Multidimensional transform
In mathematical analysis and applications, multidimensional transforms are used to analyze the frequency content of signals in a domain of two or more dimensions.
A signal or system is said to be separable if it can be expressed as a product of 1-D functions with different independent variables.
This phenomenon allows computing the FT transform as a product of 1-D FTs instead of multi-dimensional FT. if
(In the presence of round-off error, many FFT algorithms are also much more accurate than evaluating the DFT definition directly).There are many different FFT algorithms involving a wide range of mathematics, from simple complex-number arithmetic to group theory and number theory.
The discrete cosine transform (DCT) is used in a wide range of applications such as data compression, feature extraction, Image reconstruction, multi-frame detection and so on.
Boundary value problems in two or more variables characterized by partial differential equations can be solved by a direct use of the Laplace transform.
A special case (along 2 dimensions) of the multi-dimensional Laplace transform of function f(x,y) is defined[4] as
[citation needed] This special case can be used to solve the Telegrapher's equations.
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.
The DFT can also be used to perform other operations such as convolutions or multiplying large integers.
The DFT and DCT have seen wide usage across a large number of fields, we only sketch a few examples below.
There, the two-dimensional DCT-II of NxN blocks are computed and the results are quantized and 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.
The conversion from continuous time to samples (discrete-time) changes the underlying Fourier transform of x(t) into a discrete-time Fourier transform (DTFT), which generally entails a type of distortion called aliasing.
Choice of an appropriate sample-rate (see Nyquist rate) is the key to minimizing that distortion.
Similarly, the conversion from a very long (or infinite) sequence to a manageable size entails a type of distortion called leakage, which is manifested as a loss of detail (aka resolution) in the DTFT.
Choice of an appropriate sub-sequence length is the primary key to minimizing that effect.
When the available data (and time to process it) is more than the amount needed to attain the desired frequency resolution, a standard technique is to perform multiple DFTs, for example to create a spectrogram.
If the desired result is a power spectrum and noise or randomness is present in the data, averaging the magnitude components of the multiple DFTs is a useful procedure to reduce the variance of the spectrum (also called a periodogram in this context); two examples of such techniques are the Welch method and the Bartlett method; the general subject of estimating the power spectrum of a noisy signal is called spectral estimation.
A final source of distortion (or perhaps illusion) is the DFT itself, because it is just a discrete sampling of the DTFT, which is a function of a continuous frequency domain.
Thus, in the Fourier representation, differentiation is simple—we just multiply by i n. (Note, however, that the choice of n is not unique due to aliasing; for the method to be convergent, a choice similar to that in the trigonometric interpolation section above should be used.)
One then uses the inverse DFT to transform the result back into the ordinary spatial representation.
DCTs are also widely employed in solving partial differential equations by spectral methods, where the different variants of the DCT correspond to slightly different even/odd boundary conditions at the two ends of the array.
The general theory for obtaining solutions in this technique is developed by theorems on Laplace transform in n dimensions.
[11] One very important factor is that we must apply a non-destructive method to obtain those rare valuables information (from the HVS viewing point, is focused in whole colorimetric and spatial information) about works of art and zero-damage on them.
The proposed FFT-based imaging approach is diagnostic technology to ensure a long life and stable to culture arts.
The inverse multidimensional Laplace transform can be applied to simulate nonlinear circuits.
This is done so by formulating a circuit as a state-space and expanding the Inverse Laplace Transform based on Laguerre function expansion.
It is observed that a high accuracy and significant speedup can be achieved for simulating large nonlinear circuits using multidimensional Laplace transforms.
