Multidimensional system

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:

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:

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: ∑ p , q = 0 , 0 m , n a p , q y ( i − p , j − q ) = ∑ 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.

Transposing yields the transfer function 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:

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

The state-space vectors will have the following dimensions:

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 4 = [ 0 0 0 ⋯ 0 0 1 0 0 ⋯ 0 0 0 1 0 ⋯ 0 0 ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 0 ⋯ 0 0 0 0 0 ⋯ 1 0 ]