Non-linear multi-dimensional signal processing. In signal processing, nonlinear multidimensional signal processing (NMSP) covers all signal processing using nonlinear multidimensional signals and systems. Nonlinear multidimensional signal processing is a subset of signal processing (multidimensional signal processing). Nonlinear multi-dimensional systems can be used in a broad range such as imaging, teletraffic, communications, hydrology, geology, and economics. Nonlinear systems cannot be treated as linear systems, using Fourier transformation and wavelet analysis. Nonlinear systems will have chaotic behavior, limit cycle, steady state, bifurcation, multi-stability and so on. Nonlinear systems do not have a canonical representation, like impulse response for linear systems. But there are some efforts  to characterize nonlinear systems, such as Volterra and Wiener series using polynomial integrals as the use of those methods naturally extend the signal into multi-dimensions. Another example is the Empirical mode decomposition method using Hilbert transform instead of Fourier Transform for nonlinear multi-dimensional systems. This method is an empirical method and can be directly applied to data sets. Multi-dimensional nonlinear filters (MDNF) are also an important part of NMSP, MDNF are mainly used to filter noise in real data. There are nonlinear-type hybrid filters used in color image processing, nonlinear edge-preserving filters use in magnetic resonance image restoration. Those filters use both temporal and spatial information and combine the maximum likelihood estimate with the spatial smoothing algorithm.

Nonlinear analyser 

A linear frequency response function (FRF) can be extended to a nonlinear system by evaluation of higher order transfer functions and impulse response functions by Volterra series. Suppose we have a time series , which is decomposed  into components of various order

Each component is defined as

 ,

for ,  is the linear convolution.   is the generalized impulse response of order .

The 1D Fourier transform of  is

Schetzen suggested the definition of th output component as  time variables  so as to permit the application of the -dimensional Fourier transform,

Taking the inverse Fourier transform of  and  and equalizing , we obtain the following equation,

Transfer function 

Applying the th dimensional Fourier Transform to  obtain the transfer function

Multi-dimensional nonlinear filter

Nonlinear-type hybrid filters 

One example of nonlinear filters is the (generalized directional distance rational hybrid filter (GDDRHF)) for multidimensional signal processing. This filter is a two-stage type hybrid filter: 1) the stage   norm criteria and angular distance criteria to produce three output vectors with respect to the shape models; 2) the stage performs vector rational operation on the above three output vectors to produce the final output vectors.  The output vector  of the GDDRHF is the result of a vector rational function taking into account three input sub-function which form an input function set ,

where  plays an important role as an edge sensing term,  characterizes the constant vector coefficient of the input sub-functions.  and  are some positive constants. The parameter  is used to control the amount of the nonlinear effect.

Multidimensional nonlinear edge-preserving filter 

This kind of multidimensional filter has been used for MRI imaging processing. This filter uses MRI signal models to implement an approximate maximum likelihood or least squares estimate of each pixel gray level from the gray levels. It is also employs a trimmed mean spatial smoothing algorithm that uses a Euclidean distance discriminator to preserve partial volume and edge information; corresponds to using intra frame information .

Multi-dimensional ensemble empirical mode decomposition method 
A multi-dimensional ensemble empirical mode decomposition method was applied to multi-dimensional data including images and solid with variable density. The decomposition is based on the application of ensemble empirical mode decomposition (EEMD) to slices of data in each and every dimension involved. The final reconstruction of the corresponding intrinsic mode function is based on a comparable minimal scale combination principle.

For a two-dimensional signal  using EEMD, the signal is first decomposed the y-direction to obtain , each row of   is decomposed using EEMD.

Let  be sampled as 

The EEMD decomposition of the th column of  is

after all the columns are decomposed we get  th matrix being

This is the  component of the original data 

 row of  decomposition using EEMD is

rearrange the component as

So  For a multi-dimension decomposition with an -dimensional function we can use the same method above.

References 

Wavelets
Signal processing