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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.
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.
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 the constant 1 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.
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 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:
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