Report

Basis beeldverwerking (8D040) dr. Andrea Fuster Prof.dr. Bart ter Haar Romeny dr. Anna Vilanova Prof.dr.ir. Marcel Breeuwer The Fourier Transform II Contents • Fourier Transform of sine and cosine • 2D Fourier Transform • Properties of the Discrete Fourier Transform 2 Euler’s formula 3 Cosine Recall 4 Sine 5 Contents • Fourier Transform of sine and cosine • 2D Fourier Transform • Properties of the Discrete Fourier Transform 6 Discrete Fourier Transform • Forward • Inverse 7 Formulation in 2D spatial coordinates • Discrete Fourier Transform (2D) • Inverse Discrete Transform (2D) f(x,y) digital image of size M x N 8 Spatial and Frequency intervals • Inverse proportionality • Suppose function is sampled M times in x, with step , distance is covered, which is related to the lowest frequency that can be measured • And similarly for y and frequency v 9 Examples 10 Examples 11 Periodicity • 2D Fourier Transform is periodic in both directions 12 Periodicity • 2D Inverse Fourier Transform is periodic in both directions 13 Contents • Fourier Transform of sine and cosine • 2D Fourier Transform • Properties of the Discrete Fourier Transform 14 Properties of the 2D DFT 15 Real SinSin (x +(x) π/2) Real Imaginary 16 • Note: translation has no effect on the magnitude of F(u,v) 17 Symmetry: even and odd • Any real or complex function w(x,y) can be expressed as the sum of an even and an odd part (either real or complex) 18 Properties • Even function (symmetric) • Odd function (antisymmetric) 19 Properties - 2 20 FT of even and odd functions • FT of even function is real • FT of odd function is imaginary 21 Even Real Imaginary Cos (x) 22 Odd Real Imaginary Sin (x) 23 Even Real Imaginary F(Cos(x+k)) F(Cos(x)) 24 Odd Real Imaginary Sin (x)Sin(y) (x) 25 Consequences for the Fourier Transform • FT of real function is conjugate symmetric • FT of imaginary function is conjugate antisymmetric 26 Scaling property • Scaling t with a 27 • a Imaginary parts 28 Differentiation and Fourier • Let be a signal with Fourier transform • Differentiating both sides of inverse Fourier transform equation gives: 29 Examples – horizontal derivative 30 Examples – vertical derivative 31 Examples – hor and vert derivative 32 Thanks and see you next Wednesday!☺ 33