What is the difference between unwrap() function and principal value in finding the unwrapped phase?

I am working on DFT, and trying to reconstruct the image again from the magnitude and phase. I use two different ways to perform phase unwrapping:

1. by using unwrap() Matlab function.
2. by using the principal value of phase according to the following:

I found that:

• there is a difference in the unwrapped phase in each of the two methods. However, the reconstructed image was the same (the same PSNR) in both cases.
• the time of phase unwrapping by using principal value was significantly shorter than unwrap() Matlab function.

I'd appreciate it if someone could clarify the reason behind that.

Answer 1

[David Goodmanson]

Hi Ansam,

The atan2 function does not unwrap anything. It merely produces a four-quadrant theta, with

-pi < theta <= pi.

Consider an initial theta that runs from, say, 0 to 40*pi. The following demo code shows that as a function of theta, the output of atan2 is always between -pi and pi and produces a sawtooth-like result with jumps from -pi to pi. Applying unwrap removes the jumps and produces a continuuous angle so that theta2 runs from 0 to 40*pi and theta2 = theta.

If you plug angles into any trig function there is always the 2pi ambiguity, so theta and theta +- 2*pi*n give the same result. Therefore the unwrapped and not-unwrapped angles give the same result when pluggid into any trig function.

theta = 0:.01:40*pi
y = sin(theta);
x = cos(theta);
theta1 = atan2(y,x);
theta2 = unwrap(atan2(y,x));
plot(theta,theta1,theta,theta2)

[Ansam Osamah]

Thank you very much, David.