# 数学代写|图像处理作业代写Image Processing代考|BASIC DEFINITIONS AND KEY CONCEPTS

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## 数学代写|图像处理作业代写Image Processing代考|Stochastic Processes

A stochastic process, $X_{t}$, is defined as
$$X_{\mathrm{t}} \equiv X(t, \omega)$$
where
$X_{t}=$ the value of a random variable $X$ at time $t$, for some value of $\omega$;
$t=$ time, either discrete or continuous; and
$\omega=$ a random element from a sample space $\Omega$.
If $t$ is fixed, say $t=t_{0}$, then we have a random variable over the sample space. For example, if $\Omega$ is a population of people, $X_{t_{0}}$ are the values of variable $X$ for all of the individuals in the population at the moment $t=t_{0}$. If $\omega$ is fixed, say $\omega=\omega_{0}$, then we have a sample trajectory over time for individual $\omega_{0}$.

Time series data consists of a series of real numbers. However, all real numbers can be cast in the imaginary plane. In addition to its standard form (i.e., $a+b i$ ), a complex number can be placed in either polar form or exponential form, both of which have distinct advantages.

Setting $a=X \operatorname{Cos} \lambda t$ and $b=X \operatorname{Sin} \lambda t$, we have $a+b i=(X \operatorname{Cos} \lambda t)+$ $(X \operatorname{Sin} \lambda t) i$, where $X$ is the amplitude, $\lambda$ is the frequency, and $t$ is the time. The chief advantage with polar form is that it captures periodicity in time series data because of its use of the sine and cosine functions. By virtue of Euler’s formula, the polar form can be recast in exponential form as $X e^{i \lambda t}=(X \operatorname{Cos} \lambda t)+(X \operatorname{Sin} \lambda t) i$. The chief advantage of the exponential form is its greater facility in manipulating formulas within derivations and proofs. Both polar and exponential forms are used throughout the following discussion, with the particular form dependent on whichever has the greatest advantage in a particular context.

## 数学代写|图像处理作业代写Image Processing代考|Aliasing Effect

Whenever two time series are indistinguishable, there is said to be a possible aliasing effect. If time is discrete, with $t=\ldots-2,-1,0,1,2, \ldots$, the time series in Example 3 for $\lambda_{2}$ and $\lambda_{1}$ are indistinguishable if $\lambda_{2}=\lambda_{1}+2 \pi k$, where $k$ is some integer. In general, $X_{t}=X e^{i \lambda t}$ with amplitude $X$. If $X_{t}=$ $X e^{i \lambda_{2} t}$, by substitution we have
\begin{aligned} &X_{t}=X e^{i \lambda_{2} t} \ &=e^{i\left(\lambda_{1}+2 \pi k\right) t} \ &=e^{i \lambda_{1} t} e^{i 2 \pi(k t)} \ &=e^{i \lambda_{1} t} \cdot 1 \ &=e^{i \lambda_{1} t} \end{aligned}
In general, if $t=k \cdot \Delta t$, with $k$ an integer, then for $\lambda_{1}, \lambda_{2}$
$$\left|\lambda_{1}-\lambda_{2}\right|=\frac{2 \pi}{\Delta t} \cdot m$$
and there is an aliasing effect; thus, the time series is indistinguishable because
$$X e^{i \lambda_{1} t}=X e^{i \lambda_{2} t}$$

for $t=k \cdot \Delta t$. Consequently, observation of a discrete time series does not distinguish two alias frequencies.

## 数学代写|图像处理作业代写IMAGE PROCESSING代考|Spectral Representation

Any time series that is stationary in a wide sense can be represented as
$$X_{t}=\int_{-\pi}^{\pi} e^{i \lambda t} a(\lambda) d \lambda$$
where
$X_{t}=$ the value of variable $X$ at time $t$;
$e^{i \lambda t}=\operatorname{Cos}(\lambda t)+i \operatorname{Sin}(\lambda t)$, with frequency $\lambda$; and
$a(\lambda)=$ random amplitude of the time series.
with $\operatorname{Cov}\left(a\left(\lambda_{1}\right), a\left(\lambda_{2}\right)\right)=0$, where $\lambda_{1} \neq \lambda_{2}$. Because $X_{t}$ is a real number, $a(\lambda)=\frac{(a(-\lambda)}{a}$.
The spectral density, $f(\lambda)$, is
$$f(\lambda)=E\left(|a(\lambda)|^{2}\right.$$
In other words, the spectral density is the variance of the amplitude, $a(\lambda)$, over a given range of frequencies. Further, for $E\left(X_{t}\right)=0$, the variance of $X_{t}$ is
$$\operatorname{Var} X_{t} \stackrel{\text { for } E X_{t}=0}{=} E X_{t} \bar{X}{t}=\int{-\pi}^{\pi} f(\lambda) d \lambda$$
(see Figure 1.3.1).

## 数学代写|图像处理作业代写IMAGE PROCESSING代考|STOCHASTIC PROCESSES

X吨≡X(吨,ω)

X吨=随机变量的值X有时吨, 对于某个值ω;

ω=来自样本空间的随机元素Ω.

## 数学代写|图像处理作业代写IMAGE PROCESSING代考|ALIASING EFFECT

X吨=X和一世λ2吨 =和一世(λ1+2圆周率到)吨 =和一世λ1吨和一世2圆周率(到吨) =和一世λ1吨⋅1 =和一世λ1吨

|λ1−λ2|=2圆周率Δ吨⋅米

X和一世λ1吨=X和一世λ2吨

## 数学代写|图像处理作业代写IMAGE PROCESSING代考|SPECTRAL REPRESENTATION

X吨=∫−圆周率圆周率和一世λ吨一种(λ)dλ

X吨=变量的值X有时吨;

F(λ)=和(|一种(λ)|2

$$\operatorname{Var} X_{t} \stackrel{\text { for } E X_{t}=0}{=} E X_{t} \bar{X} {t}=\int {-\ pi}^{\pi} fλd \lambda$$
s和和F一世G你r和1.3.1.

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