C++ OpenCV實現圖像雙三次插值算法詳解
前言
近期在學習一些傳統的圖像處理算法，比如傳統的圖像插值算法等。傳統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上都有很詳細的介紹以及用c++編寫的代碼。但是，網上關於雙三次插值法的原理介紹雖然很多，也有對應的代碼，但是大多都不是很詳細。因此基於自己對原理的理解，自己編寫瞭圖像雙三次插值算法的c++ opencv代碼，在這裡記錄一下。
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，就是目標圖像的每一個像素都是由原圖上相對應點周圍的4×4=16個像素經過加權之後再相加得到的。這裡的加權用到的就是三次函數，這也是圖像雙三次插值算法名稱的由來（個人猜測）。用到的三次函數如下圖所示：
最關鍵的問題是，這個三次函數的輸入和輸出分別代表啥。簡單來說輸入就是原圖對應點周圍相對於這點的4×4大小區域的坐標值，大小在0~2之間，輸出就是這些點橫坐標或者縱坐標的權重。4個橫坐標、4個縱坐標，對應相乘就是4×4大小的權重矩陣，然後使用此權重矩陣對原圖相對應的區域進行相乘並相加就可以得到目標圖點的像素。
下圖可以幫助大傢更好地理解
首先，u和v是什麼呢？舉一個例子，對於一幅100×100的灰度圖像，要將其放大到500×500，那麼其縮放因子sx=500/100=5，sy=500/100=5。現在目標圖像是500×500，需要用原圖的100×100個像素值來填滿這500×500個空，根據src_x=i/sx和src_y=j/sy可以得到目標像素坐標（i，j）所對應的原圖像素坐標（src_x, src_y)，這個src_x和src_y的小數部分就是上圖中的u和v。
理解瞭u和v，就可以利用u和v來計算雙三次插值算法的權重瞭。上面說瞭三次函數的輸入是原圖對應點周圍相對於這點的4×4大小區域的坐標值，對於上面這幅圖而言，橫坐標有四個輸入，分別是1+u，u，1-u，2-u；縱坐標也有四個輸入，分別是1+v，v，1-v，2-v，根據三次函數算出權重之後兩兩相乘就是對應的4×4大小的權重矩陣。
知道瞭怎麼求權重矩陣之後，就可以遍歷整幅圖像進行插值瞭。下面是基於自己對原理的理解編寫的c++ opencv代碼，代碼沒有做優化，但是能夠讓大傢直觀地理解圖像雙三次插值算法。
二、C++ OpenCV代碼
1.計算權重矩陣
前面說瞭權重矩陣就是橫坐標的4個輸出和縱坐標的4個輸出相乘，因此隻需要求出橫坐標和縱坐標相對應的8個輸出值就行瞭。
代碼如下：
std::vector<double> getWeight(double c, double a = 0.5)
{
//c就是u和v，橫坐標和縱坐標的輸出計算方式一樣
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; //做瞭幾何中心對齊
if (src_y < 0) src_y = 0;
if (src_y > src.rows - 1) src_y = src.rows - 1;
src_y += 1;
//目標圖像點坐標對應原圖點坐標的4個縱坐標
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;
//目標圖像點坐標對應原圖點坐標的4個橫坐標
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);
//目標點像素對應原圖點像素周圍4x4區域的加權計算（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. 測試及結果
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)
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