Computer Vision CS 4495
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This 0 page Class Notes was uploaded by Alayna Veum on Monday November 2, 2015. The Class Notes belongs to CS 4495 at Georgia Institute of Technology - Main Campus taught by Staff in Fall. Since its upload, it has received 16 views. For similar materials see /class/234152/cs-4495-georgia-institute-of-technology-main-campus in ComputerScienence at Georgia Institute of Technology - Main Campus.
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Date Created: 11/02/15
The Discrete Fourier Transform Frank Dellaert 7th October 2005 The Fourier transform is essential to understanding the effects of linear lters and the phe nomenon of aliasing In class we talked about three variants of the discrete Fourier transform or DFT In all cases the idea is to write the N gtlt 1 vector f as a linear combination of basis functions fj ZCkUjk k The above is the synthesis equation as it synthezises the signal out of orthogonal basis functions 1 DFT using only Real Values This variant is explained in Chapter 8 of The Scientist and Engineer s Guide to Digital Signal Processing by Steven W Smith which can be downloaded for free at httpwwwdspguidecom For an Npoint DFT this variant uses N 2 basis functions namely N 2 l cosines and N 2 l sines N 2 N 2 f chcos27rjkN Z skcos27rjkN k0 k0 and hence the Fourier coef cients are the N 2 values 01751 for k 0N2 The reason we choose sines and cosines is because they transform very simply under convolution only their mag nitude and phase will be changed Hence we can understand the effect of any linear ltering operation on any function simply by understanding how it acts on the sine and cosine waves However it is annoying that we do not have N fourier coef cients but N 2 The apparent discrepancy is explained by the fact that we always have so sN2 0 Also while synthesis is easy analysis the DFT is not very elegant in this framework 2 Matrix Form of the Real DFT It is nicer to drop those two extra sine waves and write the synthesis in matrix notation fUc
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