直方图均衡算法(Histogram Equalized)

Lab1: Histogram Equalization

1. 实验环境(C++)

操作系统版本 MacOS Catalina 10.15
OpenCV4.0 (imgcodecs | core | highgui | imgproc)
Cmake-3.14
Clang-1100.0.33.8

2. 实验步骤

Calculate the histogram H for src

Normalize the histogram.

 std::array<double, 256> calNormalizedHist(cv::Mat& source) {
     std::array<double, 256> acc{0};
     // Calculate the histogram H for src
     for(int i = 0; i < source.rows; i++)
         for (int j = 0; j < source.cols; j++)
             acc[ source.ptr<uchar>(i)[j]] ++;
     // Normalize the histogram.
     for(int i = 0; i < acc.size(); i++)
         acc[i] /= source.rows * source.cols;
     return acc;
 }

Compute the integral of the histogram: H'

Transform the image using H′ as a look-up table: \[𝚍𝚜𝚝(x,y)=H′(𝚜𝚛𝚌(x,y)) * 255\]

 void equalizeHist(cv::Mat& source, cv::Mat& result) {
     source.copyTo(result);
     auto hist = calNormalizedHist(source);
     // Compute the integral of the histogram: H'
     for(int i = 1; i < 256; i++) hist[i] += hist[i-1];
     // Transform the image using H′ as a look-up table: 𝚍𝚜𝚝(x,y)=H′(𝚜𝚛𝚌(x,y)) * 255
     for(int i = 0; i < source.rows; i++)
         for (int j = 0; j < source.cols; j++)
             result.ptr<uchar>(i)[j] = hist[source.ptr<uchar>(i)[j]] * 255;
 }

Test Code:

 cv::Mat cv, lab, m = cv::imread("../data/lena.png", cv::IMREAD_GRAYSCALE);
 cv::equalizeHist(m, cv);
 equalizeHist(m, lab);

 cv::imwrite("../out/m.png", m);
 cv::imwrite("../out/cv.png", cv);
 cv::imwrite("../out/lab.png", lab);

 cv::imwrite("../out/cv-m.png", abs(cv-m));
 cv::imwrite("../out/lab-m.png", abs(lab-m));
 cv::imwrite("../out/lab-cv.png", abs(lab-cv));

 cv::imwrite("../out/hist.m.png", drawHist(calNormalizedHist(m)));
 cv::imwrite("../out/hist.cv.png", drawHist(calNormalizedHist(cv)));
 cv::imwrite("../out/hist.lab.png", drawHist(calNormalizedHist(lab)));

 cv::imwrite("../out/acchist.m.png", drawHist(calNormalizedHist(m), true));
 cv::imwrite("../out/acchist.cv.png", drawHist(calNormalizedHist(cv), true));
 cv::imwrite("../out/acchist.lab.png", drawHist(calNormalizedHist(lab),true));

3. 实验结果

image: Origin

image: OpenCV

image: Result

dif: Origin&OpenCV

dif: Result&Origin

dif: Result&OpenCV

Hist: Origin

Hist: OpenCV

Hist: Result

AccHist: Origin

AccHist: OpenCV

AccHist: Result

4. 实验结果分析

明显本文实现的算法与OpenCV实现的高度一致(不考虑指令集优化: SIMD|SEE4 etc.)
由Histogram与Accumulate Histogram上看, 连续变量的均匀分布意味着概率分布函数满足线性分布, 与理论推导得出的性质一致
从直方图直观观察和算法分析, 易知基于直接灰度映射(点对点映射)的经典直方图均衡算法会导致灰阶的减少.因而加剧色带(Banding)现象:
- Lookup-Table \(H’\)在定义上不是单射
所以这个意义上, 直方图的均衡是一个可以把相邻柱子合并的挪动柱子的游戏~

5 Attachment

本部分是附加部分, 主要回答实验课程上与老师的一个小小的 argument:

怎么得到均衡后的直方图 ?

比较常见的方法: 图像均衡后再统计一次直方图

比较少用的方法: 直接均衡原直方图

    std::array<double, 256> equalizeHist(std::array<double, 256>&& hist) {
        std::array<double, 256> hist_eq{0}, hist_acc{0};
        hist_acc[0] = hist[0];
        // Compute the integral of the histogram: H'
        for(int i = 1; i < 256; i++) hist_acc[i] = hist_acc[i-1] + hist[i];

        for(int i = 0; i < 256; i++)
            hist_eq[hist_acc[i]*255] += hist[i];

        return hist_eq;
    }

先计算均衡后的直方图再均衡图像对吗?
- 不对, 均衡是一种方法, 可以直接均衡图像然后计算新的直方图, 也可以直接均衡直方图, 两直方图一致
直方图均衡的均衡指什么?
- 均衡一种概率变换, 可以将线性的目标概率分布函数拓展到任意的分布函数:
  Transformation between two distribution funtion.
  
  5.* 下面给出指定累计直方图的均衡结果的部分例子
\[y=x^2\]

AccHist

Image

Hist:

Hist: Directly
\[y=x^5\]

AccHist

Image

Hist:

Hist: Directly
\[y=sin(x)\]

AccHist

Image

Hist:

Hist: Directly
\[y=\sqrt{2x-x^2}\]

AccHist

Image

Hist:

Hist: Directly
\[y=e^{x-1}\]

AccHist

Image

Hist:

Hist: Directly