1.6. TWO DIMENSIONAL FOURIER TRANSFORMATION

The resulting two-dimensional Fourier transform function describing the image, the spectrum of this image is obtained, which is defined as

, (1.6.1)

Where - spatial frequencies, and . If we denote the Fourier transform operator by then you can write

. (1.6.2)

In general, the spectrum there is a complex value. It can be decomposed into real and imaginary parts:

(1.6.3a)

or present using amplitude and phase:

, (1.6.3b)

Where

, (1.6.4a)

. (1.6.4b)

A sufficient condition for the existence of the Fourier spectrum of a function is the absolute integrability of this function, i.e. condition

. (1.6.5)

Source function can be restored by the inverse Fourier transform:

. (1.6.6a)

This ratio in the operator form can be written as

. (1.6.6b)

Since the core of the two-dimensional Fourier transform is separable, this transformation can be performed in two stages. First located

(1.6.7)

and then

(1.6.8)

Below are some useful properties of the two-dimensional Fourier transform. Their evidence can be found in books [1, 2].

Functional properties

If the function separable by spatial variables so that

, (1.6.9)

that

, (1.6.10)

Where - one-dimensional Fourier spectra of functions , . If a there is a Fourier spectrum of the function then is the Fourier spectrum of the function . (An asterisk denotes complex conjugation.) If the function symmetric, i.e. then .

Linearity

The Fourier transform operator is linear:

, (1.6.11)

Where - permanent.

Zoom

A change in the spatial scale leads to an inverse change in the scale of spatial frequencies and a proportional change in the values ​​of the spectrum:

. (1.6.12)

Therefore, the compression along one of the axes of the plane leads to stretching along the corresponding axis of the frequency plane and vice versa. There is also a proportional change in the values ​​of the spectrum.

Shift

The shift (change of coordinates) on the initial plane leads to phase changes on the frequency plane:

(1.6.13a)

On the contrary, a shift in the frequency plane causes phase changes of the original function:

(1.6.13b)

Convolution

The Fourier spectrum of a function obtained by convolving two functions is equal to the product of the spectra of the original functions:

(1.6.14)

The converse theorem states that

(1.6.15)

Parseval's theorem

Two representations of the image energy - through the function and Fourier spectrum - are related as follows:

(1.6.16)

Spectrum theorem for the autocorrelation function

The Fourier spectrum of the two-dimensional autocorrelation function of the image is equal to the square of the Fourier spectrum module of this image:

(1.6.17)

Spectra of spatial derivatives

Fourier spectra of the first spatial derivatives of the function associated with its Fourier spectrum by the following relations:

(1.6.18a)

(1.6.18b)

Therefore, the spectrum of the Laplacian is

(1.6.19)