Lecture
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)
Comments
To leave a comment
Digital image processing
Terms: Digital image processing