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1.6. TWO DIMENSIONAL FOURIER TRANSFORMATION

Lecture



The resulting two-dimensional Fourier transform function 1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION describing the image, the spectrum of this image is obtained, which is defined as

1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION , (1.6.1)

Where 1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION - spatial frequencies, and 1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION . If we denote the Fourier transform operator by 1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION then you can write

1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION . (1.6.2)

In general, the spectrum 1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION there is a complex value. It can be decomposed into real and imaginary parts:

1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION (1.6.3a)

or present using amplitude and phase:

1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION , (1.6.3b)

Where

1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION , (1.6.4a)

1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION . (1.6.4b)

A sufficient condition for the existence of the Fourier spectrum of a function 1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION is the absolute integrability of this function, i.e. condition

1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION . (1.6.5)

Source function 1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION can be restored by the inverse Fourier transform:

1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION . (1.6.6a)

This ratio in the operator form can be written as

1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION . (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.  TWO DIMENSIONAL FOURIER TRANSFORMATION (1.6.7)

and then

1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION (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 1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION separable by spatial variables so that

1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION , (1.6.9)

that

1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION , (1.6.10)

Where 1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION - one-dimensional Fourier spectra of functions 1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION , 1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION . If a 1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION there is a Fourier spectrum of the function 1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION then 1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION is the Fourier spectrum of the function 1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION . (An asterisk denotes complex conjugation.) If the function 1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION symmetric, i.e. 1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION then 1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION .

Linearity

The Fourier transform operator is linear:

1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION , (1.6.11)

Where 1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION - 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.  TWO DIMENSIONAL FOURIER TRANSFORMATION . (1.6.12)

Therefore, the compression along one of the axes of the plane 1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION 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.  TWO DIMENSIONAL FOURIER TRANSFORMATION (1.6.13a)

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

1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION (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.  TWO DIMENSIONAL FOURIER TRANSFORMATION (1.6.14)

The converse theorem states that

1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION (1.6.15)

Parseval's theorem

Two representations of the image energy - through the function 1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION and Fourier spectrum 1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION - are related as follows:

1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION (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.  TWO DIMENSIONAL FOURIER TRANSFORMATION (1.6.17)

Spectra of spatial derivatives

Fourier spectra of the first spatial derivatives of the function 1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION associated with its Fourier spectrum by the following relations:

1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION (1.6.18a)

1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION (1.6.18b)

Therefore, the spectrum of the Laplacian is

1.6.  TWO DIMENSIONAL FOURIER TRANSFORMATION (1.6.19)


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Digital image processing

Terms: Digital image processing