A few Properties of the 2-D Fourier Transform

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A few Properties of the 2-D Fourier Transform Translation Distributivity and Scaling Rotation Periodicity and Conjugate Symmetry Separability Convolution and Correlation

Translation and

Translation The past conditions mean: Multiplying f(x,y) by the showed exponential term and taking the change of the item brings about a move of the source of the recurrence plane to the point (u 0 ,v 0 ). Duplicating F(u,v) by the exponential term appeared and taking the opposite change moves the source of the spatial plane to (x 0 ,y 0 ). A move in f(x,y) doesn't influence the greatness of its Fourier change

Distributivity and Scaling Distributive over expansion yet not over increase.

Distributivity and Scaling For two scalars an and b,

Rotation Polar directions: Which implies that:

Rotation Which implies that turning f(x,y) by an edge  0 pivots F(u,v) by a similar point (and the other way around).

Periodicity & Conjugate Symmetry The discrete FT and its reverse are occasional with period N:

Periodicity & Conjugate Symmetry Although F(u,v) rehashes itself for boundlessly many estimations of u and v, just the M,N estimations of every variable in any one period are required to get f(x,y) from F(u,v). This implies just a single time of the change is important to indicate F(u,v) totally in the recurrence space (and comparably f(x,y) in the spatial area).

Periodicity & Conjugate Symmetry (moved range) move the inception of the change to u=N/2.

Periodicity & Conjugate Symmetry For genuine f(x,y), FT additionally displays conjugate symmetry: or

Periodicity & Conjugate Symmetry basically: i.e. F(u) has a time of length N and the greatness of the change is focused on the cause.

Separability The discrete FT match can be communicated in distinguishable structures which (after a few controls) can be communicated as: Where:

Separability For each estimation of x, the expression inside the sections is a 1-D change, with recurrence values v=0,1,… ,N-1. Therefore, the 2-D work F(x,v) is gotten by taking a change along each line of f(x,y) and increasing the outcome by N.

Separability The coveted outcome F(u,v) is then acquired by making a change along every segment of F(x,v).

Convolution hypothesis with FT match:

Convolution Discrete proportional: Discrete, intermittent exhibit of length M. x=0,1,2,… ,M-1 portrays a full time of f e (x)*g e (x). Summation replaces combination.

Correlation of two capacities: f(x) o g(x) Types: autocorrelation, cross-connection Used in layout coordinating

Correlation hypothesis with FT match:

Correlation Discrete proportionate: For x=0,1,2,… ,M-1

Fast Fourier Transform Number of complex increases and augmentations to actualize Fourier Transform: M 2 (M complex duplications and N-1 increases for each of the N estimations of u).

Fast Fourier Transform The disintegration of FT makes the quantity of duplications and augmentations corresponding to M log 2 M: Fast Fourier Transform or FFT calculation. E.g. on the off chance that M=1021 the standard technique will require 1000000 operations, while FFT will require 10000.