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YCbCrvoid 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 2byte 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BlockblockNum = (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mcufor (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 crfor (uint componentID = 1; componentID <= 3; componentID++) {//遍历block, 从muc中拿blockfor(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
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