Multidimensional system

In mathematical systems theory, a multidimensional system or m-D system is a system in which not only one independent variable exists (like time), but there are several independent variables.

Important problems such as factorization and stability of m-D systems (m > 1) have recently attracted the interest of many researchers and practitioners. The reason is that the factorization and stability is not a straightforward extension of the factorization and stability of 1-D systems because, for example, the fundamental theorem of algebra does not exist in the ring of m-D (m > 1) polynomials.

Applications
Multidimensional systems or m-D systems are the necessary mathematical background for modern digital image processing with many applications in biomedicine, X-ray technology and satellite communications. There are also some studies combining m-D systems with partial differential equations (PDEs).

Linear multidimensional state-space model
A state-space model is a representation of a system in which the effect of all "prior" input values is contained by a state vector. In the case of an m-d system, each dimension has a state vector that contains the effect of prior inputs relative to that dimension. The collection of all such dimensional state vectors at a point constitutes the total state vector at the point.

Consider a uniform discrete space linear two-dimensional (2d) system that is space invariant and causal. It can be represented in matrix-vector form as follows:

Represent the input vector at each point $$(i,j)$$ by $$u(i,j)$$, the output vector by $$y(i,j)$$ the horizontal state vector by $$R(i,j)$$ and the vertical state vector by $$S(i,j)$$. Then the operation at each point is defined by:



\begin{align} R(i+1,j) & = A_1R(i,j) + A_2S(i,j) + B_1u(i,j) \\ S(i,j+1) & = A_3R(i,j) + A_4S(i,j) + B_2u(i,j) \\ y(i,j) & = C_1R(i,j) +C_2S(i,j) + Du(i,j) \end{align} $$

where $$A_1, A_2, A_3, A_4, B_1, B_2, C_1, C_2$$ and $$D$$ are matrices of appropriate dimensions.

These equations can be written more compactly by combining the matrices:



\begin{bmatrix} R(i+1,j) \\ S(i,j+1) \\ y(i,j) \end{bmatrix} = \begin{bmatrix} A_1 & A_2 & B_1 \\ A_3 & A_4 & B_2 \\ C_1 & C_2 & D \end{bmatrix} \begin{bmatrix} R(i,j) \\ S(i,j) \\ u(i,j) \end{bmatrix} $$

Given input vectors $$u(i,j)$$ at each point and initial state values, the value of each output vector can be computed by recursively performing the operation above.

Multidimensional transfer function
A discrete linear two-dimensional system is often described by a partial difference equation in the form: $$\sum_{p,q=0,0}^{m,n}a_{p,q}y(i-p,j-q) = \sum_{p,q=0,0}^{m,n}b_{p,q}x(i-p,j-q)$$

where $$x(i,j)$$ is the input and $$y(i,j)$$ is the output at point $$(i,j)$$ and $$a_{p,q}$$ and $$b_{p,q}$$ are constant coefficients.

To derive a transfer function for the system the 2d Z-transform is applied to both sides of the equation above.


 * $$\sum_{p,q=0,0}^{m,n} a_{p,q}z_1^{-p}z_2^{-q}Y(z_1,z_2) = \sum_{p,q=0,0}^{m,n}b_{p,q}z_1^{-p}z_2^{-q}X(z_1,z_2)$$

Transposing yields the transfer function $$T(z_1,z_2)$$:


 * $$T(z_1,z_2) = {Y(z_1,z_2) \over X(z_1,z_2)} = {\sum_{p,q=0,0}^{m,n}b_{p,q}z_1^{-p}z_2^{-q} \over \sum_{p,q=0,0}^{m,n}a_{p,q}z_1^{-p}z_2^{-q}}$$

So given any pattern of input values, the 2d Z-transform of the pattern is computed and then multiplied by the transfer function $$T(z_1,z_2)$$ to produce the Z-transform of the system output.

Realization of a 2d transfer function
Often an image processing or other md computational task is described by a transfer function that has certain filtering properties, but it is desired to convert it to state-space form for more direct computation. Such conversion is referred to as realization of the transfer function.

Consider a 2d linear spatially invariant causal system having an input-output relationship described by:


 * $$Y(z_1,z_2) = {\sum_{p,q=0,0}^{m,n}b_{p,q}z_1^{-p}z_2^{-q} \over \sum_{p,q=0,0}^{m,n}a_{p,q}z_1^{-p}z_2^{-q}}X(z_1,z_2)$$

Two cases are individually considered 1) the bottom summation is simply the constant 1 2) the top summation is simply a constant $$k$$. Case 1 is often called the "all-zero" or "finite impulse response" case, whereas case 2 is called the "all-pole" or "infinite impulse response" case. The general situation can be implemented as a cascade of the two individual cases. The solution for case 1 is considerably simpler than case 2 and is shown below.

Example: all zero or finite impulse response

 * $$Y(z_1,z_2) = \sum_{p,q=0,0}^{m,n}b_{p,q}z_1^{-p}z_2^{-q}X(z_1,z_2)$$

The state-space vectors will have the following dimensions:


 * $$R (1 \times m),\quad S (1 \times n),\quad x (1 \times 1)$$ and $$y (1 \times 1)$$

Each term in the summation involves a negative (or zero) power of $$z_1$$ and of $$z_2$$ which correspond to a delay (or shift) along the respective dimension of the input $$x(i,j)$$. This delay can be effected by placing $$1$$’s along the super diagonal in the $$A_1$$. and $$A_4$$ matrices and the multiplying coefficients $$b_{i,j}$$ in the proper positions in the $$A_2$$. The value $$b_{0,0}$$ is placed in the upper position of the $$B_1$$ matrix, which will multiply the input $$x(i,j)$$ and add it to the first component of the $$R_{i,j}$$ vector. Also, a value of $$ b_{0,0}$$ is placed in the $$D$$ matrix which will multiply the input $$x(i,j)$$ and add it to the output $$y$$. The matrices then appear as follows:


 * $$A_1 = \begin{bmatrix}0 & 0 & 0 & \cdots & 0 & 0 \\

1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 1 & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 0 & \cdots & 1 & 0 \end{bmatrix}$$


 * $$A_2 = \begin{bmatrix}0 & 0 & 0 & \cdots & 0 & 0 \\

0 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 0 & \cdots & 0 & 0 \end{bmatrix}$$


 * $$A_3 = \begin{bmatrix}

b_{1,n} & b_{2,n} & b_{3,n} & \cdots & b_{m-1,n} & b_{m,n} \\ b_{1,n-1} & b_{2,n-1} & b_{3,n-1} & \cdots & b_{m-1, n-1} & b_{m,n-1} \\ b_{1,n-2} & b_{2,n-2} & b_{3,n-2} & \cdots & b_{m-1, n-2} & b_{m,n-2} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ b_{1,2} & b_{2,2} & b_{3,2} & \cdots & b_{m-1,2} & b_{m,2} \\ b_{1,1} & b_{2,1} & b_{3,1} & \cdots & b_{m-1,1} & b_{m,1} \end{bmatrix}$$

$$A_4 = \begin{bmatrix}0 & 0 & 0 & \cdots & 0 & 0 \\ 1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 1 & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 0 & \cdots & 1 & 0 \end{bmatrix}$$


 * $$B_1 = \begin{bmatrix}1 \\

0 \\ 0\\ 0\\ \vdots \\ 0 \\ 0 \end{bmatrix}$$


 * $$B_2 = \begin{bmatrix}

b_{0,n} \\ b_{0,n-1} \\ b_{0,n-2} \\ \vdots \\ b_{0,2} \\ b_{0,1} \end{bmatrix}$$


 * $$C_1 = \begin{bmatrix} b_{1,0} & b_{2,0} & b_{3,0} & \cdots & b_{m-1,0} & b_{m,0} \\

\end{bmatrix}$$


 * $$C_2 = \begin{bmatrix}0 & 0 & 0 & \cdots & 0 & 1 \\

\end{bmatrix}$$


 * $$D = \begin{bmatrix}b_{0,0} \end{bmatrix}$$