Importance of phase and magnitude

The definition indicates that the Fourier transform of an image can be complex. This is illustrated below in Figure (1. 4a-c).

Figure (1.4a)	Figure( 1.4b)	Figure( 1.4c )

Figure (1.4a) shows the original image , Figure (1.4b) the magnitude in a scaled form as and  Figure (1.4c) the phase .

Both the magnitude and the phase functions are necessary for the complete reconstruction of an image from its Fourier transform. Figure(1. 5a) shows what happens when Figure (1.4a) is restored solely on the basis of the magnitude information and Figure (1.5b) shows what happens when Figure (1.4a) is restored solely on the basis of the phase information.

Figure(1.5a)	Figure (1.5b)

Figure (1.5a)  figure(1.5b) shows constant

Neither the magnitude information nor the phase information is sufficient to restore the image. The magnitude-only image Figure (1.5a) is unrecognizable and has severe dynamic range problems. The phase-only image  Figure (1.5b) is barely recognizable, that is, severely degraded in quality.

  Circularly symmetric signals

An arbitrary 2D signal can always be written in a polar coordinate system as .When the 2D signal exhibits a circular symmetry this means that:

where and . As a number of physical systems such as lenses exhibit circular symmetry, it is useful to be able to compute an appropriate Fourier representation.

The Fourier transform can be written in polar coordinates and then, for a circularly symmetric signal, rewritten as a Hankel transform:

	(1.2)

where and is a Bessel function of the first kind of order zero.

The inverse Hankel transform is given by:

The Fourier transform of a circularly symmetric 2D signal is a function of only the radial frequency

.The dependence on the angular frequency  has vanished. Further if  is real, then it is automatically even due to the circular symmetry. According to equ (1.2), will then be real and even.