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Unformatted text preview: ay be reconstructed from its sample values when the sampling intervals in both directions, horizontal and vertical, are small enough (or the sampling frequencies are large enough) ECE 6258 Fall 2012 Aliasing in non‐bandlimited signals An image of a line with slope 5 Filtered by an anti‐aliasing or a directional filter • If this condition is not satisfied, the sampling process is not reversible Ghassan AlRegib ECE 6258 Fall 2012 17 CAMERAMAN 2 1 3 4 Fourier transform of the line (Should not this be a line with slope of ‐1/5?) Inversely transformed Ghassan AlRegib ECE 6258 Fall 2012 18 Generalized Periodic Sampling • In rectangular sampling, we had f [m, n] f a (mX , nY ) • For a general placement of samples, we will adopt a vector notation for indexing: n [m, n]t fa[x]: Original continuous image f[n]: 2‐D sampled image Fa (w): Fourier transform of fa[x] F(u): Fourier transform of f[n] Ghassan AlRegib 19 Ghassan AlRegib 20 5 9/3/2012 ECE 6258 Fall 2012 Generalized Periodic Sampling • Using this vector notation, the spatially continuous Fourier transform can be written as: ECE 6258 Fall 2012 Generalized Periodic Sampling where ‐ X is a 2x2 non‐singular sampling matrix with x1 and x2 are the two columns of X (i.e., basis vectors) ‐ n is a 2x1 column vector, [m,n]t A double integral w.r.t. x and y Ghassan AlRegib ECE 6258 Fall 2012 21 Ghassan AlRegib 22 Generalized Periodic Sampling Sampling lattice: an example of a lattice defined by x1 and x2 Ghassan AlRegib 23 6...
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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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