NCC(Normalized Cross Correlation)歸一化互相關

影象匹配指在已知目標基準圖的子圖集合中，尋找與實時影象最相似的子圖，以達到目標識別與定位目的的影象技術。主要方法有：基於影象灰度相關方法、基於影象特徵方法、基於神經網路相關的人工智慧方法(還在完善中)。基於影象灰度的匹配演算法簡單，匹配準確度高，主要用空間域的一維或二維滑動模版進行影象匹配，不同的演算法區別主要體現在模版及相關準則的選擇方面，但計算量大，不利於實時處理，對灰度變化、旋轉、形變以及遮擋等比較敏感；基於影象特徵的方法計算量相對較小，對灰度變化、形變及遮擋有較好的適應性，通過在原始圖中提取點、線、區域等顯著特徵作為匹配基元，進而用於特徵匹配，但是匹配精度不高。 通常又把基於灰度的匹配演算法，稱作相關匹配演算法。相關匹配演算法又分為兩類：一類強調景物之間的差別程度如平法差法(SD)和平均絕對差值法(MAD)等;另一類強調景物之間的相似程度,主要演算法又分成兩類,一是積相關匹配法,二是相關係數法。今天我們就來說說歸一化互相關係數法（NCC).

1. NCC原理： 假設兩幅進行匹配計算的影象中的小影象為g,大小為m×n,大影象為S，大小為M×N.用Sx,y表示S中以（x,y)為左上角點與g大小相同的子塊。利用相關係數公式計算實時圖和基準圖之間的相關係數，得到相關係數矩陣ρ(x,y),通過對相關係數矩陣的分析，判斷兩幅影象是否相關。 ρ(x,y)的定義為 隨機變數X和Y的(Pearson)相關係數） 式中： 是Sx,y和g的協方差；

Dx,y為Sx,y的方差

D為g的方差， g的灰度均值 影象Sx,y的灰度均值 將Dx,y和D代入式得到： 相關係數滿足： 在[-1,1]絕對尺度範圍之間衡量兩者的相似性。相關係數刻畫了兩者之間的近似程度的線性描述。一般說來，越接近於1，兩者越近似的有線性關係。 2. C++程式碼實現

    Mat image1 = imread("E:xx.tif", IMREAD_GRAYSCALE);
	Mat image2 = imread("E:yy.tif", IMREAD_GRAYSCALE); 
	int overlap = 350;
	float pearsonCorrelationCoefficientMax = 0;
	int overlapMaxCorrelationCoefficient = 0;
	for (int overlap = 350; overlap < 650; overlap += 50)
	{
	//****************************************//
		Mat imageTemp = image2(Rect(0, 0, overlap, image1.rows));
		long double tempTotalcount = 0;
		long double tempTotalPixel = 0;
		for (int i = 0; i < overlap; i++)
		{
			for (int j = 0; j < image1.rows; j++)
			{
				tempTotalcount += 1;
				//cout << i<<","<<j<<"："<<int(imageTemp.at<uchar>(j,i)) << ",";
				tempTotalPixel += float(imageTemp.at<uchar>(j, i));
			}
			cout << endl;
		}
		float tempAvg = tempTotalPixel / tempTotalcount;
		//**************************************//	
		long double tempSubstract = 0;
		for (int i = 0; i < overlap; i++)
		{
			for (int j = 0; j < image1.rows; j++)
			{
				long double tempSquare = (long double(imageTemp.at<uchar>(j, i)) - tempAvg)* (long double(imageTemp.at<uchar>(j, i)) - tempAvg);
				tempSubstract = tempSubstract + tempSquare;
			}
			cout << endl;
		}
		float tempVariance = sqrt(tempSubstract / tempTotalcount);
	//***********************************************//
		Mat imageBase = image1(Rect(image1.cols-overlap, 0, overlap, image1.rows));
		int baseTotalcount = 0;
		int baseTotalPixel = 0;
		for (int i = 0; i < overlap; i++)
		{
			for (int j = 0; j < image1.rows; j++)
			{
				baseTotalcount += 1;
				//cout << i<<","<<j<<"："<<int(imageTemp.at<uchar>(j,i)) << ",";
				baseTotalPixel += float(imageBase.at<uchar>(j, i));
			}
			cout << endl;
		}
		float baseAvg = baseTotalPixel / baseTotalcount;

		//*****************************************//	
		long double baseSubstract = 0;
		for (int i = 0; i < overlap; i++)
		{
			for (int j = 0; j < image1.rows; j++)
			{
				long double baseSquare = (long double(imageBase.at<uchar>(j, i)) - baseAvg)* (long double(imageBase.at<uchar>(j, i)) - baseAvg);
				baseSubstract = baseSubstract + baseSquare;
			}
			cout << endl;
		}
		float baseVariance = sqrt(baseSubstract / baseTotalcount);
		//***************************************//
		long double dotMul = 0;
		for (int i = 0; i < overlap; i++)
		{
			for (int j = 0; j < image1.rows; j++)
			{
				dotMul += abs((long double(imageBase.at<uchar>(j, i)) - baseAvg)*(long double(imageTemp.at<uchar>(j, i)) - tempAvg));
			}
			cout << endl;
		}
		float dotMulAvg = dotMul / baseTotalcount;

		float pearsonCorrelationCoefficient=dotMulAvg / (baseVariance*tempVariance);
		if (pearsonCorrelationCoefficientMax < pearsonCorrelationCoefficient)
		{
			pearsonCorrelationCoefficientMax = pearsonCorrelationCoefficient;
			overlapMaxCorrelationCoefficient = overlap;
		}

	}
	cout << "最大相關係數" << pearsonCorrelationCoefficientMax << endl;
	cout << "最大相關係數時重疊區域" << overlapMaxCorrelationCoefficient << endl;