Multidimensional transform: Difference between revisions

Revision as of 06:46, 27 September 2014

Revision as of 00:41, 28 October 2014

Line 3:

== Multidimensional Fourier transform ==

One mostof the more popular multidimensional transformtransforms is the [[Fourier transform]], which converts thea signal from a time/spatialspace ___domain representation to a frequency ___domain representation.<ref name="Smith">Smith,W. Handbook of Real-Time Fast Fourier Transforms:Algorithms to Product Testing, Wiley_IEEE Press, edition 1, pages 73–80, 1995</ref> The discrete-___domain multidimensional Fourier transform (FT) can be computed as follows:

:<math> F(w_1,w_2,\dots,w_m) = \sum_{n_1=-\infty}^\infty \sum_{n_2=-\infty}^\infty \cdots \sum_{n_m=-\infty}^\infty f(n_1,n_2,\dots,n_m) e^{-j w_1 n_1 -j w_2 n_2 \cdots -j w_m n_m}</math>

Line 31:

== Multidimensional discrete cosine transform ==

The discrete cosine transform finds(DCT) is used in a wide range of applications such as data [[Data compression|compression]], [[feature extraction]], [[Image reconstruction]], multi-frame [[detection]] and so on. The give size multidimensional DCT is given by:

:<math> Fx(K_1,K_2,\ldots,K_r ) = \sum_{n_1=0}^{N_1-1} \sum_{n_2=0}^{N_2-1} \cdots \sum_{n_r=0}^{N_r-1} fx(n_1,n_2,\ldots,n_r) \cos { \frac{ \pi (2n_1+1) K_1}{2N_1}} \cos { \frac{ \pi (2n_r+1) K_r}{2N_r}}</math>