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[1][2]:
Represent the input vector at each point
by
, the output vector by
the horizontal state vector by
and the vertical state vector by
. Then the operation at each point is defined by:
where
and
are matrices of appropriate dimensions.
These equations can be written more compactly by combining the matrices:
Given input vectors
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:
where
is the input and
is the output at point
and
and
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
:
So given any pattern of input values, the 2d Z-transform of the pattern is computed and then multiplied by the transfer function
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
2)the top summation is simply a constant
. 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.
Case 1 - all zero or finite impulse response
The state-space vectors will have the following dimensions:
and
Each term in the summation involves a negative (or zero) power of
and of
which correspond to a delay (or shift) along the respective dimension of the input
. This delay can be effected by placing
’s along the super diagonal in the
. and
matrices and the multiplying coefficients
in the proper positions in the
. The value
is placed in the upper position of the
matrix, which will multiply the input
and add it to the first component of the
vector. Also, a value of
is placed in the
matrix which will multiply the input
and add it to the output
.
The matrices then appear as follows: