图像的灰度直方图、直方图均衡化、直方图规定化（匹配）

p(r_k)=\frac{n_k}{MN}

void calcHist( const Mat* images, int nimages,
                          const int* channels, InputArray mask,
                          OutputArray hist, int dims, const int* histSize,
                          const float** ranges, bool uniform = true, bool accumulate = false );

class Histogram1D
{
private:
    int histSize[1]; // 项的数量
    float hranges[2]; // 统计像素的最大值和最小值
    const float* ranges[1];
    int channels[1]; // 仅计算一个通道

public:
    Histogram1D()
    {
        // 准备1D直方图的参数
        histSize[0] = 256;
        hranges[0] = 0.0f;
        hranges[1] = 255.0f;
        ranges[0] = hranges;
        channels[0] = 0;
    }

    MatND getHistogram(const Mat &image)
    {
        MatND hist;
        // 计算直方图
        calcHist(&image ,// 要计算图像的
            1,                // 只计算一幅图像的直方图
            channels,        // 通道数量
            Mat(),            // 不使用掩码
            hist,            // 存放直方图
            1,                // 1D直方图
            histSize,        // 统计的灰度的个数
            ranges);        // 灰度值的范围
        return hist;
    }

    Mat getHistogramImage(const Mat &image)
    {
        MatND hist = getHistogram(image);

        // 最大值，最小值
        double maxVal = 0.0f;
        double minVal = 0.0f;

        minMaxLoc(hist, &minVal, &maxVal);

        //显示直方图的图像
        Mat histImg(histSize[0], histSize[0], CV_8U, Scalar(255));

        // 设置最高点为nbins的90%
        int hpt = static_cast<int>(0.9 * histSize[0]);
        //每个条目绘制一条垂直线
        for (int h = 0; h < histSize[0]; h++)
        {
            float binVal = hist.at<float>(h);
            int intensity = static_cast<int>(binVal * hpt / maxVal);
            // 两点之间绘制一条直线
            line(histImg, Point(h, histSize[0]), Point(h, histSize[0] - intensity), Scalar::all(0));
        }
        return histImg;
    }
};

    Histogram1D hist;
    Mat histImg;
    histImg = hist.getHistogramImage(image);

    imshow("Image", image);
    imshow("Histogram", histImg);

P(S_k) = \frac{n_k}{n}

CDF(S_k) = \sum\limits^k_{i=0}\frac{n_i}{n}=\sum\limits^k_{i=0}P_s(S_i)

D_j = L\cdot CDF(S_i)

CDF(S_i)

void equalization_self(const Mat &src, Mat &dst)
{
    Histogram1D hist1D;
    MatND hist = hist1D.getHistogram(src);

    hist /= (src.rows * src.cols); // 对得到的灰度直方图进行归一化
    float cdf[256] = { 0 }; // 灰度的累积概率
    Mat lut(1, 256, CV_8U); // 灰度变换的查找表
    for (int i = 0; i < 256; i++)
    {
        // 计算灰度级的累积概率
        if (i == 0)
            cdf[i] = hist.at<float>(i);
        else
            cdf[i] = cdf[i - 1] + hist.at<float>(i);

        lut.at<uchar>(i) = static_cast<uchar>(255 * cdf[i]); // 创建灰度的查找表
    }

    LUT(src, lut, dst); // 应用查找表，进行灰度变化，得到均衡化后的图像

}

s = T(r)

v = G(z)

s = v,且 z = G^{-1}(v) = G^{-1}(T(r))

P_r(r)

s_k = T(r_k) = L \cdot \sum\limits_{i=0}^{i=k}P_r(r_k)

v_k = G(z_m) = L \cdot \sum\limits_{j=0}^{j=m}P_z(z_m)

s_k = v_m

\begin{array}{c} 
s_k = v_k \ L \cdot \sum\limits_{i=0}^{i=k}P_r(r_k) = L \cdot \sum\limits_{j=0}^{j=m}P_z(z_m) \ \sum\limits_{i=0}^{i=k}P_r(r_k) = \sum\limits_{j=0}^{j=m}P_z(z_m) 
\end{array}

void hist_specify(const Mat &src, const Mat &dst,Mat &result)
{
    Histogram1D hist1D;
    MatND src_hist = hist1D.getHistogram(src);
    MatND dst_hist = hist1D.getHistogram(dst);

    float src_cdf[256] = { 0 };
    float dst_cdf[256] = { 0 };

    // 源图像和目标图像的大小不一样，要将得到的直方图进行归一化处理
    src_hist /= (src.rows * src.cols);
    dst_hist /= (dst.rows * dst.cols);

    // 计算原始直方图和规定直方图的累积概率
    for (int i = 0; i < 256; i++)
    {
        if (i == 0)
        {
            src_cdf[i] = src_hist.at<float>(i);
            dst_cdf[i] = dst_hist.at<float>(i);
        }
        else
        {
            src_cdf[i] = src_cdf[i - 1] + src_hist.at<float>(i);
            dst_cdf[i] = dst_cdf[i - 1] + dst_hist.at<float>(i);
        }
    }

    // 累积概率的差值
    float diff_cdf[256][256];
    for (int i = 0; i < 256; i++)
        for (int j = 0; j < 256; j++)
            diff_cdf[i][j] = fabs(src_cdf[i] - dst_cdf[j]);

    // 构建灰度级映射表
    Mat lut(1, 256, CV_8U);
    for (int i = 0; i < 256; i++)
    {
        // 查找源灰度级为ｉ的映射灰度
        //　和ｉ的累积概率差值最小的规定化灰度
        float min = diff_cdf[i][0];
        int index = 0;
        for (int j = 1; j < 256; j++)
        {
            if (min > diff_cdf[i][j])
            {
                min = diff_cdf[i][j];
                index = j;
            }
        }
        lut.at<uchar>(i) = static_cast<uchar>(index);
    }

    // 应用查找表，做直方图规定化
    LUT(src, lut, result);
}