### 数学代写|信息论作业代写information theory代考|Introduction to Image Compression

statistics-lab™ 为您的留学生涯保驾护航 在代写信息论information theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写信息论information theory代写方面经验极为丰富，各种代写信息论information theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|信息论作业代写information theory代考|Introduction to Image Compression

Earlier in this chapter we discussed the coding of data sets for compression. By applying these techniques we can store or transmit all of the information content of a string of data with fewer bits than are in the source data. The data stream can represent the original data to an extent that is satisfactory to the most discerning eye. Since we can represent a picture by something between a thousand and a million bytes of data, we should be able to apply the techniques studied earlier directly to the task of compressing that data for storage and transmission. First, we consider the following points:
$\begin{array}{ll}\text { DID YOU } & \text { 1. High quality images are represented by very large data sets. A photographic quality image } \ \text { KNOW } & \text { may require } 40 \text { to } 100 \text { million bits for representation. These large file sizes drive the need } \ \text { for extremely high compression ratios to make storage and transmission (particularly of }\end{array}$ movies) practical.

1. Applications that involve imagery seem to be inherently linked to immediate human consumption, and so need to be fast in execution on computers and in transmission. Television, movies, computer graphical user interfaces, and the World Wide Web are examples of applications in which imagery must be moved from storage or across some kind of distribution network very quickly for immediate human intake.
2. Imagery has the quality of higher redundancy than we can generally expect in arbitrary data. For example, a pair of adjacent horizontal lines in an image are nearly identical (typically), while, two adjacent lines in a book have essentially no commonality.
The first two points indicate that we will almost always want to apply the highest level of compression technology available for the movement and storage of image data. The third factor indicates that compression ratios will usually be quite high. The third factor also says that some special compression techniques may be possible that will take advantage of the structure and properties of image data. The close relationship between neighboring pixels in an image can be exploited to improve the compression ratios. This is very important for the task of coding and decoding image data for real-time applications.

Another interesting point to note is that the human eye is very tolerant to approximation error in an image. Thus, it may be possible to compress the image data in a manner in which the less important information (to the human eye) can be dropped. That is, by trading off some of the quality of the image we might obtain a significantly reduced data size. This technique is called lossy compression, as opposed to the lossless compression techniques discussed earlier. This sentiment, however, can never be expressed with regards to, say, financial data or even textual data! Lossy compression can only be applied to data such as images and audio for which human beings will tolerate some loss of fidelity.

## 数学代写|信息论作业代写information theory代考|The JPEG Standard for Lossless Compression

The two lossless JPEG compression options differ only in the form of the entropy code that is applied to the innovations data. The user can choose to use either a Huffman code or an Arithmetic code.

DID YOU We have seen earlier that Arithmetic code, like Huffman code, achieves compression by using KNOW = the probabilistic nature of the data to render the information with fewer bits than used in the original data stream. Its primary advantage over the Huffman code is that it can come closer to the Shannon entropy limit of compression for data streams that involve a relatively small alphabet. The reason is that Huffman codes work best (highest compression ratios) when the probabilities of the symbols can be expressed as fractions of powers of two. The Arithmetic code construction is not closely tied to these particular values, as is the Huffman code. The computation of coding and decoding Arithmetic codes is more costly than that of Huffiman codes. Typically a 5 to $10 \%$ reduction in file size is seen with the application of Arithmetic codes over that obtained with Huffman coding.

Some compression can be achieved if we can predict the next pixel using the previous pixels. In this way we just have to transmit the prediction coefficients (or difference in the values) instead of the entire pixel. The predictive process that is used in the lossless JPEG coding schemes to form the innovations data is also variable. However, in this case, the variation is not based upon the user’s choice, but rather, for any image on a line-by-line basis. The choice is made according to that prediction method that yields the best prediction overall for the entire line.

There are eight prediction methods available in the JPEG coding standards. One of the eight (which is the no prediction option) is not used for the lossless coding option that we are examining here. The other seven may be divided into the following categories:

1. Predict the next pixel on the line as having the same value as the last one.
2. Predict the next pixel on the line as having the same value as the pixel in this position on the previous line (that is, above it).
3. Predict the next pixel on the line as having a value related to a combination of the previous, above and previous to the above pixel values. One such combination is simply the average of the other three.

## 数学代写|信息论作业代写information theory代考|The JPEG Standard for Lossy Compression

The JPEG standard includes a set of sophisticated lossy compression options which resulted from much experimentation by the creators of JPEG with regard to human acceptance of types of image distortion. The JPEG standard was the result of years of effort by the JPEG which was formed as a joint effort by two large, standing, standards organizations, the CCITT (The European telecommunications standards organization) and the ISO (International Standards Organization).

The JPEG lossy compression algorithm consists of an image simplification stage, which removes the image complexity at some loss of fidelity, followed by a lossless compression step based on predictive filtering and Huffman or Arithmetic coding.

The lossy image simplification step, which we will call the image reduction, is based on the exploitation of an operation known as the Discrete Cosine Transform (DCT), defined as follows.
$$Y(k, l)=\sum_{i=0}^{N-1} \sum_{j=0}^{M-1} 4 y(i, j) \cos \left(\frac{\pi k}{2 N}(2 i+1)\right) \cos \left(\frac{\pi l}{2 M}(2 j+1)\right)$$
DID YOU where the input image is $N$ pixels by $M$ pixels, $y(i, j)$ is the intensity of the pixel in row $i$ and KNOW $=$ column $j, Y(k, l)$ is the DCT coefficient in row $k$ and column $l$ of the DCT matrix. All DCT multiplications are real. This lowers the number of required multiplications, as compared to the discrete Fourier transform. For most images, much of the signal energy lies at low frequencies, which appear in the upper left corner of the DCT. The lower right values represent higher frequencies, and are often small (usually small enough to be neglected with little visible distortion). The DCT is, unfortunately, computationally very expensive and its complexity increases as $O\left(N^{2}\right)$. Therefore, images compressed using DCT are first divided into blocks.

In the JPEG image reduction process, the DCT is applied to 8 by 8 pixel blocks of the image. Hence, if the image is 256 by 256 pixels in size, we break it into 32 by 32 square blocks of 8 by 8 pixels and treat each one independently. The 64 pixel values in each block are transformed by the DCT into a new set of 64 values. These new 64 values, known also as the DCT coefficients, form a whole new way of representing an image. The DCT coefficients represent the spatial frequency of the image sub-block. The upper left corner of the DCT matrix has low frequency components and the lower right-corner the high frequency components (see Fig. 1.19). The top left coefficient is called the DC coefficient. Its value is proportional to the average value of the 8 by 8 block of pixels. The rest are called the $\mathbf{A C}$ coefficients.

So far we have not obtained any reduction simply by taking the DCT. However, due to the nature of most natural images, maximum energy (information) lies in low frequency as opposed to high frequency. We can represent the high frequency components coarsely, or drop them altogether, without strongly affecting the quality of the resulting image reconstruction. This leads to a lot of compression (lossy). The JPEG lossy compression algorithm does the following operations:

1. First the lowest weights are trimmed by setting them to zero.
2. The remaining weights are quantized (that is, rounded off to the nearest of some number of discrete code represented values), some more coarsely than others according to observed levels of sensitivity of viewers to these degradations.

## 数学代写|信息论作业代写information theory代考|Introduction to Image Compression

你是否  1. 高质量的图像由非常大的数据集表示。照片质量的图像   知道  可能需要 40 至 100 百万位用于表示。这些大文件大小推动了需求   以极高的压缩比进行存储和传输（尤其是 电影）实用。

1. 涉及图像的应用程序似乎与直接的人类消费有着内在的联系，因此需要在计算机上快速执行和传输。电视、电影、计算机图形用户界面和万维网都是应用程序的示例，在这些应用程序中，图像必须非常快速地从存储设备或某种分发网络中移动，以便立即供人使用。
2. 图像具有比我们通常在任意数据中预期的更高冗余的质量。例如，图像中的一对相邻水平线（通常）几乎相同，而一本书中的两条相邻线基本上没有共性。
前两点表明我们几乎总是希望应用最高级别的压缩技术来移动和存储图像数据。第三个因素表明压缩比通常会很高。第三个因素还表明，一些特殊的压缩技术可能会利用图像数据的结构和属性。可以利用图像中相邻像素之间的密切关系来提高压缩比。这对于实时应用的图像数据编码和解码任务非常重要。

## 数学代写|信息论作业代写information theory代考|The JPEG Standard for Lossless Compression

JPEG 编码标准中有八种预测方法可用。八种之一（即无预测选项）不用于我们在此处检查的无损编码选项。其他七种可分为以下几类：

1. 预测线上的下一个像素与最后一个像素具有相同的值。
2. 预测该行上的下一个像素与上一行（即在其上方）该位置的像素具有相同的值。
3. 将线上的下一个像素预测为具有与上述像素值的先前、上方和先前的组合相关的值。一种这样的组合只是其他三种组合的平均值。

## 数学代写|信息论作业代写information theory代考|The JPEG Standard for Lossy Compression

JPEG 标准包括一组复杂的有损压缩选项，这些选项源于 JPEG 的创建者在人类接受图像失真类型方面的大量实验。JPEG 标准是 JPEG 多年努力的结果，它是由两个大型的常设标准组织 CCITT（欧洲电信标准组织）和 ISO（国际标准组织）共同努力形成的。

JPEG 有损压缩算法由图像简化阶段组成，该阶段在保真度损失的情况下消除图像复杂性，然后是基于预测滤波和霍夫曼或算术编码的无损压缩步骤。

1. 首先，通过将最低权重设置为零来修剪它们。
2. 剩余的权重被量化（即，四舍五入到最接近的一些离散代码表示值），根据观察到的观众对这些退化的敏感度，一些权重比其他权重更粗略。

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。