# 数学代写|傅里叶分析代写Fourier analysis代考|MAST20026

## 数学代写|傅里叶分析代写Fourier analysis代考|Object Detection in Images

The normalized cross correlation (correlation coefficient) of images $x(m, n)$ and $y(m, n)$ is defined as
$$r n_{x y}(m, n)=\frac{\sum_{k=-\infty}^{\infty} \sum_{l=-\infty}^{\infty}\left(x(k, l)-\bar{x}l\right)(y(k-m, l-n)-\bar{y})}{\sqrt{\sum{k=-\infty}^{\infty} \sum_{l=-\infty}^{\infty}\left(x(k, l)-\bar{x}l\right)^2 \sum{k=-\infty}^{\infty} \sum_{l=-\infty}^{\infty}(y(k-m, l-n)-\bar{y})^2}}$$
The differences between this version and cross-correlation are: (i) the correlation is computed using the local mean-subtracted versions of the two inputs and (ii) the output is normalized to the range $-1-1$. One consequence is that the template cannot be composed of uniform values. The correlation coefficient is assigned the value zero, if the variance of the image over the overlapping portion with the template is zero. The higher the value of the coefficients, the better is the match between the template and the image. The fluctuating part of the values of the inputs is used in computing the correlation. The numerator is cross-correlation with the means subtracted. The denominator is a normalizing factor. It is the square root of the product of the variances of the overlapping samples of the inputs.

Consider the computation of the cross-correlation coefficients between $x(\mathrm{~m}, n)$ and $y(m, n)$.

$$y(m, n)=\left[\begin{array}{rrr} 2 & 1 & 3 \ 1 & 1 & -2 \ 1 & 3 & 1 \end{array}\right] \quad x(m, n)=\left[\begin{array}{rrrr} 2 & 1 & 4 & 3 \ 1 & -2 & 3 & 1 \ 2 & -2 & 1 & -1 \ 1 & 1 & -2 & 2 \end{array}\right]$$
Subtracting the mean, $1.2222$ from $y(m, n)$, we get
\begin{aligned} y m(m, n) &=\left[\begin{array}{rrr} 2 & 1 & 3 \ 1 & 1 & -2 \ 1 & 3 & 1 \end{array}\right]-\left[\begin{array}{lll} 1,2222 & 1,2222 & 1,2222 \ 1,2222 & 1,2222 & 1,2222 \ 1,2222 & 1,2222 & 1,2222 \end{array}\right] \ &=\left[\begin{array}{rrr} 0.7778 & -0.2222 & 1.7778 \ -0.2222 & -0.2222 & -3.2222 \ -0.2222 & 1.7778 & -0.2222 \end{array}\right] \end{aligned}
The variance of $y m(m, n)$ is $17.5556$.
Subtracting the mean, $0.8889$, from part of $x(m, n)$ for a neighborhood $x(1: 3,2: 4)$.

## 数学代写|傅里叶分析代写Fourier analysis代考|Orthogonal Frequency Division Modulation

The purpose of a communication system is to transmit messages from source to destination. The source generates the message, such as human voice, an image, an email message, or some kind of information. These signals are, as in other engineering applications, mostly nonelectrical (e.g., speech signal, image, pressure, vibration, and room temperature). These signals are converted by transducers to electrical signals for the purpose of efficient processing, storage, and transmission. Typical examples of transducers are microphone, thermocouple, strain gauge, computer keyboard, and CCD cameras.

The message signals are of lowpass nature and are also called baseband signals. Over short distances, the baseband signal from a source can reach the destination without much degradation. When we talk with people, this is what happens. However, it is not practically possible to transmit a baseband signal over long distances, a major reason being the requirement of an antenna of large size.

Therefore, the baseband signals are embedded (called modulation) in a carrier signal of suitable high frequency and that signal is transmitted. For example, we can walk to a place that is nearby. To travel long distances, we need a carrier, such as a car. Of course, a car can carry some number of people. Similarly, using a set of highfrequency carriers, we can transmit more than one baseband signal simultaneously over a channel. Therefore, modulation makes the transmission of baseband signals over long distances possible, in addition to the ability to transmit a set of signals at the same time. The transmitted signals pass through a channel, such as a pair of twisted copper wires, a coaxial cable, an optical fiber, or a radio link.

At the receiver, the distorted signal, in passing through the channel along with the added noise, is processed to reduce the distortion and noise to acceptable levels. Then, the modifications done to the message at the transmitter are reversed (called demodulation) to get the receiver output. This signal is converted to its original form by an appropriate transducer.

One of the ways a set of signals is transmitted through a channel, over nonoverlapping frequency bands, is called frequency division multiplexing (FDM). Multiplexing is combining several signals in to a single signal. The bandwidth of the channel of transmission is divided into nonoverlapping parts, with each part carrying a baseband signal. At the receiver, bandpass filters are used to separate the signals. Practical filters have a transition band. Therefore, sufficient frequency gap must be left between two modulated signals. A recent digital communication method that provides significant advantages is the orthogonal frequency division modulation (OFDM). The input to a digital communication system is a sequence of digits. The source of input could be inherently digital or digitized analog signal.

## 数学代写|傅里叶分析代写傅里叶分析代考|图像中的物体检测

.图像中的物体检测

$$r n_{x y}(m, n)=\frac{\sum_{k=-\infty}^{\infty} \sum_{l=-\infty}^{\infty}\left(x(k, l)-\bar{x}l\right)(y(k-m, l-n)-\bar{y})}{\sqrt{\sum{k=-\infty}^{\infty} \sum_{l=-\infty}^{\infty}\left(x(k, l)-\bar{x}l\right)^2 \sum{k=-\infty}^{\infty} \sum_{l=-\infty}^{\infty}(y(k-m, l-n)-\bar{y})^2}}$$

$$y(m, n)=\left[\begin{array}{rrr} 2 & 1 & 3 \ 1 & 1 & -2 \ 1 & 3 & 1 \end{array}\right] \quad x(m, n)=\left[\begin{array}{rrrr} 2 & 1 & 4 & 3 \ 1 & -2 & 3 & 1 \ 2 & -2 & 1 & -1 \ 1 & 1 & -2 & 2 \end{array}\right]$$减去平均值， $1.2222$ 从 $y(m, n)$，我们得到
\begin{aligned} y m(m, n) &=\left[\begin{array}{rrr} 2 & 1 & 3 \ 1 & 1 & -2 \ 1 & 3 & 1 \end{array}\right]-\left[\begin{array}{lll} 1,2222 & 1,2222 & 1,2222 \ 1,2222 & 1,2222 & 1,2222 \ 1,2222 & 1,2222 & 1,2222 \end{array}\right] \ &=\left[\begin{array}{rrr} 0.7778 & -0.2222 & 1.7778 \ -0.2222 & -0.2222 & -3.2222 \ -0.2222 & 1.7778 & -0.2222 \end{array}\right] \end{aligned}

## 数学代写|傅里叶分析代写傅里叶分析代考|正交频分调制

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