数据压缩（九）——DPCM压缩系统的实现与分析

实验内容：将预测误差图像写入文件并将该文件输入Huffman编码器，得到输出码流、给出概率分布图并计算压缩比。最后比较两种系统（1.DPCM+熵编码和2.仅进行熵编码）之间的编码效率（压缩比和图像质量）。压缩质量以PSNR进行计算。

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（一）DPCM编解码原理

  其中，$d_n$是预测误差，$Q$是量化器，$P$中存储的是重建值。

关键代码

DPCM()函数

void DPCM(unsigned char* YOrigi, unsigned char* YError, unsigned char* YRestr, int height, int width, int bits)
{
	for (int i = 0; i < height; i++)
		for (int j = 0; j < width; j++)
			DPCM_Pixel(YOrigi, YError, YRestr, i * width, j, bits);//for each pixel,use the function DPCM_Pixel() to calculate the YRestr[i] and YError[i] 
}

DPCM_Pixel()函数

void DPCM_Pixel(unsigned char* YOrigi, unsigned char* YError, unsigned char* YRestr, int pre, int j, int bits)
{
	/*for the first Pixel,suppose YError[0]=128-YOrigi[0],
	then quantify it and store*/
	int E_temp, R_temp;
	switch (j)
	{
	case 0:
		E_temp = 128 - YOrigi[pre + j];
		E_temp = ErrorQuantity(E_temp, bits);//quantity
		YError[pre + j] = E_temp;
		E_temp = invErrorQuantity(E_temp, bits);//inverse quantization
		R_temp = 128 - E_temp;
		R_temp = OverflowX(R_temp, 255, 0);//prevent overflow
		YRestr[pre + j] = (unsigned char)R_temp;
		break;
	default:
		E_temp = YOrigi[pre + j] - YRestr[pre + j - 1];
		E_temp = ErrorQuantity(E_temp, bits);//quantity
		YError[pre + j] = E_temp;
		E_temp = invErrorQuantity(E_temp, bits);//inverse quantization
		R_temp = E_temp + YRestr[pre + j - 1];
		R_temp = OverflowX(R_temp, 255, 0);//prevent overflow
		YRestr[pre + j] = (unsigned char)R_temp;
		break;
	}
}

（二）PSNR原理

  $PSNR$(Peak signal-to-noise ratio,常缩写为$PSNR$)是一个表示信号最大可能功率的比值的工程术语。由于许多信号都有非常宽的动态范围，峰值信噪比常用对数单位(db)来表示。
  计算$PSNR$要先知道$MSE$的计算。$MSE$是均方误差。若有两个$m\times n$单色图像$I$和$K$,他们的均方误差定义为:$$MSE=\frac{1}{mn}{\Sigma }_{i=0}^{m-1}{\Sigma }_{j=0}^{n-1}[I(i,j)-K(i.j){]}^2$$
  $PSNR$定义为：
$$PSNR=10\times lo{g}_{10}(\frac{MA{X}_I^2}{MSE})$$
  其中，$MA{X}_I$表示图像点颜色的最大数值。

一般来说
PSNR>40dB:图像质量非常好（非常接近原始图像）
30 20 PSNR<20:图像不可接受

关键代码

PSNR()函数

double PSNR(unsigned char* YOrigi, unsigned char* YRestr, int height, int width)
{
	int fmax = pow(2, 8) - 1;
	int a = fmax * fmax;
	double mean_se = MSE(YOrigi, YRestr, height, width);
	double peak_SNR = 10 * log10((double)a / mean_se);
	return peak_SNR;
}

MSE()函数

double MSE(unsigned char* YOrigi, unsigned char* YRestr, int height, int width)
{
	int size = height * width;
	long long int sum = 0;
	double mean;
	for (int i = 0; i < size; i++)
	{
		long long int temp = (long long int)(YOrigi[i] - YRestr[i]) * (long long int)(YOrigi[i] - YRestr[i]);
		sum += temp;
	}
	mean = (double)sum / (double)size;
	return mean;
}

（三）Huffman编码原理

  $Huffman$编码在很多地方都有提到，在此不再赘述。对此进行一个总结：$Huffman$编码是一种无失真的编码方法，不引入任何失真。并且所编出的码字为即时码。$Huffman$编码非常的实用。

对Huffman编码的结果分析

  老师给了一个.exe文件，用来存放$Huffman$编码的结果。

  其中第一列是信源符号$a$，第二列是信源符号出现的次数$b$，第三列是编码的码字$c$。
  对原文件和DPCM编码后输出的error文件都进行此操作，即可判断DPCM+熵编码和只进行熵编码两种方案的区别，压缩比直接由.huff文件大小比上原.yuv文件大小计算得。

（四）完整实验过程

4.1 DPCM编码 && PSNR

完整代码

  解决方案资源管理器如下：

main.cpp

#include 
#include 
#include 
#include "DPCM_code.h"

using namespace std;

int main(int argc, char** argv)
{
	//moon图片464 538,使用的是4:4:4的yuv文件
	int width = atoi(argv[1]);
	int height = atoi(argv[2]);
	int bits = atoi(argv[3]);
	int Ysize = height * width;
	int Esize = height * width * 2;

	ifstream OrigiFile(argv[4], ios::binary);
	ofstream ErrorFile(argv[5], ios::binary);
	ofstream RestrFile(argv[6], ios::binary);
	if (!OrigiFile) { cout << "error to open OrigiFile!" << endl; }
	if (!ErrorFile) { cout << "error to open ErrorFile!" << endl; }
	if (!RestrFile) { cout << "error to open RestrFile!" << endl; }

	unsigned char* YOrigi = new unsigned char[Ysize];
	unsigned char* YError = new unsigned char[Ysize];
	unsigned char* YRestr = new unsigned char[Ysize];
	unsigned char* E_File = new unsigned char[Esize];

	//读入Y分量和其余分量
	OrigiFile.read((char*)YOrigi, Ysize);
	OrigiFile.read((char*)E_File, Esize);

	//进行预测和量化
	DPCM(YOrigi, YError, YRestr, height, width, bits);
	
	//对重建图像进行重置，用以判断解码端工作效果
	for (int i = 0; i < Ysize; i++)
			YRestr[i] = 0;
	
	//从YError解出量化后的yuv文件
	RDPCM(YError, YRestr, height, width, bits);
	double peak_SNR = PSNR(YOrigi, YRestr, height, width);
	cout << peak_SNR << endl;
	
	ErrorFile.write((char*)YError, Ysize);
	ErrorFile.write((char*)E_File, Esize);
	RestrFile.write((char*)YRestr, Ysize);
	RestrFile.write((char*)E_File, Esize);

	OrigiFile.close();
	ErrorFile.close();
	RestrFile.close();
	delete[]YOrigi;
	delete[]YError;
	delete[]YRestr;
	delete[]E_File;
	return 0;
}

DPCM_Code.h

#pragma once
void DPCM(unsigned char* YOrigi, unsigned char* YError, unsigned char* YRestr, int height, int width, int bits);
void DPCM_Pixel(unsigned char* YOrigi, unsigned char* YError, unsigned char* YRestr, int pre, int j, int bits);
void RDPCM(unsigned char* YError, unsigned char* YRestr, int height, int width, int bits);
void RDPCM_Pixel(unsigned char* YError, unsigned char* YRestr, int pre, int j, int bits);
double PSNR(unsigned char* YOrigi, unsigned char* YRestr, int height, int width);

DPCM_Code.cpp

#include 
#include 
#include 
#include 
#include "DPCM_code.h"

using namespace std;

//double sign(double x)
//{
//	if (x < 0)
//		return -1;
//	else if (x > 0)
//		return 1;
//	else
//		return 0;
//}

int OverflowX(int x, int High, int Low)
{
	if (x > High)
		return High;
	else if (x < Low)
		return Low;
	else
		return x;
}

double MSE(unsigned char* YOrigi, unsigned char* YRestr, int height, int width)
{
	int size = height * width;
	long long int sum = 0;
	double mean;
	for (int i = 0; i < size; i++)
	{
		long long int temp = (long long int)(YOrigi[i] - YRestr[i]) * (long long int)(YOrigi[i] - YRestr[i]);
		sum += temp;
	}
	mean = (double)sum / (double)size;
	return mean;
}

int ErrorQuantity(int X, int bits)
{
	X = X + 255;//化为无符号数
	X = X / 2;//将误差范围归一到[0,255]
	X = floor(X / pow(2, 8 - bits));
	X = X * pow(2, 8 - bits);
	X = OverflowX(X, 255, 0);
	return X;
}

int invErrorQuantity(int X, int bits)
{
	X = X * 2;//将误差放回原来的范围
	X = X - 255;//将无符号数转为有符号数
	return X;
}

void DPCM_Pixel(unsigned char* YOrigi, unsigned char* YError, unsigned char* YRestr, int pre, int j, int bits)
{
	/*for the first Pixel,suppose YError[0]=128-YOrigi[0],
	then quantify it and store*/
	int E_temp, R_temp;
	switch (j)
	{
	case 0:
		E_temp = 128 - YOrigi[pre + j];
		E_temp = ErrorQuantity(E_temp, bits);//quantity
		YError[pre + j] = E_temp;
		E_temp = invErrorQuantity(E_temp, bits);//inverse quantization
		R_temp = 128 - E_temp;
		R_temp = OverflowX(R_temp, 255, 0);//prevent overflow
		YRestr[pre + j] = (unsigned char)R_temp;
		break;
	default:
		E_temp = YOrigi[pre + j] - YRestr[pre + j - 1];
		E_temp = ErrorQuantity(E_temp, bits);//quantity
		YError[pre + j] = E_temp;
		E_temp = invErrorQuantity(E_temp, bits);//inverse quantization
		R_temp = E_temp + YRestr[pre + j - 1];
		R_temp = OverflowX(R_temp, 255, 0);//prevent overflow
		YRestr[pre + j] = (unsigned char)R_temp;
		break;
	}
}

void RDPCM_Pixel(unsigned char* YError, unsigned char* YRestr, int pre, int j, int bits)
{
	int E_temp, R_temp;
	switch (j)
	{
	case 0:
		E_temp = YError[pre + j];
		E_temp = invErrorQuantity(E_temp, bits);
		R_temp = 128 - E_temp;
		R_temp = OverflowX(R_temp, 255, 0);
		YRestr[pre + j] = (unsigned char)R_temp;
		break;
	default:
		E_temp = YError[pre + j];
		E_temp = invErrorQuantity(E_temp, bits);
		R_temp = E_temp + YRestr[pre + j - 1];
		R_temp = OverflowX(R_temp, 255, 0);
		YRestr[pre + j] = (unsigned char)R_temp;
		break;
	}
	
}

void DPCM(unsigned char* YOrigi, unsigned char* YError, unsigned char* YRestr, int height, int width, int bits)
{
	for (int i = 0; i < height; i++)
		for (int j = 0; j < width; j++)
			DPCM_Pixel(YOrigi, YError, YRestr, i * width, j, bits);//for each pixel,use the function DPCM_Pixel() to calculate the YRestr[i] and YError[i] 
}

void RDPCM(unsigned char* YError, unsigned char* YRestr, int height, int width, int bits)
{
	for (int i = 0; i < height; i++)
		for (int j = 0; j < width; j++)
			RDPCM_Pixel(YError, YRestr, i * width, j, bits);
}

double PSNR(unsigned char* YOrigi, unsigned char* YRestr, int height, int width)
{
	int fmax = pow(2, 8) - 1;
	int a = fmax * fmax;
	double mean_se = MSE(YOrigi, YRestr, height, width);
	double peak_SNR = 10 * log10((double)a / mean_se);
	return peak_SNR;
}

PSNR输出结果表

  $PSNR$的结果总结成表格，结果如下：

  可以发现，量化bit数越大，重建的图像质量越好，与常识相符。
  用$YUV$文件查看器打开文件进行验证。（由于图片数较多，以下就仅使用8bit、4bit、1bit量化的文件）

比特数	结果
8bit量化
4bit量化
1bit量化

  可以发现，8bit量化的时候基本可以还原原图像，而4bit量化就出现了轻微的块效应，1bit量化图像就完全不可用了，符合预期。

信源符号概率分布图

  由于要画概率分布图，在DPCM的main()函数代码中添加如下代码段，用来输出

void Pro(unsigned char* YOrigi, int Ysize)
{
	for (int i = 0; i < Ysize; i++)
		Probability[YOrigi[i]] = Probability[YOrigi[i]] + 1;
	for (int i = 0; i < 256; i++)
		Probability[i] = Probability[i] / 256;
}

  完整的main.cpp代码修改如下：

#include 
#include 
#include 
#include "DPCM_code.h"

using namespace std;

double Probability[256] = { 0 };
void Pro(unsigned char* YOrigi, int Ysize)
{
	for (int i = 0; i < Ysize; i++)
		Probability[YOrigi[i]] = Probability[YOrigi[i]] + 1;
	for (int i = 0; i < 256; i++)
		Probability[i] = Probability[i] / 256;
}

int main(int argc, char** argv)
{
	//moon图片464 538,使用的是4:4:4的yuv文件
	int width = atoi(argv[1]);
	int height = atoi(argv[2]);
	int bits = atoi(argv[3]);
	int Ysize = height * width;
	int Esize = height * width * 2;

	ifstream OrigiFile(argv[4], ios::binary);
	ofstream ErrorFile(argv[5], ios::binary);
	ofstream RestrFile(argv[6], ios::binary);
	ofstream ProbaFile(argv[7], ios::binary);
	if (!OrigiFile) { cout << "error to open OrigiFile!" << endl; }
	if (!ErrorFile) { cout << "error to open ErrorFile!" << endl; }
	if (!RestrFile) { cout << "error to open RestrFile!" << endl; }
	if (!ProbaFile) { cout << "error to open ProbaFile!" << endl; }

	unsigned char* YOrigi = new unsigned char[Ysize];
	unsigned char* YError = new unsigned char[Ysize];
	unsigned char* YRestr = new unsigned char[Ysize];
	unsigned char* E_File = new unsigned char[Esize];

	//读入Y分量和其余分量
	OrigiFile.read((char*)YOrigi, Ysize);
	OrigiFile.read((char*)E_File, Esize);

	//进行预测和量化
	DPCM(YOrigi, YError, YRestr, height, width, bits);
	
	//对重建图像进行重置，用以判断解码端工作效果
	for (int i = 0; i < Ysize; i++)
			YRestr[i] = 0;
	
	//从YError解出量化后的yuv文件
	RDPCM(YError, YRestr, height, width, bits);
	double peak_SNR = PSNR(YOrigi, YRestr, height, width);
	cout << peak_SNR << endl;

	//计算概率分布图
	if (argv[8][0] == 'O')
		Pro(YOrigi, Ysize);
	else if (argv[8][0] == 'E')
		Pro(YError, Ysize);

	//输出概率分布
	for (int i = 0; i < 256; i++)
	{
		ProbaFile << i << "\t" << Probability[i] << endl;
		//cout << i << "\t" << Probability[i] << endl;
	}

	ErrorFile.write((char*)YError, Ysize);
	ErrorFile.write((char*)E_File, Esize);
	RestrFile.write((char*)YRestr, Ysize);
	RestrFile.write((char*)E_File, Esize);

	OrigiFile.close();
	ErrorFile.close();
	RestrFile.close();
	delete[]YOrigi;
	delete[]YError;
	delete[]YRestr;
	delete[]E_File;
	return 0;
}

  为了简便起见，以4bit量化和8bit量化的error概率分布图和原图的概率分布图为例，命令行输入如下：

  做出的图表如下（原文件，1bit量化，2bit量化，4bit量化，8bit量化）：

4.2 Huffman编码

压缩比输出结果表

  将8bit至1bit量化输出的E.yuv文件和原图HisSeed.yuv输入$Huffman$编码器。命令行输入如下：

  输出是.huff文件和.txt文件。比较生成的.huff文件的文件大小。文件大小和压缩比计算如下表所示：

  可以发现，进行了量化后的文件的压缩比随着量化比特数的增加而增大，即量化比特数越大，压缩效果越好。

（五）结果汇总

  以2bit量化、8bit量化为例。

bit数	error图查看	重建图查看	PSNR值	压缩比	信源符号概率分布图
2			12.0588dB	38.31%
8			51.1307dB	83.06%