Cisusefulforfast implementationoffiltering notation m

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Unformatted text preview: 2 1 MN M 1 N 1 F [k , l ] 2 k 0 l 0 Ghassan AlRegib © Georgia Tech 16 4 9/13/2012 Modulation ECE 6258 Fall 2012 f [m, n]WM mk0WN nl0 F ((k k0 )) M , ((l l0 )) N Circular Convolution ECE 6258 Fall 2012 • Circular convolution (C. C.) is useful for fast implementation of filtering • Notation: M 1 N 1 g h f f [i, j ]h ((m i )) M , ((n j )) N i 0 j 0 M 1 N 1 g f h h[i, j ] f ((m i )) M , ((n j )) N i 0 j 0 Ghassan AlRegib © Georgia Tech ECE 6258 Fall 2012 17 Circular Convolution • Observation: ① The C. C. depends on the size of DFT, i.e. , N M ② The size of DFT (i.e. ) is assumed to be M N f equal to the support regions of and h Ghassan AlRegib © Georgia Tech ECE 6258 Fall 2012 18 Circular Convolution • Let us decouple the size of DFT from the support region of and f h f [m, n] Let be a image with support region PQ PQ h[m, n] Let be a array with support region RS RS [m, n] And be the result of the linear convolution between f R 1 S 1 and , thus h [m, n] h[i, j ] f [m i, n j ] linearly filtered image i 0 j 0 [ m, n ] Notice that has an enlarged region of support: 0 m P R 1 0 n Q S 1 Ghassan AlRegib © Georgia Tech 19 Ghassan AlRegib © Georgia Tech 20 5 9/13/2012 Question ECE 6258 Fall 2012 • Q: How can we compute the linear convolution in term of IDFT? • A: Continuing: • Consider f ((m i)) M , ((n j )) N f [m i rM , n j sN ] r s g[m, n] • Let us substitute the expression in above R 1 S 1 g[m, n] f h h[i, j ] f ((m i )) M , (( n j )) N M 1 N 1 g[m, n] f h h[i, j ] f [m i rM , n j sN ] r s i 0 j 0 i 0 j 0 M P R 1 N Q S 1 g[m, n] f h [m rM , n sN ] [m, n] MN for where r Ghassan AlRegib © Georgia Tech 21 Analysis ECE 6258 Fall 2012 • The C. C. is a spatially aliased version of the linear convolution in the region MN • If we carefully choose and to be large M N M P R 1 enough, and , then N Q S 1 [m, n] the replicate of will not overlap and thus g[m, n] [m, n] for [m, n] MN Ghassan AlRegib © Georgia Tech Question ECE 6258 Fall 2012 Ghassan AlRegib © Georgia Tech ECE 6258 Fall 2012 [m, n] MN for s 22 Steps to Compute Linear Convolution • Steps to compute the linear convolution: M P R 1 M N ① Choose and , such that N Q S 1 ② Zero‐pad h[m, n] ③ Zero‐pad f [m, n] f h[m, n] M N ④ Compute the DFT of both and [m, n] ⑤ Form the product H [k , l ] F [k , l ] M N ⑥ Comput the inverse DFT of H [k , l ] F [k , l ] desired linear convolution 23 Ghassan AlRegib © Georgia Tech 24 6 9/13/2012 Example ECE 6258 Fall 2012 1 0 f [m, n]...
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This note was uploaded on 02/14/2014 for the course ECE 6258 taught by Professor Staff during the Fall '08 term at Georgia Institute of Technology.

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