multidimensional systems

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:

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

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:

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

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:
$\sum_^a_y(i-p,j-q) = \sum_^b_x(i-p,j-q)$

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

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

: $\sum_^ a_z_1^z_2^Y(z_1,z_2) = \sum_^b_z_1^z_2^X(z_1,z_2)$

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

: $T(z_1,z_2) = =$

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:

: $Y(z_1,z_2) = X(z_1,z_2)$

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

: $Y(z_1,z_2) = \sum_^b_z_1^z_2^X(z_1,z_2)$

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 $b_$ in the proper positions in the $A_2$. The value $b_$ 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 $R_$ vector. Also, a value of $b_$ 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:

: $A_1 = \begin0 & 0 & 0 & \cdots & 0 & 0 \\ 1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 1 & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 0 & \cdots & 1 & 0 \end$

: $A_2 = \begin0 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 0 & \cdots & 0 & 0 \end$

: $A_3 = \begin b_ & b_ & b_ & \cdots & b_ & b_ \\ b_ & b_ & b_ & \cdots & b_ & b_ \\ b_ & b_ & b_ & \cdots & b_ & b_ \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ b_ & b_ & b_ & \cdots & b_ & b_ \\ b_ & b_ & b_ & \cdots & b_ & b_ \end$

$A_4 = \begin0 & 0 & 0 & \cdots & 0 & 0 \\ 1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 1 & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 0 & \cdots & 1 & 0 \end$

: $B_1 = \begin1 \\ 0 \\ 0\\ 0\\ \vdots \\ 0 \\ 0 \end$

: $B_2 = \begin b_ \\ b_ \\ b_ \\ \vdots \\ b_ \\ b_ \end$

: $C_1 = \begin b_ & b_ & b_ & \cdots & b_ & b_ \\ \end$

: $C_2 = \begin0 & 0 & 0 & \cdots & 0 & 1 \\ \end$

: $D = \beginb_ \end$

References

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Digital imaging
Partial differential equations
Stability theory
Multidimensional signal processing