Rather than assuming that what an image should be compressed into is a
set of edges, the phase congruency model of feature detection assumes
that the compressed image format should be high in *information*
(or *entropy*), and low in redundancy. Thus, instead of searching
for points where there are sharp changes in intensity, this model
searches for patterns of *order* in the phase component of the
Fourier transform. Phase is chosen because the experiments of
Oppenheim and Lim [9] demonstrated that it is crucial to the
perception of visual features. Further physiological
evidence [6] indicates that the human visual system responds
strongly to points in an image where the phase information is highly
ordered. Thus the phase congruency model defines *features* as
points in an image with high phase order.

The phase congruency model is a *frequency-based* model of
visual processing. It supposes that, instead of processing visual
data spatially, the visual system is capable of performing
calculations using the phase and amplitude of the individual frequency
components in a signal. Thus, the underlying computational tool is the
Fourier transform, or one of its equivalents. To this end, let us
suppose that we can represent our image signal in the Fourier
domain. To simplify the presentation, we will assume a simple
one-dimensional signal, representing a 1D slice through an image. Such
a signal, say *f*(*x*), is reconstructed from its Fourier transform by

For example, if the image signal were a simple step edge, then

and, at the point of the step edge (
We need to make precise what it is we mean by phase congruency. This is
done by defining the *phase congruency function*, *PC*(*x*), at each
point *x* in the signal. We have

Although the definition of *PC* captures precisely what it is we want
to measure, it is an awkward function to implement. Luckily, some
simple trigonometric manipulations suffice to prove that *PC* is
proportional to a well-known computation in biological vision, namely
the *local energy* in a signal.

The local energy of a signal is defined in terms of the signal
and its *Hilbert transform*. The Hilbert transform has a somewhat
complicated definition in the spatial domain, namely

We can define a vector **E** at a point *x* by

The magnitude of the vector **E** is called the *local energy*
of the signal (sometimes also called the *envelope* of the
signal) and it is defined as

The argument

gives the angle at which the phase congruency occurs, and can be used to define the feature type.
As was mentioned earlier, the human visual system has the capacity to
simulate convolution by odd and even symmetric filters in quadrature.
That the filters are in quadrature means not only that they form an
odd and even symmetric pair (that is, the output of convolution by one
filter is a 90^{o} phase shift of the output of the other), but also
that they both have a zero mean value and the same sum-of-squares
value. More specifically, if *M*_{e} represents the even filter and
*M*_{o} represents the odd filter, then

The filters *M*_{e} and *M*_{o} need to be carefully designed. The even
symmetric filter is chosen so that it covers as much of the frequency
spectrum as possible, whilst eliminating the d.c. term. The odd symmetric
filter is then just a 90^{o} phase shift filter of the even filter. Thus

It is now simple to see why , for

So, in order to search for local maxima in the phase congruency function, one equivalently searches for local maxima in the local energy function. These local maxima will occur at step edges of either parity (up or down), lines and bar edges, and other types of features such as the illusory Mach bands [8].

To illustrate how this works, figure 5 shows a simple test image that
contains a variety of features at different contrasts. Figure 6 shows
the output of a simple gradient-based edge detector (here, the Sobel
operator). Note that the output depends on the relative contrast of
the edge, and that the output for line features is *two* edges,
one on either side of the line. Figure 7 shows the output of the local
energy (or phase congruency) detector. Here we note that the output is
a uniform response, regardless of the type or contrasts of the feature
involved.

Figures 8 and 9 illustrate the local energy map and the detected features for the mandrill image.