前言

近期在学习一些传统的图像处理算法，比如传统的图像插值算法等。传统的图像插值算法包括邻近插值法、双线性插值法和双三次插值法，其中邻近插值法和双线性插值法在网上都有很详细的介绍以及用c++编写的代码。但是，网上关于双三次插值法的原理介绍虽然很多，也有对应的代码，但是大多都不是很详细。因此基于自己对原理的理解，自己编写了图像双三次插值算法的c++ opencv代码，在这里记录一下。

一、图像双三次插值算法原理

首先是原理部分。图像双三次插值的原理，就是目标图像的每一个像素都是由原图上相对应点周围的4x4=16个像素经过加权之后再相加得到的。这里的加权用到的就是三次函数，这也是图像双三次插值算法名称的由来（个人猜测）。用到的三次函数如下图所示：

最关键的问题是，这个三次函数的输入和输出分别代表啥。简单来说输入就是原图对应点周围相对于这点的4x4大小区域的坐标值，大小在0~2之间，输出就是这些点横坐标或者纵坐标的权重。4个横坐标、4个纵坐标，对应相乘就是4x4大小的权重矩阵，然后使用此权重矩阵对原图相对应的区域进行相乘并相加就可以得到目标图点的像素。
这里有一篇文章可以帮助大家更好地理解，《实现图像缩放功能的Matlab插值算法研究与比较》，这篇文章的系数a=-1。

首先，u和v是什么呢？举一个例子，对于一幅100x100的灰度图像，要将其放大到500x500，那么其缩放因子sx=500/100=5，sy=500/100=5。现在目标图像是500x500，需要用原图的100x100个像素值来填满这500x500个空，根据src_x=i/sx和src_y=j/sy可以得到目标像素坐标（i，j）所对应的原图像素坐标（src_x, src_y)，这个src_x和src_y的小数部分就是上图中的u和v。
理解了u和v，就可以利用u和v来计算双三次插值算法的权重了。上面说了三次函数的输入是原图对应点周围相对于这点的4x4大小区域的坐标值，对于上面这幅图而言，横坐标有四个输入，分别是1+u，u，1-u，2-u；纵坐标也有四个输入，分别是1+v，v，1-v，2-v，根据三次函数算出权重之后两两相乘就是对应的4x4大小的权重矩阵。
知道了怎么求权重矩阵之后，就可以遍历整幅图像进行插值了。下面是基于自己对原理的理解编写的c++ opencv代码，代码没有做优化，但是能够让大家直观地理解图像双三次插值算法。

二、C++ OpenCV代码

1.计算权重矩阵

前面说了权重矩阵就是横坐标的4个输出和纵坐标的4个输出相乘，因此只需要求出横坐标和纵坐标相对应的8个输出值就行了。

代码如下：

std::vector<double> getWeight(double c, double a = 0.5)
{//c就是u和v，横坐标和纵坐标的输出计算方式一样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|<2std::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.遍历插值

代码如下：

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);//处理灰度图if (src.channels() == 1){for (int i = 1; i < dst_rows + 1; ++i){int src_y = (i + 0.5) / sy - 0.5; //做了几何中心对齐if (src_y < 0) src_y = 0;if (src_y > src.rows - 1) src_y = src.rows - 1;src_y += 1;//目标图像点坐标对应原图点坐标的4个纵坐标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;//目标图像点坐标对应原图点坐标的4个横坐标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);//目标点像素对应原图点像素周围4x4区域的加权计算（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);}}}//处理彩色图像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. 测试及结果

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);
}

彩色图像(放大两倍)

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