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  demonstrated that it is crucial to the perception of visual features. Further physiological evidence  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
where, for each frequency , is the amplitude of the cosine wave and is the phase offset of that wave. The term T is related to the size of the image window, and from now on we will assume it is 1.
For example, if the image signal were a simple step edge, then
and, at the point of the step edge (x = 0), all the phase terms are aligned at . This is the only place in the signal where there is congruency in the phase values; at all other points, the phase values of individual frequency components assume differing values between 0 and .
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
To understand this definition, we can think of our signal f(x), at any point x in the image, as being made up of the sum of various sine waves at different amplitudes and phase angles, which we plot on a vector map.
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
But this corresponds to simple phase shifting in the frequency domain. Specifically, the positive frequency terms are phase shifted by 90o and the negative frequency terms are phase shifted by -90o; the zero frequency, or d.c. term, is set to zero, since phase shifting has no meaning here. Thus, in the frequency domain,
where sgn is the function that returns the sign of its argument. Thus, if
We can define a vector E at a point x by
where and are the unit vectors along the x and y axes. At any particular point x in the signal, E is the vector sum (or integral) of the Fourier terms. The nth component is a vector of length an making an angle of with the x axis.
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
Local peaks in the local energy function correspond to local peaks in the phase congruency function.
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 90o 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 Me represents the even filter and Mo represents the odd filter, then
Moreover, the human visual system has the capacity to compute the sum of squares of the output from convolution with the odd and even symmetric filters, that is, by combining the output of the simple and complex cell responses, it computes a local energy for the signal. This is made more precise by defining
The filters Me and Mo 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 90o 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 .
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.