Phase congruency is a measure of feature significance in computer images, a method of

edge detection Edge detection includes a variety of mathematical methods that aim at identifying edges, curves in a digital image at which the image brightness changes sharply or, more formally, has discontinuities. The same problem of finding discontinuitie ...

that is particularly robust against changes in illumination and contrast.

Foundations

Phase congruency reflects the behaviour of the image in the

frequency domain In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a signa ...

. It has been noted that edgelike features have many of their frequency components in the same phase. The concept is similar to coherence, except that it applies to functions of different wavelength. For example, the

Fourier decomposition A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...

of a

square wave A square wave is a non-sinusoidal periodic waveform in which the amplitude alternates at a steady frequency between fixed minimum and maximum values, with the same duration at minimum and maximum. In an ideal square wave, the transitions b ...

consists of

sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...

functions, whose frequencies are odd multiples of the fundamental frequency. At the rising edges of the square wave, each sinusoidal component has a rising phase; the phases have maximal congruency at the edges. This corresponds to the human-perceived edges in an image where there are sharp changes between light and dark.

Definition

Phase congruency compares the weighted alignment of the Fourier components of a signal

A_

with the sum of the Fourier components. :

PC(t) = \max_ \frac

where

\phi_

is the local or instantaneous phase as can be calculated using the

Hilbert transform In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, of a real variable and produces another function of a real variable . This linear operator is given by convolution with the functi ...

and

A_

are the local amplitude, or energy, of the signal. When all the phases are aligned, this is equal to 1. Several ways of implementing phase congruency have been developed, of which two versions are available in open source, one written for Matlab and the other written in Java as a plugin for the ImageJ software. Given the different notations used for its formulation, a unified version has been recently presented, where a methodology for the parameter tuning is also presented.

Advantages

The square-wave example is naive in that most edge detection methods deal with it equally well. For example, the

first derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

has a maximal magnitude at the edges. However, there are cases where the perceived edge does not have a sharp step or a large derivative. The method of phase congruency applies to many cases where other methods fail. A notable example is an image feature consisting of a single line, such as the letter "l". Many edge-detection algorithms will pick up two adjacent edges: the transitions from white to black, and black to white. On the other hand, the phase congruency map has a single line. A simple Fourier analogy of this case is a

triangle wave A triangular wave or triangle wave is a non-sinusoidal waveform named for its triangular shape. It is a periodic, piecewise linear, continuous real function. Like a square wave, the triangle wave contains only odd harmonics. However, the ...

. In each of its crests there is a congruency of crests from different sinusoidal functions.

Disadvantages

Calculating the phase congruency map of an image is very computationally intensive, and sensitive to image noise. Techniques of

noise reduction Noise reduction is the process of removing noise from a signal. Noise reduction techniques exist for audio and images. Noise reduction algorithms may distort the signal to some degree. Noise rejection is the ability of a circuit to isolate an und ...

are usually applied prior to the calculation.

References

{{Reflist

External links

Research page of Peter Kovesi
list of research papers, example images and implementations.
Lecture by Professor Michael Brady
Image processing Computer vision