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直方圖均衡算法(Histogram Equalized)

本文轉載自   查看原文   2019-10-31 14:14   913 

Lab1: Histogram Equalization

1. 實驗環境(C++)

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

2. 實驗步驟

  1. Calculate the histogram H for src

  2. 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;
	}

  3. Compute the integral of the histogram: H'

  4. 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;
	}

  5. 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. 實驗結果


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. 實驗結果分析

  • 明顯本文實現的算法與OpenCV實現的高度一致(不考慮指令集優化: SIMD|SEE4 etc.)

  • 由Histogram與Accumulate Histogram上看, 連續變量的均勻分布意味着概率分布函數滿足線性分布, 與理論推導得出的性質一致

  • 從直方圖直觀觀察和算法分析, 易知基於直接灰度映射(點對點映射)的經典直方圖均衡算法會導致灰階的減少.因而加劇色帶(Banding)現象:

    • Lookup-Table \(H’\)在定義上不是單射

    所以這個意義上, 直方圖的均衡是一個可以把相鄰柱子合並的挪動柱子的游戲~

5 Attachment

本部分是附加部分, 主要回答實驗課程上與老師的一個小小的 argument:

  • 怎么得到均衡后的直方圖 ?

    • 比較常見的方法: 圖像均衡后再統計一次直方圖

    • 比較少用的方法: 直接均衡原直方圖

      	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;
	}

  • 先計算均衡后的直方圖再均衡圖像對嗎?

    • 不對, 均衡是一種方法, 可以直接均衡圖像然后計算新的直方圖, 也可以直接均衡直方圖, 兩直方圖一致

  • 直方圖均衡的均衡指什么?

    • 均衡一種概率變換, 可以將線性的目標概率分布函數拓展到任意的分布函數:
      ​ Transformation between two distribution funtion.

5.* 下面給出指定累計直方圖的均衡結果的部分例子

  • \[y=x^2 \]


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 
#include <iostream>
#include <opencv2/imgcodecs.hpp>
#include <opencv2/imgproc.hpp>
#include <opencv2/highgui.hpp>
#include <array>


cv::Mat drawHist(std::array<double, 256>&& acc, bool accumulate = false) {
	if(accumulate) for(int i = 1; i < acc.size(); i++) acc[i] += acc[i-1];
	
	double max_v = *std::max_element(acc.begin(), acc.end());
	
	cv::Mat visual = cv::Mat::zeros(256, 256, CV_8UC3);
	for(int i = 0; i < 256; i++)
		cv::line(visual, cv::Point(i, 256.0*(1-acc[i]/max_v)), cv::Point(i, 256), 255);
	
	return visual;
}

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;
}


std::array<double, 256> equalizeHist(std::array<double, 256>&& hist,
										std::function<double(double)> && f=[](double x){return x;}) {
	std::array<double, 256> hist_eq{0};
	
	std::array<int, 256> H{0};
	
	double sum = 0;
	for(int i = 0, j = 0; i < 256; i++)
		for(;sum < f(i/255.0); j++) {
			sum += hist[j]; H[j] = i;
		}

	for(int i = 0; i < 256; i++)
		hist_eq[H[i]] += hist[i];
	
	return hist_eq;
}


// Only for single channel
void equalizeHist(cv::Mat& source, cv::Mat& result,
                  std::function<double(double)> && f=[](double x){return x;}) {
	source.copyTo(result);

	auto hist = calNormalizedHist(source);
	
	std::array<int, 256> H{0};
	
	// Compute the integral of the histogram: H'
	double sum = 0;
	for(int i = 0, j = 0; i < 256; i++)
		for(;sum < f(i/255.0); j++) {
			sum += hist[j]; H[j] = i;
		}

	// directly gray mapping
	// 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] = H[source.ptr<uchar>(i)[j]];
}
	

int main() {

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

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

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


	cv::imwrite("../out/acchist.m.bmp", drawHist(calNormalizedHist(m), true));
	cv::imwrite("../out/acchist.cv.bmp", drawHist(calNormalizedHist(cv), true));
	cv::imwrite("../out/acchist.lab.bmp", drawHist(calNormalizedHist(lab),true));
	
	{
		auto f = [&](std::string f_name, std::function<double(double)>&& f) {
			cv::Mat m_f;
			equalizeHist(m, m_f, std::move(f));
			cv::imwrite("../out/lab."+f_name+".bmp", m_f);
			cv::imwrite("../out/hist."+f_name+".bmp", drawHist(calNormalizedHist(m_f)));
			cv::imwrite("../out/acchist."+f_name+".bmp", drawHist(calNormalizedHist(m_f), true));
			cv::imwrite("../out/hist."+f_name+"_direct.bmp",drawHist(equalizeHist(calNormalizedHist(m), std::move(f))))    ;
		};
		

		f("x^2", [](double x){return x*x;});
		f("sin(x)", [](double x){return sin(3.1415926 * 0.5 * x);});
		f("sqrt(x*(2-x))", [](double x){return sqrt(x*(2-x));});
		f("exp(x-1)", [](double x){return std::exp(x-1);});
		f("x^5", [](double x){return std::pow(x,5);});
	}
	
//	cv::imshow("lena", m);
//	cv::imshow("lena: equalizeHist", cv);
//	cv::imshow("lena: equalizeHist zlb", lab);
//
//
//	cv::imshow("origin", drawHist(calNormalizedHist(m)));
//	cv::imshow("cvHist", drawHist(calNormalizedHist(cv), true));
//	cv::imshow("zlbHist", drawHist(calNormalizedHist(lab), true));
//	cv::waitKey();
	return 0;
}


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