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C++ OpenCV實(shí)現(xiàn)圖像雙三次插值算法詳解

更新時(shí)間：2021年12月02日 16:19:37 作者：怎么今天你又不開心了呢

圖像雙三次插值的原理，就是目標(biāo)圖像的每一個(gè)像素都是由原圖上相對(duì)應(yīng)點(diǎn)周圍的4x4=16個(gè)像素經(jīng)過加權(quán)之后再相加得到的。本文主要介紹了通過C++ OpenCV實(shí)現(xiàn)圖像雙三次插值算法，需要的可以參考一下

前言

近期在學(xué)習(xí)一些傳統(tǒng)的圖像處理算法，比如傳統(tǒng)的圖像插值算法等。傳統(tǒng)的圖像插值算法包括鄰近插值法、雙線性插值法和雙三次插值法，其中鄰近插值法和雙線性插值法在網(wǎng)上都有很詳細(xì)的介紹以及用c++編寫的代碼。但是，網(wǎng)上關(guān)于雙三次插值法的原理介紹雖然很多，也有對(duì)應(yīng)的代碼，但是大多都不是很詳細(xì)。因此基于自己對(duì)原理的理解，自己編寫了圖像雙三次插值算法的c++ opencv代碼，在這里記錄一下。

一、圖像雙三次插值算法原理

首先是原理部分。圖像雙三次插值的原理，就是目標(biāo)圖像的每一個(gè)像素都是由原圖上相對(duì)應(yīng)點(diǎn)周圍的4x4=16個(gè)像素經(jīng)過加權(quán)之后再相加得到的。這里的加權(quán)用到的就是三次函數(shù)，這也是圖像雙三次插值算法名稱的由來（個(gè)人猜測(cè)）。用到的三次函數(shù)如下圖所示：

最關(guān)鍵的問題是，這個(gè)三次函數(shù)的輸入和輸出分別代表啥。簡(jiǎn)單來說輸入就是原圖對(duì)應(yīng)點(diǎn)周圍相對(duì)于這點(diǎn)的4x4大小區(qū)域的坐標(biāo)值，大小在0~2之間，輸出就是這些點(diǎn)橫坐標(biāo)或者縱坐標(biāo)的權(quán)重。4個(gè)橫坐標(biāo)、4個(gè)縱坐標(biāo)，對(duì)應(yīng)相乘就是4x4大小的權(quán)重矩陣，然后使用此權(quán)重矩陣對(duì)原圖相對(duì)應(yīng)的區(qū)域進(jìn)行相乘并相加就可以得到目標(biāo)圖點(diǎn)的像素。

下圖可以幫助大家更好地理解

首先，u和v是什么呢？舉一個(gè)例子，對(duì)于一幅100x100的灰度圖像，要將其放大到500x500，那么其縮放因子sx=500/100=5，sy=500/100=5?，F(xiàn)在目標(biāo)圖像是500x500，需要用原圖的100x100個(gè)像素值來填滿這500x500個(gè)空，根據(jù)src_x=i/sx和src_y=j/sy可以得到目標(biāo)像素坐標(biāo)（i，j）所對(duì)應(yīng)的原圖像素坐標(biāo)（src_x, src_y)，這個(gè)src_x和src_y的小數(shù)部分就是上圖中的u和v。

理解了u和v，就可以利用u和v來計(jì)算雙三次插值算法的權(quán)重了。上面說了三次函數(shù)的輸入是原圖對(duì)應(yīng)點(diǎn)周圍相對(duì)于這點(diǎn)的4x4大小區(qū)域的坐標(biāo)值，對(duì)于上面這幅圖而言，橫坐標(biāo)有四個(gè)輸入，分別是1+u，u，1-u，2-u；縱坐標(biāo)也有四個(gè)輸入，分別是1+v，v，1-v，2-v，根據(jù)三次函數(shù)算出權(quán)重之后兩兩相乘就是對(duì)應(yīng)的4x4大小的權(quán)重矩陣。

知道了怎么求權(quán)重矩陣之后，就可以遍歷整幅圖像進(jìn)行插值了。下面是基于自己對(duì)原理的理解編寫的c++ opencv代碼，代碼沒有做優(yōu)化，但是能夠讓大家直觀地理解圖像雙三次插值算法。

二、C++ OpenCV代碼

1.計(jì)算權(quán)重矩陣

前面說了權(quán)重矩陣就是橫坐標(biāo)的4個(gè)輸出和縱坐標(biāo)的4個(gè)輸出相乘，因此只需要求出橫坐標(biāo)和縱坐標(biāo)相對(duì)應(yīng)的8個(gè)輸出值就行了。

代碼如下：

std::vector<double> getWeight(double c, double a = 0.5)
{
	//c就是u和v，橫坐標(biāo)和縱坐標(biāo)的輸出計(jì)算方式一樣
	std::vector<double> temp(4);
	temp[0] = 1 + c; temp[1] = c; 
	temp[2] = 1 - c; temp[3] = 2 - c;
	
	//y(x) = (a+2)|x|*|x|*|x| - (a+3)|x|*|x| + 1   |x|<=1
	//y(x) = a|x|*|x|*|x| - 5a|x|*|x| + 8a|x| - 4a  1<|x|<2
	std::vector<double> weight(4);
	weight[0] = (a * pow(abs(temp[0]), 3) - 5 * a * pow(abs(temp[0]), 2) + 8 * a * abs(temp[0]) - 4 * a);
	weight[1] = (a + 2) * pow(abs(temp[1]), 3) - (a + 3) * pow(abs(temp[1]), 2) + 1;
	weight[2] = (a + 2) * pow(abs(temp[2]), 3) - (a + 3) * pow(abs(temp[2]), 2) + 1;
	weight[3] = (a * pow(abs(temp[3]), 3) - 5 * a * pow(abs(temp[3]), 2) + 8 * a * abs(temp[3]) - 4 * a);

	return weight;
}

2.遍歷插值

代碼如下：

void bicubic(cv::Mat& src, cv::Mat& dst, int dst_rows, int dst_cols)
{
	dst.create(dst_rows, dst_cols, src.type());
	double sy = static_cast<double>(dst_rows) / static_cast<double>(src.rows);
	double sx = static_cast<double>(dst_cols) / static_cast<double>(src.cols);
	cv::Mat border;
	cv::copyMakeBorder(src, border, 1, 1, 1, 1, cv::BORDER_REFLECT_101);

	//處理灰度圖
	if (src.channels() == 1)
	{
		for (int i = 1; i < dst_rows + 1; ++i)
		{
			int src_y = (i + 0.5) / sy - 0.5; //做了幾何中心對(duì)齊
			if (src_y < 0) src_y = 0;
			if (src_y > src.rows - 1) src_y = src.rows - 1;
			src_y += 1;
			//目標(biāo)圖像點(diǎn)坐標(biāo)對(duì)應(yīng)原圖點(diǎn)坐標(biāo)的4個(gè)縱坐標(biāo)
			int i1 = std::floor(src_y);
			int i2 = std::ceil(src_y);
			int i0 = i1 - 1;
			int i3 = i2 + 1;
			double u = src_y - static_cast<int64>(i1);
			std::vector<double> weight_x = getWeight(u);

			for (int j = 1; j < dst_cols + 1; ++j)
			{
				int src_x = (j + 0.5) / sy - 0.5;
				if (src_x < 0) src_x = 0;
				if (src_x > src.rows - 1) src_x = src.rows - 1;
				src_x += 1;
				//目標(biāo)圖像點(diǎn)坐標(biāo)對(duì)應(yīng)原圖點(diǎn)坐標(biāo)的4個(gè)橫坐標(biāo)
				int j1 = std::floor(src_x);
				int j2 = std::ceil(src_x);
				int j0 = j1 - 1;
				int j3 = j2 + 1;
				double v = src_x - static_cast<int64>(j1);
				std::vector<double> weight_y = getWeight(v);

				//目標(biāo)點(diǎn)像素對(duì)應(yīng)原圖點(diǎn)像素周圍4x4區(qū)域的加權(quán)計(jì)算（插值）
				double pix = weight_x[0] * weight_y[0] * border.at<uchar>(i0, j0) + weight_x[1] * weight_y[0] * border.at<uchar>(i0, j1)
					+ weight_x[2] * weight_y[0] * border.at<uchar>(i0, j2) + weight_x[3] * weight_y[0] * border.at<uchar>(i0, j3)
					+ weight_x[0] * weight_y[1] * border.at<uchar>(i1, j0) + weight_x[1] * weight_y[1] * border.at<uchar>(i1, j1)
					+ weight_x[2] * weight_y[1] * border.at<uchar>(i1, j2) + weight_x[3] * weight_y[1] * border.at<uchar>(i1, j3)
					+ weight_x[0] * weight_y[2] * border.at<uchar>(i2, j0) + weight_x[1] * weight_y[2] * border.at<uchar>(i2, j1)
					+ weight_x[2] * weight_y[2] * border.at<uchar>(i2, j2) + weight_x[3] * weight_y[2] * border.at<uchar>(i2, j3)
					+ weight_x[0] * weight_y[3] * border.at<uchar>(i3, j0) + weight_x[1] * weight_y[3] * border.at<uchar>(i3, j1)
					+ weight_x[2] * weight_y[3] * border.at<uchar>(i3, j2) + weight_x[3] * weight_y[3] * border.at<uchar>(i3, j3);
				if (pix < 0) pix = 0;
				if (pix > 255)pix = 255;

				dst.at<uchar>(i - 1, j - 1) = static_cast<uchar>(pix);
			}
		}
	}
	//處理彩色圖像
	else if (src.channels() == 3)
	{
		for (int i = 1; i < dst_rows + 1; ++i)
		{
			int src_y = (i + 0.5) / sy - 0.5;
			if (src_y < 0) src_y = 0;
			if (src_y > src.rows - 1) src_y = src.rows - 1;
			src_y += 1;
			int i1 = std::floor(src_y);
			int i2 = std::ceil(src_y);
			int i0 = i1 - 1;
			int i3 = i2 + 1;
			double u = src_y - static_cast<int64>(i1);
			std::vector<double> weight_y = getWeight(u);

			for (int j = 1; j < dst_cols + 1; ++j)
			{
				int src_x = (j + 0.5) / sy - 0.5;
				if (src_x < 0) src_x = 0;
				if (src_x > src.rows - 1) src_x = src.rows - 1;
				src_x += 1;
				int j1 = std::floor(src_x);
				int j2 = std::ceil(src_x);
				int j0 = j1 - 1;
				int j3 = j2 + 1;
				double v = src_x - static_cast<int64>(j1);
				std::vector<double> weight_x = getWeight(v);

				cv::Vec3b pix;

				pix[0] = weight_x[0] * weight_y[0] * border.at<cv::Vec3b>(i0, j0)[0] + weight_x[1] * weight_y[0] * border.at<cv::Vec3b>(i0, j1)[0]
					+ weight_x[2] * weight_y[0] * border.at<cv::Vec3b>(i0, j2)[0] + weight_x[3] * weight_y[0] * border.at<cv::Vec3b>(i0, j3)[0]
					+ weight_x[0] * weight_y[1] * border.at<cv::Vec3b>(i1, j0)[0] + weight_x[1] * weight_y[1] * border.at<cv::Vec3b>(i1, j1)[0]
					+ weight_x[2] * weight_y[1] * border.at<cv::Vec3b>(i1, j2)[0] + weight_x[3] * weight_y[1] * border.at<cv::Vec3b>(i1, j3)[0]
					+ weight_x[0] * weight_y[2] * border.at<cv::Vec3b>(i2, j0)[0] + weight_x[1] * weight_y[2] * border.at<cv::Vec3b>(i2, j1)[0]
					+ weight_x[2] * weight_y[2] * border.at<cv::Vec3b>(i2, j2)[0] + weight_x[3] * weight_y[2] * border.at<cv::Vec3b>(i2, j3)[0]
					+ weight_x[0] * weight_y[3] * border.at<cv::Vec3b>(i3, j0)[0] + weight_x[1] * weight_y[3] * border.at<cv::Vec3b>(i3, j1)[0]
					+ weight_x[2] * weight_y[3] * border.at<cv::Vec3b>(i3, j2)[0] + weight_x[3] * weight_y[3] * border.at<cv::Vec3b>(i3, j3)[0];
				pix[1] = weight_x[0] * weight_y[0] * border.at<cv::Vec3b>(i0, j0)[1] + weight_x[1] * weight_y[0] * border.at<cv::Vec3b>(i0, j1)[1]
					+ weight_x[2] * weight_y[0] * border.at<cv::Vec3b>(i0, j2)[1] + weight_x[3] * weight_y[0] * border.at<cv::Vec3b>(i0, j3)[1]
					+ weight_x[0] * weight_y[1] * border.at<cv::Vec3b>(i1, j0)[1] + weight_x[1] * weight_y[1] * border.at<cv::Vec3b>(i1, j1)[1]
					+ weight_x[2] * weight_y[1] * border.at<cv::Vec3b>(i1, j2)[1] + weight_x[3] * weight_y[1] * border.at<cv::Vec3b>(i1, j3)[1]
					+ weight_x[0] * weight_y[2] * border.at<cv::Vec3b>(i2, j0)[1] + weight_x[1] * weight_y[2] * border.at<cv::Vec3b>(i2, j1)[1]
					+ weight_x[2] * weight_y[2] * border.at<cv::Vec3b>(i2, j2)[1] + weight_x[3] * weight_y[2] * border.at<cv::Vec3b>(i2, j3)[1]
					+ weight_x[0] * weight_y[3] * border.at<cv::Vec3b>(i3, j0)[1] + weight_x[1] * weight_y[3] * border.at<cv::Vec3b>(i3, j1)[1]
					+ weight_x[2] * weight_y[3] * border.at<cv::Vec3b>(i3, j2)[1] + weight_x[3] * weight_y[3] * border.at<cv::Vec3b>(i3, j3)[1];
				pix[2] = weight_x[0] * weight_y[0] * border.at<cv::Vec3b>(i0, j0)[2] + weight_x[1] * weight_y[0] * border.at<cv::Vec3b>(i0, j1)[2]
					+ weight_x[2] * weight_y[0] * border.at<cv::Vec3b>(i0, j2)[2] + weight_x[3] * weight_y[0] * border.at<cv::Vec3b>(i0, j3)[2]
					+ weight_x[0] * weight_y[1] * border.at<cv::Vec3b>(i1, j0)[2] + weight_x[1] * weight_y[1] * border.at<cv::Vec3b>(i1, j1)[2]
					+ weight_x[2] * weight_y[1] * border.at<cv::Vec3b>(i1, j2)[2] + weight_x[3] * weight_y[1] * border.at<cv::Vec3b>(i1, j3)[2]
					+ weight_x[0] * weight_y[2] * border.at<cv::Vec3b>(i2, j0)[2] + weight_x[1] * weight_y[2] * border.at<cv::Vec3b>(i2, j1)[2]
					+ weight_x[2] * weight_y[2] * border.at<cv::Vec3b>(i2, j2)[2] + weight_x[3] * weight_y[2] * border.at<cv::Vec3b>(i2, j3)[2]
					+ weight_x[0] * weight_y[3] * border.at<cv::Vec3b>(i3, j0)[2] + weight_x[1] * weight_y[3] * border.at<cv::Vec3b>(i3, j1)[2]
					+ weight_x[2] * weight_y[3] * border.at<cv::Vec3b>(i3, j2)[2] + weight_x[3] * weight_y[3] * border.at<cv::Vec3b>(i3, j3)[2];

				for (int i = 0; i < src.channels(); ++i)
				{
					if (pix[i] < 0) pix = 0;
					if (pix[i] > 255)pix = 255;
				}
				dst.at<cv::Vec3b>(i - 1, j - 1) = static_cast<cv::Vec3b>(pix);
			}
		}	
	}	
}

3. 測(cè)試及結(jié)果

int main()
{
	cv::Mat src = cv::imread("C:\\Users\\Echo\\Pictures\\Saved Pictures\\bilateral.png");
	cv::Mat dst;
	bicubic(src, dst, 309/0.5, 338/0.5);
	cv::imshow("gray", dst);
	cv::imshow("src", src);
	cv::waitKey(0);
}

彩色圖像(放大兩倍)