Linear Operators

Linear System Modeling by Linear Operators

Functioning of an imaging system with particular conditions can be described by the system description operator by abstract way. Consequently, it is practical to give a short summary about the properties of operators and enhance that operator set being well apply able in the model description on both imaging and image processing fields. One of the important condition is, the system description operators should be linear i.e. should map between the elements of linear space.

Definition:

Operator is a rule, mapping, transformation between the elements of set. Operator associates with the elements of same or other set by a prescribed transformation. In the following model description the mapping is in the linear space i.e. only those operator will be considered, which associates with the elements of the linear space.

Let , be subset of $$\ell$ linear space i.e. $$X \subset \ell, Y \subset \ell$ . Operator $$\textbf{\underline{T}}$ performs such an association between the and set, where $$\forall \textbf{\underline{x}} \in \mathfrak{D}$ elements of $$\mathfrak{D} \subseteq X$ subset will be mapped the $$\textbf{\underline{T}}\underline{x}$ elements of linear space. (Figure 4.).

Figure 4.

The $$\mathfrak{D}$ set is called the domain of the operator, while $$\mathfrak{D}_{T} \left \{ \textbf{\underline{T}} \underline{x};\underline{x} \in \mathfrak{D} \right \}$ is the image of the operator.

Definition: If the domain and the image of the $$\textbf{\underline{T}}$ operator are in same linear space, then $$\textbf{\underline{T}}$ is the operator of linear space.

Definition: If the mapping of the elements of $$\forall \underline{x} \in \mathfrak{D}$ subset by $$\textbf{\underline{T}}$ operator results scalar value set as an image, i.e. $$\forall \underline{y} \in \mathfrak{D} \subseteq Y\subset\Gamma$ , where $\Gamma$ is the set of the real and complex numbers, then
$$\textbf{\underline{T}}$ is called linear functional.

Operation rules between the operators:

$$ a.) \quad (\lambda \textbf{\underline{T}})\underline{x} = \lambda(\textbf{\underline{T}} \underline{x}),\text{ where }\lambda \in \Gamma\text{ and }\underline{x} \in X \subset \ell$

$$ b.) \quad (\textbf{\underline{T}}+\textbf{\underline{S}})\underline{x} = \textbf{\underline{T}}\underline{x} + \textbf{\underline{S}} \underline{x}$

$$ c.) \quad (\textbf{\underline{T}}\textbf{\underline{S}})\underline{x}=\textbf{\underline{T}}(\textbf{\underline{S}} \underline{x}) \neq \textbf{\underline{S}}(\textbf{\underline{T}}\underline{x}).\text{ If }\textbf{\underline{S}}(\textbf{\underline{T}}) = \textbf{\underline{T}}(\textbf{\underline{S}}),\text{ then }\textbf{\underline{S}}\text{ and }\textbf{\underline{T}}\text{ commutative operators. }$

$$ d.) \quad \textbf{\underline{R}}(\textbf{\underline{T}}\textbf{\underline{S}})\underline{x} = (\textbf{\underline{R}} \textbf{\underline{T}})\textbf{\underline{S}} \underline{x}$

Definition $$\textbf{\underline{T}}$ operator is bijective between the and set, for $$\forall \underline{x}_{l}$ , and $$\underline{x}_{k}$ elements, where all the $$l \neq k$ cases $$\underline{x}_{l} \neq \underline{x}_{k}$ then $$\textbf{\underline{T}}\underline{x}_{l} \neq \textbf{\underline{T}} \underline{x}_{k}$ .

Definition Let $$\textbf{\underline{T}}$ be a bijective operator. Consequently, $$\textbf{\underline{T}}^{-1}$ operator is the inverse of $$\textbf{\underline{T}}$ , if the domain of $$\textbf{\underline{T}}^{-1}$ is the image of $$\textbf{\underline{T}}$ and $$\textbf{\underline{T}}\textbf{\underline{T}}^{-1} \underline{x} = \underline{x}$ , in case of $$\forall \underline{x} \in \mathfrak{D}$ .

Linear Operators:

$$\textbf{\underline{T}}$ is linear operator, if
$$ \left.\begin{matrix} \textbf{\underline{T}}(\lambda_{i} \underline{x}) = \lambda_{i}(\textbf{\underline{T}} \underline{x})\\ \\ \textbf{\underline{T}} \left(\sum_{i=1}^{n} \lambda_{i} \underline{x}_{i}\right) = \sum_{i=1}^{n} \lambda_{i}\left(\textbf{\underline{T}} \underline{x}_{i}\right) \end{matrix}\right\}\text{ conditions are satisfied, where }\lambda_{i} \in \Gamma \text{ and } \underline{x}_{i} \in \mathfrak{D} \text{ for } \forall i.$

Linear operators are playing important role both in the abstract mathematical modeling and in the evaluation and description of the imaging systems. Imaging methods as well as the analysis of the time and space dependent processes (such as various bio-chemical processes and time dependent signal analysis) described by linear shift invariant operators are high priority modeling procedures. Denote the set of linear shift invariant operators $$\textbf{\underline{L}}$ , where $$\textbf{\underline{L}} \subset \ell$ . The description of space and time dependent process can be expressed in the following way:
$$ y(\textbf{\underline{r}},t) = \textbf{\underline{L}} \left \{} {f(\textbf{\underline{r}},t) \right \}$
$$\textbf{\underline{L}}$ linear operator is shift invariant, if,
$$ y(\textbf{\underline{r}}-\textbf{\underline{r}}_{0} ,t-t_{0}) = \textbf{\underline{L}} \left \{f(\textbf{\underline{r}}-\textbf{\underline{r}}_{0},t-t_{0}) \right \}$

Time and space independent linear shift invariant systems can be described by linear shift invariant operators - $$\textbf{\underline{L}}$ -. Systems described by $$\textbf{\underline{L}}$ are called linear system by constant coefficients (where the coefficients represent the system parameters being time and space independent). If a running process in the linear shift invariant system is time independent, then the process equation is $$y(\textbf{\underline{r}}-\textbf{\underline{r}}_{0}) = \textbf{\underline{L}} \left \{f(\textbf{\underline{r}}-\textbf{\underline{r}}_{0}) \right \}$ , if space independent, then $$y(t-t_{0}) = \textbf{\underline{L}} \left \{f(t-t_{0}) \right \}$ . The relation between the input and output variables will be described by single parameter (denoted by „”) way in the followings due to the more simple formulas (where „” may represent any variables - time, space coordinate, ….etc. -)(Figure 5.).

Figure 5. Linear invariant system ('w' transfer function represents the signal transfer). F(t) is the input (exciting) function, while y(t) is the system response function.

The relation between the input and output function can be described in the followings: $$y(t)=\textbf{\underline{w}}F(t)$ by system transfer way, or $$F(t)=\textbf{\underline{L}}y(t)$ y(t) by the system description operators.
$$\textbf{\underline{L}}$ operator can be expressed for linear systems having constant coefficients as follow:
$$ \textbf{\underline{L}} = a_{n} \frac{d^{n}}{dt^{n}} + a_{(n-1)} \frac{d^{(n-1)}}{dt^{(n-1)}} + \cdots + a_{1} \frac{d}{dt} + a_{0} = \sum_{i=0}^{n}a_{i} \frac{d^{i}}{dt^{i}},\text{ where }a_{n}, a_{n-1},\cdots,a_{1},a_{0} \in \Gamma$
are the system parameters being independent of the „” variable ( $$\textbf{\underline{L}}$ is invariant).

Consequently, the system description equation is: $$\textbf{\underline{L}}y(t) = F(t)$
The output response i.e. the general solution of the equation can be derived by the followings: $$y(t)=y_{0}(t)+y_{p}(t)$ , where $$y_{0}(t)$ is the complementary function (it is the response of zero input, i.e. general solution of the homogenous equation), while $$y_{p}(t)$ is the particular solution i.e. the response of the particular input function (i.e. exciting of the system) by the existing initial conditions.

$$ \left.\begin{matrix} \textbf{\underline{L}}y_{0}(t)=0\\ \\ \textbf{\underline{L}}y_{p}(t)=F(t) \end{matrix}\right\} \textbf{\underline{L}} \left[y_{0}(t)+y_{p}(t)\right] = F(t)$

In case of the system described by $$\textbf{\underline{L}}$ : the complementary function - $$y_{0}(t)$ - gives the transient response of the system, while $$y_{p}(t)$ is the particular solution describing the stationary response (state) of the system by a particular input (exciting) function.

Let’s consider the following $$F_{1}(t),F_{2}(t),\cdots,F_{n}(t)$ independent input functions of the $$\textbf{\underline{L}}$ system operator, where the obtained responses are: $$y_{1}(t),y_{2}(t),\cdots,y_{n}(t)$ .
It is known:
$$ \left.\begin{matrix} \textbf{\underline{L}}y_{1}(t) = F_{1}(t)\\ \textbf{\underline{L}}y_{2}(t) = F_{2}(t)\\ \vdots \\ \textbf{\underline{L}}y_{n}(t) = F_{n}(t) \end{matrix}\right\} \textbf{\underline{L}}y_{1}(t) + \textbf{\underline{L}} y_{2}(t) + \cdots+\textbf{\underline{L}}y_{n}(t) = F_{1}(t) + F_{2}(t) + \cdots+F_{n}(t)$

$$\textbf{\underline{L}}\left[\sum_{i=1}^{n}y_{i}(t) \right] = \sum_{i=1}^{n}F_{i}(t),$ where the principle of superposition for linear systems have been derived.

Linear invariant operators -L- physical phenomenon description

Let’s consider two simple physical examples showing on Figures 6. and 7. The system description linear invariant operator $$\textbf{\underline{L}}$ will be determined for both phenomenon in the followings:

Figure 6. Mechanical example	Figure 7. Electric circuit example
A mass „m” is mounted on a spring having „k” spring constant and $\delta$ attenuation factor. F(t) is the excited force. Differential equation of mechanical model: $$m\frac{d^2x(t)}{dt^2}+2\delta m \frac{dx(t)}{dt}+kx(t) = F(t)$ $$\frac{d^2x(t)}{dt^2}+2\delta \frac{dx(t)}{dt}+\frac{k}{m}x(t) = \frac{F(t)}{m}$	„R” resistor, „C” capacitor and L inductivity are serial connected through „K” switch to U(t) excited input voltage. Differential equation of electric model: $$L\frac{d^2q(t)}{dt^2}+R \frac{dq(t)}{dt}+\frac{1}{C}q(t) = U(t)$ $$\frac{d^2q(t)}{dt^2}+ \frac{R}{L} \frac{dq(t)}{dt}+\frac{1}{LC}q(t) = \frac{U(t)}{L}$

Apply the following denoting: $$\omega^2 = \frac{k}{m}$ ; and $$\omega^2=\frac{1}{LC}$

$$\frac{d^2x(t)}{dt^2}+2\delta \frac{dx(t)}{dt}+\omega^2x(t) = \frac{F(t)}{m}=f(t)$	$$\frac{d^2q(t)}{dt^2}+ \frac{R}{L} \frac{dq(t)}{dt}+\omega^2q(t) = \frac{U(t)}{L}=u(t)$

System description operators of both physical phenomenon are mathematically identical:
$$\textbf{\underline{L}}=\frac{d^2}{dt^2}+a_{1}\frac{d}{dt}+a_{0}$ operator describes both physical phenomenon.
Operator of the mechanical model:
$$\underline{\textbf{L}}_{\textbf{(M)}}x(t)=f(t)\text{, where }\frac{d^2}{dt^2}+2\delta \frac{d}{dt}+\omega^2=\underline{\textbf{L}}_{\textbf{(M)}}$
Operator of the electric circuit model:
$$\underline{\textbf{L}}_{\textbf{(E)}}q(t)=u(t)\text{, where }\frac{d^2}{dt^2}+\frac{R}{L} \frac{d}{dt}+\omega^2=\underline{\textbf{L}}_{\textbf{(E)}}$

The two simple examples illustrate very well the determination of the system description operators by analytical way, as well as the system response analysis for different input functions. Unfortunately most of the cases the internal structures of the system are unknown. By means of the system responses obtained by various characteristic input functions is possible to make survey of the system properties and to create the system description operator by means of the obtained system responses. All of the methods and tools will be discussed in the following chapters.

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Linear Operators

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