# What is phase congruency?

I am beginner in Image processing.I am studying the importance of phase in signal.Can anybody explain what is phase congruency ?

• you have asked questions on DSP stack exhange, and you keep doing. Most of them have received good answers, could you consider marking them as answered if they were helpfull to you ? – Antoine Bassoul Apr 30 '15 at 13:04

## 2 Answers

Phase congruency is where the phases of a quadrature filter responses at different scales are similar.

A quadrature filter consists of two parts:

An even (symmetrical) part and an odd (anti-symmetrical) part, which are the Hilbert (or Riesz) transforms of each other. See below:

Let $u(x)$ be an even function with finite energy, and $v(x) = \mathcal{H}(u)(x)$ be its Hilbert transform. The functions are in quadrature.

We can combine them as one filter by multiplying the odd part by $i$ and adding them together: $g(x) = u(x) + i v(x)$

Convolving the filter with a signal, $f(x)$, we thus get a complex response where the even part is the response to the even filter and the odd part is the response to the odd filter: $f_A(x) = (f \ast g)(x)$

Phase is the argument of this complex response $\phi = \arg f_A(x)$ and is separate from the amplitude of the response $A = |f_A(x)|$. Phase describes the shape of the signal at a point - odd or even or somewhere in-between - while amplitude describes the strength. This independence is important as it means signals with different strengths can be compared.

If the phase is the same at multiple scales, then there is likely to be a signal feature there. This is phase congruency. Proof and examples of this are in the papers below. The big result is that phase congruency can find signal features independent of their strength. An extremely useful collection of phase related MATLAB routines:

http://www.csse.uwa.edu.au/~pk/Research/MatlabFns/index.html

Some links:

Papers:

• Oppenheim, A. V., & Lim, J. S. (1981). The importance of phase in signals.
• Kovesi, P. (1995). Image Features From Phase Congruency.
• Kovesi, P. (1996). Invariant measures of image features from phase information.
• Kovesi, P. (1997). Symmetry and asymmetry from local phase.
• Kovesi, P. (1999). Image features from phase congruency.
• Kovesi, P. (2002). Edges are not just steps.
• Kovesi, P. (2003). Phase congruency detects corners and edges.
• Morrone, M. C., & Burr, D. C. (1988). Feature detection in human vision: a phase-dependent energy model.
• Morrone, M. C., & Owens, R. a. (1987). Feature detection from local energy.
• Perona, P., & Malik, J. (1990). Detecting and Localizing Edges Composed of Steps, Peaks and Roofs.
• Reisfeld, D. (1996). Constrained phase congruency: simultaneous detection of interest points and of their orientational scales.
• Robbins, B., & Owens, R. (1997). 2D feature detection via local energy.
• Venkatesh, S., & Owens, R. (1990). On the classification of image features.
• sorry sir but I don't know about "Quadrature filter response". – pandu Apr 19 '15 at 11:51
• @pandu Answer updated. I suggest starting with "Feature detection from local energy". Also see csse.uwa.edu.au/~pk/Research/MatlabFns/index.html – geometrikal Apr 19 '15 at 12:32

The more that sinewaves that have a zero crossing in the same direction at the same point add up in magnitude, the greater the contrast at that point. e.g. a possible edge. Phase congruency is another name (actually a superset) for the amount of sinewaves that line up with zero crossings at (nearly) the same point or points.

Whereas a random distribution of phases is more likely to result in something that a human would see as uniform noise rather than an edge.

• ... . up arrow. – robert bristow-johnson Apr 12 '15 at 2:00
• Sounds like a good exercise for the student. IFFT synthesize an image using only phase 0 even index sine waves (positive zero crossing at the center). iFFT synthesize some images using random phases but the same magnitudes. Compare. Bonus exercise: see how much random noise you can add to the phases for any edge to visually disappear. – hotpaw2 Apr 13 '15 at 18:55
• hot's answer was pretty good. and i don't do image processing. i think i understand image processing enough that if you ran that image (first each row, then each column) through an all-pass filter with non-linear phase response (i.e. an APF that is not just a delay), some or all of those nice edges will get fuzzy. the reason is as hotpaw says: when you break the raster function down into sinusoids of various frequencies, a bunch of those sinusoids are lining up their zero-crossings in order to add up to be a step transition. – robert bristow-johnson Apr 18 '15 at 14:24
• Note that the specific frequencies associated with a coherent edge may relate to the contrast or sharpness of the transition. – hotpaw2 Apr 19 '15 at 0:09