我们已经从BMP图中拿到了需要压缩RGB的数据,我们需要对原数据从RGB域转变YCbCr域,之后对YCbCr数据进行下采样(down sampling)。对于不需要看文章的同学,这边直接给出源代码。https://github.com/Cheemion/JPEG_COMPRESS

图片引用"Compressed Image File Formats JPEG, PNG, GIF, XBM, BMP - John Miano"[1]
1.RGB域和YCbCr域
RGB代表红绿蓝,通过3种颜色的叠加来得到我们看到的颜色。0-到255分别代表颜色从浅到深。
Y = 0.299 * red + 0.587 * green + 0.114 * blue;
Cb = -0.1687 * red - 0.3313 * green + 0.5 * blue + 128;
Cr = 0.5 * red - 0.4187 * green - 0.0813 * blue + 128;
Y是RGB的加权平均值,称之为亮度(luminance)
Cb是B分量和亮度的差值, 称为Chrominance(Cb)
Cr是R分量和亮度的差值,称为Chrominance(Cr)
以下代码将RGB转为YCbCr。为什么将RGB转为YCbCr? 因为人眼对亮度(Y)的变化更敏感,所以我可以对Cr和Cb进行下采样(压缩,比如本来1个字节代表一个pixel的数据,压缩后用1个字节代表4个pixels的数据),尽可能保留完整的Y分量。通过这样子我们可以进一步的压缩数据。
- void JPG::convertToYCbCr() {
- for(uint i = 0; i < height; i++) {
- for(uint j = 0; j < width; j++) {
- YCbCr temp = BMPData[i * width + j];
- BMPData[i * width + j].Y = 0.299 * temp.red + 0.587 * temp.green + 0.114 * temp.blue;
- BMPData[i * width + j].Cb = -0.1687 * temp.red - 0.3313 * temp.green + 0.5 * temp.blue + 128;
- BMPData[i * width + j].Cr = 0.5 * temp.red - 0.4187 * temp.green - 0.0813 * temp.blue + 128;
- }
- }
- }
2.sampling(采样)
采样通常是对连续信号进行采样,比如下图蓝色是连续信号x(t),红色是对信号进行采样后得到的信号x[n]=x(T*n), T是采样间隔,1/T是采样频率。

而在JPEG中,我们是对已经离散的数据进行采样,并且JPEG中的采样数值是相对采样数值。相对于最高采样频率的采样数值。
如下左图
Y(luminance)分量的水平采样频率(H, Horizantal sampling frequency)和垂直采样频率(V, vertical sampling frequency)都是4,是最高的采样频率。最高的采样频率就相当于保留原图的Y分量,不进行下采样。
Cb分量的水平和垂直的采样频率都是2,等于最高采样频率的一半。所以水平每2个点采样一次,垂直每2个点采样一次。
Cr分量的水平和垂直采样频率都是1,等于最高采样频率的1/4。所以水平和垂直每4个点采样一个点。
3个分量的量叠加就得到了我们的像素的值。

图片引用"Compressed Image File Formats JPEG, PNG, GIF, XBM, BMP - John Miano"[1]
2.YCbCr数据在JPEG中的存储
JPEG规定所有的数据都是以8*8的一个block(data unit)的形式进行离散余弦变化和存储的.可以把这8*8的block看成是最小存储单元。
MCU是Y,Cb,Cr的完整的block组成的能够完整还原一个范围的色彩的最小单元。啥意思?
假设我们的图片是10*10的大小
.
若Y,Cb,Cr的水平和垂直的采样频率都为1,则原图由4个mcu(4种颜色分别代表一个MCU)组成(每个mcu包含1个y的block,一个cb的block,一个cr的block, 每个mcu的大小为8*8),边缘空白的地方可用0替代,也可以重复边缘的值。
左上角那块4*4的小block的值分别
pixel[0,0] = y[0,0] + cb[0,0] + cr[0,0]
pixel[0,1] = y[0,1] + cb[0,1] + cr[0,1]
pixel[1,0] = y[1,0] + cb[1,0] + cr[1,0]
pixel[1,1] = y[1,1] + cb[1,1] + cr[1,1]

若Y的水平和垂直采样频率为2, cb和cr的采样频率为1, 则原图由1个mcu组成(大小为16*16)。mcu中包含4个y的block(2*2),一个cb,一个cr。总共6个block,大小只占原来block的一半。
左上角那块4*4的小block的值分别
pixel[0,0] = y[0,0] + cb[0,0] + cr[0,0]
pixel[0,1] = y[0,1] + cb[0,0] + cr[0,0]
pixel[1,0] = y[1,0] + cb[0,0] + cr[0,0]
pixel[1,1] = y[1,1] + cb[0,0] + cr[0,0]

总结:mcu大小= 垂直最大采样值 * 水平最大采样值, 一个mcu包含y的水平采样值*y的垂直采样值个的y个block(y的水平采样为2,垂直为2,则一个muc有4个yblock)。其他分量同理
1.3定义JPG class代码
- //定义Block
using Block = int[64];
//定义YCbCr,同时这个结构用来展示存放rgb数据 - struct YCbCr {
- union
- {
- double Y;
- double red;
- };
- union
- {
- double Cb;
- double green;
- };
- union {
- double Cr;
- double blue;
- };
- };
struct MCU {- Block* y;
- Block* cb;
- Block* cr;
- };
//定义JPG类,用于压缩图片- class JPG
- {
- public:
//rgb转到YCbCr - void convertToYCbCr();
//下采样 - void subsampling();
//变化 - void discreteCosineTransform();
//量化 - void quantization();
//哈夫曼 - void huffmanCoding();
//输出 - void output(std::string path);
- public:
MCU* data;
Block* blocks;
//BMPData存放的是bmp图片的RGB数据 - YCbCr* BMPData;
- uint blockNum;
- //原图的像素
- uint width;
- uint height;
- //mcu 有多少个 长度是多少
- uint mcuWidth;
- uint mcuHeight;
- //一个完整的muc的水平和垂直像素个数
- uint mcuVerticalPixelNum;
- uint mcuHorizontalPixelNum;
- //用于subsampling
- // only support 1 or 2
- byte YVerticalSamplingFrequency;
- byte YHorizontalSamplingFrequency;
- byte CbVerticalSamplingFrequency;
- byte CbHorizontalSamplingFrequency;
- byte CrVerticalSamplingFrequency;
- byte CrHorizontalSamplingFrequency;
- byte maxVerticalSamplingFrequency;
- byte maxHorizontalSamplingFrequency;
- public:
- JPG(uint width, uint height,const RGB* const rgbs,
- byte YVerticalSamplingFrequency, byte YHorizontalSamplingFrequency,
- byte CbVerticalSamplingFrequency, byte CbHorizontalSamplingFrequency,
- byte CrVerticalSamplingFrequency, byte CrHorizontalSamplingFrequency
- )
- :width(width), height(height),
- YVerticalSamplingFrequency(YVerticalSamplingFrequency), YHorizontalSamplingFrequency(YHorizontalSamplingFrequency),
- CbVerticalSamplingFrequency(CbVerticalSamplingFrequency), CbHorizontalSamplingFrequency(CbHorizontalSamplingFrequency),
- CrVerticalSamplingFrequency(CrVerticalSamplingFrequency), CrHorizontalSamplingFrequency(CrHorizontalSamplingFrequency)
- {
- maxHorizontalSamplingFrequency = std::max({YHorizontalSamplingFrequency, CbHorizontalSamplingFrequency, CrHorizontalSamplingFrequency});
- maxVerticalSamplingFrequency = std::max({YVerticalSamplingFrequency, CbVerticalSamplingFrequency, CrVerticalSamplingFrequency});
- //mcu的个数
- mcuWidth = (width + (maxHorizontalSamplingFrequency * 8 - 1)) / (maxHorizontalSamplingFrequency * 8);
- mcuHeight = (height + (maxVerticalSamplingFrequency * 8 - 1)) / (maxVerticalSamplingFrequency * 8);
-
- mcuVerticalPixelNum = maxVerticalSamplingFrequency * 8;
- mcuHorizontalPixelNum = maxHorizontalSamplingFrequency * 8;
- //总共多少个MCU
- data = new MCU[mcuWidth * mcuHeight];
- //一个MCU有多少个Block
- blockNum = (YVerticalSamplingFrequency * YHorizontalSamplingFrequency + CbVerticalSamplingFrequency * CbHorizontalSamplingFrequency + CrHorizontalSamplingFrequency * CrVerticalSamplingFrequency);
-
- //分配block内存空间
- blocks = new Block[mcuHeight * mcuHeight * blockNum];
- //把内存映射到对于的结构中
- for (uint i = 0; i < mcuHeight; i++) {
- for (uint j = 0; j < mcuWidth; j++) {
- data[i * mcuWidth + j].y = &blocks[(i * mcuWidth + j) * blockNum];
- data[i * mcuWidth + j].cb = data[i * mcuWidth + j].y + YVerticalSamplingFrequency * YHorizontalSamplingFrequency;
- data[i * mcuWidth + j].cr = data[i * mcuWidth + j].cb + CbVerticalSamplingFrequency * CbHorizontalSamplingFrequency;
- }
- }
- //BMP数据用于存放,bmp的原图的数据
- BMPData = new YCbCr[width * height];
//把bmp数据暂时存放在BMPdata中 - for(uint i = 0; i < height; i++) {
- for(uint j = 0; j < width; j++) {
- BMPData[i * width + j].red = static_cast<double>(rgbs[i * width + j].red);
- BMPData[i * width + j].blue = static_cast<double>(rgbs[i * width + j].blue);
- BMPData[i * width + j].green = static_cast<double>(rgbs[i * width + j].green);
- }
- }
- }
- ~JPG() {
- delete[] data;
- delete[] blocks;
- delete[] BMPData;
- }
- };
1.6下采样代码
- //这里直接把左上的点 当作subsampling的点了
- //也可以取平均值
- void JPG::subsampling() {
- //遍历mcu
- for (uint i = 0; i < mcuHeight; i++) {
- for (uint j = 0; j < mcuWidth; j++) {
//拿到mcu - MCU& currentMCU = data[i * mcuWidth + j];
//每个mcu起始的坐标点 - uint heightOffset = i * maxVerticalSamplingFrequency * 8;
- uint widthOffset = j * maxHorizontalSamplingFrequency * 8;
- //iterate over 每一个component Y, cb cr
- for (uint componentID = 1; componentID <= 3; componentID++) {
- //遍历block, 从muc中拿block
- for(uint ii = 0, yOffSet = heightOffset; ii < getVerticalSamplingFrequency(componentID); ii++, yOffSet = yOffSet + 8) {
- for(uint jj = 0, xOffset = widthOffset; jj < getHorizontalSamplingFrequency(componentID); jj++, xOffset = xOffset + 8) {
//拿到具体的block对象 - Block& currentBlock = currentMCU[componentID][ii * getHorizontalSamplingFrequency(componentID) + jj];
- //遍历Block every pixels 像素, 并且采样赋值
- for(uint y = 0; y < 8; y++) {
- for(uint x = 0; x < 8; x++) {
//得到被采样的那个点的坐标 - uint sampledY = yOffSet + y * maxVerticalSamplingFrequency / getVerticalSamplingFrequency(componentID);
- uint sampledX = xOffset + x * maxHorizontalSamplingFrequency / getHorizontalSamplingFrequency(componentID);
- //cannot find in original pictures;
- if(sampledX >= width || sampledY >= height) {
- currentBlock[y * 8 + x] = 0;
- } else {
- currentBlock[y * 8 + x] = BMPData[sampledY * width + sampledX][componentID];
- }
- }
- }
- }
- }
- }
- }
- }
- }
完整代码 https://github.com/Cheemion/JPEG_COMPRESS/tree/main/Day2
完结
Thanks for reading.
wish you have a good day.
>>>> JPG学习笔记3(附完整代码)
参考资料
[1]https://github.com/Cheemion/JPEG_COMPRESS/blob/main/resource/Compressed%20Image%20File%20Formats%20JPEG%2C%20PNG%2C%20GIF%2C%20XBM%2C%20BMP%20-%20John%20Miano.pdf