Multidimensional transform: Difference between revisions

Revision as of 21:41, 15 November 2016

Revision as of 07:25, 16 November 2016

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=== Properties of Fourier transform ===

Similar properties of the 1-D FT transform apply, but instead of the input parameter being just a single entry, it’sit's a Multi-dimensional (MD) array or vector. Hence, it’sit's x(n1,…,nM) instead of x(n).

====Linearity====

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<math>\sum_{n_1=-\infty}^\infty ... \sum_{n_M =-\infty}^\infty |x_1 (n_1,...,n_M)|^2 {=} \frac{1}{(2\pi)^M} \int\limits_{-\pi}^{\pi} ... \int\limits_{-\pi}^{\pi}|X_1(\omega_1,...,\omega_M)|^2 d\omega_1...d\omega_M</math>

A special case of the Parseval’sParseval's theorem is when the two multi-dimensional signals are the same. In this case, the theorem portrays the energy conservation of the signal and the term in the summation or integral is the energy-density of the signal.

====Separability====

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: <math> \ln|z_1| \ge \ln|z_{01}| \text{ and } \ln|z_2| \ge L \ln|z_1| + \{ \ln|z_{02}| - L\ln|z_{01}| \} </math>

where ''L'' is the slope. </math>

The [[2D Z-transform]], similar to the Z-transform, is used in multidimensional signal processing to relate a two-dimensional discrete-time signal to the complex frequency ___domain in which the 2D surface in 4D space that the Fourier transform lies on is known as the unit surface or unit bicircle.

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=== Image processing ===

[[ImageFile:Dctjpeg.png|thumb|250px|Two-dimensional DCT frequencies from the [[JPEG#Discrete cosine transform|JPEG DCT]]]]

The DCT is used in [[JPEG]] image compression, [[MJPEG]], [[MPEG]], [[DV]], [[Daala]], and [[Theora]] [[video compression]]. There, the two-dimensional DCT-II of ''N''x''N'' blocks are computed and the results are [[Quantization (signal processing)|quantized]] and [[Entropy encoding|entropy coded]]. In this case, ''N'' is typically 8 and the DCT-II formula is applied to each row and column of the block. The result is an 8x8 transform coefficient array in which the: (0,0) element (top-left) is the DC (zero-frequency) component and entries with increasing vertical and horizontal index values represent higher vertical and horizontal spatial frequencies, as shown in the picture on the right.