Fundamental Properties of 2D DFT

DFT Properties:

1. Separability
2 Translation or Shifting
3 Periodicity and Conjugate Symmetry
4 Rotation
5. Distributivity
6 Scalling
7 Convolution
8 Correlation
9 Projection Slice Theorem

1. Separability

The 2D FT can be implemented as two consecutive 1D FTs: first in the x direction, then in the y direction (or vice versa). Symbolically:


The computation of the 2-D Fourier transform as a series of 1-D transforms. 


2. Translation or Shifting

  The FT of a shifted function is unaltered except for a linearly varying phase factor.


3. Periodicity and Conjugate Symmetry

The DFT and IDFT are periodic with period N  ; that is:



4. Rotation

Simply stated: if a function is rotated, then its Fourier transform rotates an equal amount.



5. Distributivity

The Fourier transform and its inverse are distributive over addition but not over multiplication.


This property is best summarized by "a contraction in one domain produces corresponding expansion in the Fourier domain". 



The convolution of two functions f(x) and g(x) is defined by the integral:

The Convolution Theorem tells us that convolution in the spatial domain corresponds to multiplication in the frequency domain, and vice versa. 


8. Correlation

One of the principal applications of correlation in image processing is in the area of template or prototype matching i.e. finding the closest match between an unknown image and a set of known images.



9. Projection-Slice Theorem

Again this is a property which has no useful analogy in one dimension. The theorem can be stated briefly: the (1D) Fourier transform of the projection of a 2D function is the central slice of the Fourier transform of that function.

If F(u,0) is the central slice, and defines the projection of f(x,y) on the x axis then:



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