Non-linear multi-dimensional signal processing

Nonlinear 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 $y(t)$, which is decomposed $y(t)$ into components of various order

$y(t)=y_{0}+y_{1}(t)+y_{2}(t)+\cdots +y_{n}(t)$,

Each component is defined as

$y_{n}(t)=\int _{-\infty }^{+\infty }\cdots \int _{-\infty }^{+\infty }h_{n}(\tau _{1},\tau _{2},\cdots ,\tau _{n})\displaystyle \prod _{i=1}^{n}x(t-\tau _{i})d\tau _{i}$ ,

For $n=1$, linear convolution. The $h_{n}(\tau _{1},\tau _{2},\cdots ,\tau _{n})$ is the generalized impulse response of order $n$. The above formula is using delay time series to reconstruction nonlinear system. However, we can also using multi-dimensional signal instead of the delay time series.

Transfer function

Applying the $n$th dimensional FT to $h_{n}(\tau _{1},\tau _{2},\cdots ,\tau _{n})$ obtain the transfer function

$H_{n}(\omega ,\cdots ,\omega )=\int _{-\infty }^{+\infty }\cdots \int _{-\infty }^{+\infty }h_{n}(\tau _{1},\cdots ,\tau _{n})\exp(-j\omega _{1}\tau _{1}-\cdots -j\omega _{n}\tau _{n})d\tau _{1}\cdots d\tau _{n}$

A nonlinear multi-dimensional (frequency) analyser represents

Multi-dimensional nonlinear filter

Nonlinear-type hybrid filters

Nonlinear filters (generalized directional distance rational hybrid filters(GDDRHF)) for multidimensional signal processing. This filter is a two-stage type hybrid filters, combined first stage $L_{p}$ norm criteria and angular distance criteria to produce three output vectors with respect to the shape models, second stage a vector rational operation acts on the above three output vectors to produce the final output vectors. The output vector ${\underline {y}}({\textbf {x}}_{i})$ of the GDDRHF, is result of vector rational function taking into account three input sub-functions which form an input functions set $\{{\underline {y}}_{1},{\underline {y}}_{2},{\underline {y}}_{3}\}$,

${\underline {y}}({\textbf {x}}_{i})={\underline {y}}_{2}({\textbf {x}}_{i})+{\frac {\sum _{j=1}^{3}\beta _{j}{\underline {y}}_{j}({\textbf {x}}_{i})}{h+kD[{\underline {y}}_{1}({\textbf {x}}_{i}),{\underline {y}}_{3}({\textbf {x}}_{i})]}}$

where ,$D[\cdot ]$ is a function of scalar output which plays an important role in rational function as an edge sensing term, $\beta =[\beta _{1},\beta _{2},\beta _{3}]$ characterizes the constant vector coefficient of the input sub-functions. h and k are some positive constants.

The parameter k is used to control the amount of the nonlinear effect.

Multidimensional nonlinear edge-preserving filter

This kind of multidimensional filter has been used on 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 for the

same location in sequence; this corresponds to using inter frame information. It is also employs a trimmed mean spatial smoothing algorithm that uses a Euclidean distance discriminator to preserve partial volume and edge information; this

corresponds to using intra frame information .

Multi-dimensional ensemble empirical mode decomposition method

Multi-dimensional ensemble empirical mode decomposition for multi-dimensional data (images or solid with variable density). The decomposition is based on application s 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 two-dimensional signal using EEMD, $f(x,y)$ is spatially two-dimensional data or an image, after it is decompsed in y-direction, we obtain $g_{j}(m,n)$ , further decompose each row of $g_{j}(m,n)$ using EEMD.

$f(x,y)$ sampled as $f(m,n)=\left({\begin{array}{cccc}f_{1,1}&f_{2,1}&\cdots &f_{M,1}\\f_{1,2}&f_{2,2}&\cdots &f_{M,2}\\\cdots &\cdots &\cdots &\cdots \\f_{1,n}&f_{2,N}&\cdots &f_{M,N}\end{array}}\right),$

$m$th column of $f(m,n)$ decompositions using EMD is

$f(m,\sim )=\sum _{j=1}^{J}C_{j}(m,\sim )=\sum _{j=1}^{J}\left({\begin{array}{c}c_{m,1,j}\\c_{m,2,j}\\\cdots \\c_{m,N,j}\\\end{array}}\right)$

after all the columns of original are decomposed we get $J$ th matrix being

$g_{j}(m,n)=\left({\begin{array}{cccc}c_{1,1,j}&c_{2,1,j}&\cdots &c_{M,1,j}\\c_{1,2,j}&c_{2,2,j}&\cdots &c_{M,2,j}\\\cdots &\cdots &\cdots &\cdots \\c_{1,N,j}&c_{2,N,j}&\cdots &c_{M,N,j}\\\end{array}}\right)$

This is the $J$ components of the original data $f(m,n)$

$n$th row of $J$ th component of $g_{j}(m,n)$ decompositions using EEMD is

$g_{j}(\sim ,n)=\sum _{k=1}^{K}D_{j,k}(\sim ,n)=\sum _{k=1}^{K}\left({\begin{array}{cccc}d_{1,n,j,k}&c_{2,n,j,k}&\cdots &c_{M,n,j,k}\\\end{array}}\right)$

rearrange the component as

$h_{j,k}(m,n)=\left({\begin{array}{cccc}d_{1,1,j,k}&d_{2,1,j,k}&\cdots &d_{M,1,j,k}\\d_{1,2,j,k}&d_{2,2,j,k}&\cdots &d_{M,2,j,k}\\\cdots &\cdots &\cdots &\cdots \\d_{1,N,j,k}&d_{2,1,N,k}&\cdots &d_{M,N,j,k}\\\end{array}}\right)$

So $f(m,n)=\sum _{k=1}^{K}\sum _{j=1}^{j}h_{j,k}(m,n)$, for multi-dimension decomposition we can use the same methods above just change our system to be $n$-dimensions $f(x_{1},x_{2},\cdots ,x_{n})$

The MDEEMD for a picture C1,C2,C3,C4,C5 is five mode component after decompositions.