model of the
complex cell, based on Gaussian derivatives. It is most important to account
for the insensitivity of the complex response to small shifts of the image. The
new model uses a linear combination of the first few derivative filters, at a
single position, to approximate the first derivative filter, at a series of
adjacent positions. The maximum response, over all positions, gives a signal
that is insensitive to small shifts of the image. This model, unlike previous
approaches, is based on the scale space theory of visual processing. In
particular, the complex cell is built from filters that respond to the \twod\
differential structure of the image. The computational aspects of the new model
are studied in one and two dimensions, using the steerability of the Gaussian
derivatives. The response of the model to basic images, such as edges and
gratings, is derived formally. The response to natural images is also
evaluated, using statistical measures of shift insensitivity. The relevance of
the new model to the cortical image representation is discussed.
