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l14 - 1 CMPEN/EE455 Lecture 14 Topics today 1 Wraparound...

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1 CMPEN/EE455 Lecture 14 Topics today: 1. Wraparound error in 2-D (circular) convolution using the DFT 2. Fast Fourier Transform (FFT) algorithm (G&W Ch. 4.11) 3 Example — No zero padding Filter ( A × B ) image f ( x, y ) with small ( C × D ) operator g ( x, y ) symmetrical about origin (i.e., assume A >> C and B >> D ) “Blindly” use ( A × B ) transforms to get h ( x, y ) = f ( x, y ) * g ( x, y ) (NOTE: This is very useful for filtering N × N images with small mask [e.g., 3 × 3] filters — avoids overhead of zero padding.) Question: What are good and bad points of h ( x, y )?
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CMPEN/EE455 – Lect 14 2 f ( x, y ) = 0 outside 0 x A - 1 and 0 y B - 1 nonzero only in an A × B rectangle g ( x, y ) = 0 outside - C - 1 2 x C - 1 2 and - D - 1 2 y D - 1 2 nonzero only in a C × D rectangle; assume C, D odd integers To do filtering, a. f e ( x, y ) = f ( x, y ) (We won’t zero pad f ( x, y )!) b. Make g e ( x, y ) = A × B zero-padded version of g ( x, y ) Requires special data organization; see NEXT PAGE! So, main period for (DFT) filtered image is h e ( x, y ) within 0 x A - 1 and 0 y B - 1 where h e ( x, y ) = IDFT ( A × B ) DFT ( A × B ) { f e ( x, y ) } · DFT ( A × B ) { g e ( x, y ) }
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CMPEN/EE455 – Lect 14 3 Picture of g e ( x, y ) for g ( x, y ) being a 5 × 5 operator
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CMPEN/EE455 – Lect 14 4 Picture of h e ( x, y ) 0 s main period
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