Assortment of topics of interest to me:

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Linear Multidimensional State-Space Model

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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

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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

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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[1][2]

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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:

 

 

 

 

 

 

 

 

 

References

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  1. ^ a b Tzafestas, S.G., ed. (1986). Multidimensional Systems: Techniques and Applications. New York: Marcel-Dekker.
  2. ^ a b Kaczorek, T. (1985). Two-Dimensional Linear Systems. Lecture Notes Contr. and Inform. Sciences. Vol. 68. Springer-Verlag.