ft-2d.slides.printing.6

ft-2d.slides.printing.6 - The 2-D Fourier Transform The 2-D...

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The 2-D Fourier Transform CS 450: Introduction to Digital Signal and Image Processing Bryan Morse BYU Computer Science The 2-D Fourier Transform Introduction Two-Dimensional Continous Fourier Transform Basis functions are product of ± generalized sinusoids with frequency u in the x direction ± generalized sinusoids with frequency v in the y direction b ( u , v ) = e i 2 π ux e i 2 π vy = e i 2 π ( ux + vy ) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 u = 4 , v = 0 u = 0 , v = 5 u = 4 , v = 5 (real parts) The 2-D Fourier Transform Continuous Transform Two-Dimensional Continous Fourier Transform The transform now becomes: F ( u , v ) = ± -∞ ± -∞ f ( x , y ) e - i 2 π ( ux + vy ) dx dy Similar process for the inverse: f ( x , y ) = ± -∞ ± -∞ F ( u , v ) e i 2 π ( ux + vy ) du dv The 2-D Fourier Transform Examples Example Image Fourier Transform (magnitude) The 2-D Fourier Transform Examples One-Dimensional Fourier Transform:
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ft-2d.slides.printing.6 - The 2-D Fourier Transform The 2-D...

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