Multidimensional?

 The term Multidimensional is used in Multi-Dimensional Science and implies the existence of many worlds or rather an infinite number of them. In the context of MDS they are essentially non-physical. 

Multi-Dimensional Science

The following is from vocabulary.com

he adjective multidimensional describes anything with many different parts or aspects. You might talk about your relationship with the next door neighbor as multidimensional if, say, he's also your teacher, and if his son is married to your older sister.

Describing something as multidimensional implies that it's complex. You could talk about a multidimensional book filled with intricate themes, characters, plots, and symbols; or you could even call a person multidimensional if she had a particularly complicated personality. The word dimension forms the root of multidimensional, so if you imagine "many dimensions," you'll have a clear idea of what the word means.

Definitions of multidimensional

1. adjective

    having or involving or marked by several dimensions or aspects
   “multidimensional problems”

   “a multidimensional proposition”

   “a multidimensional personality”

   Synonyms:

   dimensional

   having dimension--the quality or character or stature proper to a person

   2-dimensional, flat, two-dimensional

   lacking the expected range or depth; not designed to give an illusion or depth

   3-dimensional, third-dimensional, three-d, three-dimensional

   involving or relating to three dimensions or aspects; giving the illusion of depth

   4-dimensional, four-dimensional

   involving or relating to the fourth dimension or time

   see less

   Antonyms:

   one-dimensional, unidimensional

   relating to a single dimension or aspect; having no depth or scope

   
   

   
   

   
   

Below is in part a mathmatical presentation of a so-called multidimensional system...

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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.

Contents

• 1Applications
• 2Linear multidimensional state-space model
• 3Multidimensional transfer function
• 4Realization of a 2d transfer function
  • 4.1Example: all zero or finite impulse response
• 5References

Applications[edit]

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.^[1][2] There are also some studies combining m-D systems with partial differential equations (PDEs).

Linear multidimensional state-space model[edit]

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:^[3][4]

Represent the input vector at each point ${\displaystyle (i,j)}$ by ${\displaystyle u(i,j)}$, the output vector by ${\displaystyle y(i,j)}$ the horizontal state vector by ${\displaystyle R(i,j)}$ and the vertical state vector by ${\displaystyle S(i,j)}$. Then the operation at each point is defined by:

${\displaystyle {\begin{aligned}R(i+1,j)&=A_{1}R(i,j)+A_{2}S(i,j)+B_{1}u(i,j)\\S(i,j+1)&=A_{3}R(i,j)+A_{4}S(i,j)+B_{2}u(i,j)\\y(i,j)&=C_{1}R(i,j)+C_{2}S(i,j)+Du(i,j)\end{aligned}}}$

where ${\displaystyle A_{1},A_{2},A_{3},A_{4},B_{1},B_{2},C_{1},C_{2}}$ and ${\displaystyle D}$ are matrices of appropriate dimensions.

These equations can be written more compactly by combining the matrices:

${\displaystyle {\begin{bmatrix}R(i+1,j)\\S(i,j+1)\\y(i,j)\end{bmatrix}}={\begin{bmatrix}A_{1}&A_{2}&B_{1}\\A_{3}&A_{4}&B_{2}\\C_{1}&C_{2}&D\end{bmatrix}}{\begin{bmatrix}R(i,j)\\S(i,j)\\u(i,j)\end{bmatrix}}}$

Given input vectors ${\displaystyle u(i,j)}$ 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[edit]

A discrete linear two-dimensional system is often described by a partial difference equation in the form: ${\displaystyle \sum _{p,q=0,0}^{m,n}a_{p,q}y(i-p,j-q)=\sum _{p,q=0,0}^{m,n}b_{p,q}x(i-p,j-q)}$

where ${\displaystyle x(i,j)}$ is the input and ${\displaystyle y(i,j)}$ is the output at point ${\displaystyle (i,j)}$ and ${\displaystyle a_{p,q}}$ and ${\displaystyle b_{p,q}}$ are constant coefficients.

To derive a transfer function for the system the 2d Z-transform is applied to both sides of the equation above.

${\displaystyle \sum _{p,q=0,0}^{m,n}a_{p,q}z_{1}^{-p}z_{2}^{-q}Y(z_{1},z_{2})=\sum _{p,q=0,0}^{m,n}b_{p,q}z_{1}^{-p}z_{2}^{-q}X(z_{1},z_{2})}$

Transposing yields the transfer function ${\displaystyle T(z_{1},z_{2})}$:

${\displaystyle T(z_{1},z_{2})={Y(z_{1},z_{2}) \over X(z_{1},z_{2})}={\sum _{p,q=0,0}^{m,n}b_{p,q}z_{1}^{-p}z_{2}^{-q} \over \sum _{p,q=0,0}^{m,n}a_{p,q}z_{1}^{-p}z_{2}^{-q}}}$

So given any pattern of input values, the 2d Z-transform of the pattern is computed and then multiplied by the transfer function ${\displaystyle T(z_{1},z_{2})}$ to produce the Z-transform of the system output.

Realization of a 2d transfer function[edit]

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:

${\displaystyle Y(z_{1},z_{2})={\sum _{p,q=0,0}^{m,n}b_{p,q}z_{1}^{-p}z_{2}^{-q} \over \sum _{p,q=0,0}^{m,n}a_{p,q}z_{1}^{-p}z_{2}^{-q}}X(z_{1},z_{2})}$

Two cases are individually considered 1) the bottom summation is simply the constant 1 2) the top summation is simply a constant ${\displaystyle k}$. 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[edit]

${\displaystyle Y(z_{1},z_{2})=\sum _{p,q=0,0}^{m,n}b_{p,q}z_{1}^{-p}z_{2}^{-q}X(z_{1},z_{2})}$

The state-space vectors will have the following dimensions:

${\displaystyle R(1\times m),\quad S(1\times n),\quad x(1\times 1)}$ and ${\displaystyle y(1\times 1)}$

Each term in the summation involves a negative (or zero) power of ${\displaystyle z_{1}}$ and of ${\displaystyle z_{2}}$ which correspond to a delay (or shift) along the respective dimension of the input ${\displaystyle x(i,j)}$. This delay can be effected by placing ${\displaystyle 1}$’s along the super diagonal in the ${\displaystyle A_{1}}$. and ${\displaystyle A_{4}}$ matrices and the multiplying coefficients ${\displaystyle b_{i,j}}$ in the proper positions in the ${\displaystyle A_{2}}$. The value ${\displaystyle b_{0,0}}$ is placed in the upper position of the ${\displaystyle B_{1}}$ matrix, which will multiply the input ${\displaystyle x(i,j)}$ and add it to the first component of the ${\displaystyle R_{i,j}}$ vector. Also, a value of ${\displaystyle b_{0,0}}$ is placed in the ${\displaystyle D}$ matrix which will multiply the input ${\displaystyle x(i,j)}$ and add it to the output ${\displaystyle y}$. The matrices then appear as follows:

${\displaystyle A_{1}={\begin{bmatrix}0&0&0&\cdots &0&0\\1&0&0&\cdots &0&0\\0&1&0&\cdots &0&0\\\vdots &\vdots &\vdots &\ddots &\vdots &\vdots \\0&0&0&\cdots &0&0\\0&0&0&\cdots &1&0\end{bmatrix}}}$

${\displaystyle A_{2}={\begin{bmatrix}0&0&0&\cdots &0&0\\0&0&0&\cdots &0&0\\0&0&0&\cdots &0&0\\\vdots &\vdots &\vdots &\ddots &\vdots &\vdots \\0&0&0&\cdots &0&0\\0&0&0&\cdots &0&0\end{bmatrix}}}$

${\displaystyle A_{3}={\begin{bmatrix}b_{1,n}&b_{2,n}&b_{3,n}&\cdots &b_{m-1,n}&b_{m,n}\\b_{1,n-1}&b_{2,n-1}&b_{3,n-1}&\cdots &b_{m-1,n-1}&b_{m,n-1}\\b_{1,n-2}&b_{2,n-2}&b_{3,n-2}&\cdots &b_{m-1,n-2}&b_{m,n-2}\\\vdots &\vdots &\vdots &\ddots &\vdots &\vdots \\b_{1,2}&b_{2,2}&b_{3,2}&\cdots &b_{m-1,2}&b_{m,2}\\b_{1,1}&b_{2,1}&b_{3,1}&\cdots &b_{m-1,1}&b_{m,1}\end{bmatrix}}}$

${\displaystyle A_{4}={\begin{bmatrix}0&0&0&\cdots &0&0\\1&0&0&\cdots &0&0\\0&1&0&\cdots &0&0\\\vdots &\vdots &\vdots &\ddots &\vdots &\vdots \\0&0&0&\cdots &0&0\\0&0&0&\cdots &1&0\end{bmatrix}}}$

${\displaystyle B_{1}={\begin{bmatrix}1\\0\\0\\0\\\vdots \\0\\0\end{bmatrix}}}$

${\displaystyle B_{2}={\begin{bmatrix}b_{0,n}\\b_{0,n-1}\\b_{0,n-2}\\\vdots \\b_{0,2}\\b_{0,1}\end{bmatrix}}}$

${\displaystyle C_{1}={\begin{bmatrix}b_{1,0}&b_{2,0}&b_{3,0}&\cdots &b_{m-1,0}&b_{m,0}\\\end{bmatrix}}}$

${\displaystyle C_{2}={\begin{bmatrix}0&0&0&\cdots &0&1\\\end{bmatrix}}}$

${\displaystyle D={\begin{bmatrix}b_{0,0}\end{bmatrix}}}$

^[3][4]

References[edit]

1. ^ Bose, N.K., ed. (1985). Multidimensional Systems Theory, Progress, Directions and Open Problems in Multidimensional Systems. Dordre http, Holland: D. Reidel Publishing Company.
2. ^ Bose, N.K., ed. (1979). Multidimensional Systems: Theory and Applications. IEEE Press.
3. ^ Jump up to:^a b Tzafestas, S.G., ed. (1986). Multidimensional Systems: Techniques and Applications. New York: Marcel-Dekker.
4. ^ Jump up to:^a b Kaczorek, T. (1985). Two-Dimensional Linear Systems. Lecture Notes Contr. and Inform. Sciences. Vol. 68. Springer-Verlag.

Categories: 

• Digital imaging
• Partial differential equations
• Stability theory
• Multidimensional signal processing