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. 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. 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). 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 ) {displaystyle (i,j)} by u ( i , j ) {displaystyle u(i,j)} , the output vector by y ( i , j ) {displaystyle y(i,j)} the horizontal state vector by R ( i , j ) {displaystyle R(i,j)} and the vertical state vector by S ( i , j ) {displaystyle 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 {displaystyle A_{1},A_{2},A_{3},A_{4},B_{1},B_{2},C_{1},C_{2}} and D {displaystyle D} are matrices of appropriate dimensions.