box filter的三种实现（c++）

box filter简单解释

box filter的作用很简单，即对局部区域求平均，并把值赋给某个点，一般我们赋给区域中心。用公式表达如下：
$$result(row,col)=\frac{1}{n^2}\sum _i^{patch}\sum _j^{patch}image(i,j)$$
其中 $patch$是以$(row,col)$为中心的一块区域。

为了跟后面的公式及程序对应，我们做如下定义：

• $r$：patch的半径。半径在宽高方向可以不相等，但是本文目的不在于对半径的处理，所以简单起见设为相等。
• $n$：patch的长度，等于$(2\ast r+1）$。
• $(rows,cols)$：图像的尺寸，行数和列数。
• $(row,col)$：对完整图像的索引。
• $(i,j)$：对图像patch的索引
• $k$：对通道的索引。

1. 暴力实现——四循环

外层两个循环是关于完整图像$(row,col)$的循环，内层两个循环是关于图像patch$(i,j)$的循环。

注意：如果图像是多通道的话实际上还有一个通常维度的循环，但是通道数不是本文优化的重心，所以本文不再赘述这个因素，后文也不再提，并且在计算量的估计中也会把这个因素省略掉。

这个实现比较简单，需要做的计算有：

• $rows\ast cols\ast n\ast n$ 次加法，内层循环的计算量$o(n^2)$，非常大。
• $rows\ast cols$ 次除法：除法为了求平均

2. 行列分离

patch的平均可以进行行列分离，也就是先对行方向做平均，并缓存结果，再对缓存的结果做列方向的平均。以公式的形式表达如下：
$$\frac{1}{n^2}\sum _i\sum _{j}image(i,j)=\frac{1}{n}\sum _i\frac{1}{n}\sum _{j}image(i,j)$$

举个例子展开写会容易理解，比如3*3的patch，共9个数：
$$\begin{array}{rl}mean(a_{ij})& =\frac{1}{9}(a_{00}+a_{01}+a_{02}+a_{10}+a_{11}+a_{12}+a_{20}+a_{21}+a_{22})\\ & =\frac{1}{3}(\frac{1}{3}(a_{00}+a_{01}+a_{02})+\frac{1}{3}(a_{10}+a_{11}+a_{12})+\frac{1}{3}(a_{20}+a_{21}+a_{22}))\\ & =\frac{1}{3}(\frac{1}{3}\sum _j{a}_{0j}+\frac{1}{3}\sum _j{a}_{1j}+\frac{1}{3}\sum _j{a}_{2j})\\ & =\frac{1}{3}\sum _i(\frac{1}{3}\sum _j{a}_{ij})\end{array}$$

这种方式的计算量：

• $2\ast rows\ast cols\ast n$ 次加法，相对于暴力版本，内层循环降低了一个数量级的算力，变成$o(n)$了
• $2\ast rows\ast cols$ 次除法

3. 行列分离优化版

第二种实现可以对求和做进一步优化。在单个维度做求和时，可以对当前一维patch的和做一个缓存，当中心点移动后，减去弹出像素的值，加上新增像素的值，这样就避免了重复性求和操作。

这种方案需要对patch的和做一个初始化和缓存，该方案的计算量为：

• $2\ast rows\ast cols$ 次减法，$2\ast rows\ast cols$ 次加法，内层循环的计算变为$o(1)$了，进一步降低了一个数量级算力。
• $2\ast rows\ast cols$ 次除法

代码

上面做计算量估计的时候没有考虑边界条件，在具体代码实现的时候需要仔细处理边界，防止数组访问越界。

代码同时跟opencv做了个效果和性能的对比，第三种方式虽然仍然比opencv慢，但性能基本处于同一量级了，opencv可能还做了一些其他跟算法无关的优化，比如指令集、并行化之类的。

注意：下面为了方便比较，opencv boxFilter的边界处理参数选择BORDER_CONSTANT。即使是边界处patch不满覆盖的情况下，opencv仍然除以$n^2$，也就是说除以的数字有点大了，所以边界会逐渐发黑，特别是kernel_size（对应于radius）比较大时候视觉效果更明显。

#include 
#include 
#include 
#include 
#include 


using namespace std;
using namespace cv;


Mat BoxFilter_1(const Mat& image, int radius);
Mat BoxFilter_2(const Mat& image, int radius);
Mat BoxFilter_3(const Mat& image, int radius);


int main()
{
    clock_t time_beg;
    clock_t time_end;

    Mat image = imread("lena_std.bmp", IMREAD_UNCHANGED);
    image.convertTo(image, CV_32FC3);
    image /= 255.0f;

    int radius = 9;
    int ksize = radius * 2 + 1;

    Mat image_box_filter_cv;
    time_beg = clock();
    boxFilter(image, image_box_filter_cv, -1, Size(ksize, ksize), Point(-1, -1), true, BORDER_CONSTANT);
    time_end = clock();
    cout << "box-filter-cv time cost: " << time_end - time_beg << endl;

    Mat image_box_filter_1 = BoxFilter_1(image, radius);
    Mat image_box_filter_2 = BoxFilter_2(image, radius);
    Mat image_box_filter_3 = BoxFilter_3(image, radius);

    

    namedWindow("original_image", 1);
    imshow("original_image", image);
    namedWindow("cv_box_filter", 1);
    imshow("cv_box_filter", image_box_filter_cv);
    namedWindow("box_filter-1", 1);
    imshow("box_filter-1", image_box_filter_1);
    namedWindow("box_filter-2", 1);
    imshow("box_filter-2", image_box_filter_2);
    namedWindow("box_filter-3", 1);
    imshow("box_filter-3", image_box_filter_3);

    Mat diff;
    cv::absdiff(image_box_filter_2, image_box_filter_3, diff);
    namedWindow("diff", 1);
    imshow("diff", 50 * diff);

    waitKey(0);
    destroyAllWindows();

    return 0;
}


Mat BoxFilter_1(const Mat& image, int radius)
{
    int cols = image.cols;
    int rows = image.rows;
    int channels = image.channels();
    int row_bound = rows - 1;
    int col_bound = cols - 1;
    Mat result(rows, cols, CV_32FC3);

    clock_t time_beg;
    clock_t time_end;
    time_beg = clock();

    for (int row = 0; row < rows; ++row) {
        int row_beg = max(row - radius, 0);
        int row_end = min(row + radius, row_bound);
        for (int col = 0; col < cols; ++col) {
            int col_beg = max(col - radius, 0);
            int col_end = min(col + radius, col_bound);

            vector<float> sums(channels, 0.0f);
            int count = 0;
            for (int i = row_beg; i <= row_end; ++i) {
                for (int j = col_beg; j <= col_end; ++j) {
                    count++;
                    for (int k = 0; k < channels; ++k) {
                        sums[k] += image.at<Vec3f>(i, j)[k];
                    }
                }
            }

            for (int k = 0; k < channels; ++k) {
                result.at<Vec3f>(row, col)[k] = sums[k] / static_cast<float>(count);

                // opencv BORDER_CONSTANT:
                /*float COUNT = (float)(2 * radius + 1) * (2 * radius + 1);
                result.at(row, col)[k] = sums[k] / COUNT;*/
            }
        }
    }
    result = cv::max(cv::min(result, 1.0), 0.0);
    time_end = clock();
    cout << "box-filter-1 time cost: " << time_end - time_beg << endl;

    return result;
}


Mat BoxFilter_2(const Mat& image, int radius)
{
    int cols = image.cols;
    int rows = image.rows;
    int channels = image.channels();
    int row_bound = rows - 1;
    int col_bound = cols - 1;
    Mat result(rows, cols, CV_32FC3);

    clock_t time_beg;
    clock_t time_end;
    time_beg = clock();

    // compute mean for row-wise
    Mat row_result(rows, cols, CV_32FC3);
    for (int row = 0; row < rows; ++row) {
        for (int col = 0; col < cols; ++col) {
            int col_beg = max(col - radius, 0);
            int col_end = min(col + radius, col_bound);

            vector<float> sums(channels, 0.0f);
            int count = 0;
            for (int j = col_beg; j <= col_end; ++j) {
                count++;
                for (int k = 0; k < channels; ++k) {
                    sums[k] += image.at<Vec3f>(row, j)[k];
                }
            }
            for (int k = 0; k < channels; ++k) {
                row_result.at<Vec3f>(row, col)[k] = sums[k] / static_cast<float>(count);
            }
        }
    }

    // compute mean for column-wise
    for (int col = 0; col < cols; ++col) {
        for (int row = 0; row < rows; ++row) {
            int row_beg = max(row - radius, 0);
            int row_end = min(row + radius, row_bound);

            vector<float> sums(channels, 0.0f);
            int count = 0;
            for (int i = row_beg; i <= row_end; ++i) {
                count++;
                for (int k = 0; k < channels; ++k) {
                    sums[k] += row_result.at<Vec3f>(i, col)[k];
                }
            }
            for (int k = 0; k < channels; ++k) {
                result.at<Vec3f>(row, col)[k] = sums[k] / static_cast<float>(count);
            }
        }
    }
    result = cv::max(cv::min(result, 1.0), 0.0);
    time_end = clock();
    cout << "box-filter-2 time cost: " << time_end - time_beg << endl;

    return result;
}


Mat BoxFilter_3(const Mat& image, int radius)
{
    int cols = image.cols;
    int rows = image.rows;
    int channels = image.channels();
    Mat result(rows, cols, CV_32FC3);

    clock_t time_beg;
    clock_t time_end;
    time_beg = clock();

    // compute mean for row-wise
    Mat row_result(rows, cols, CV_32FC3);
    for (int row = 0; row < rows; ++row) {
        // initialize sums for row
        vector<float> sums(channels, 0.0f);
        int count = 0;
        for (int col = 0; col < radius; ++col) {
            if (col < cols) {
                count++;
                for (int k = 0; k < channels; ++k) {
                    sums[k] += image.at<Vec3f>(row, col)[k];
                }
            }
        }
        // process row
        for (int col = 0; col < cols; ++col) {
            int left = col - radius - 1;
            int right = col + radius;
            if (left >= 0) {
                count--;
                for (int k = 0; k < channels; ++k) {
                    sums[k] -= image.at<Vec3f>(row, left)[k];
                }
            }
            if (right < cols) {
                count++;
                for (int k = 0; k < channels; ++k) {
                    sums[k] += image.at<Vec3f>(row, right)[k];
                }
            }
            for (int k = 0; k < channels; ++k) {
                row_result.at<Vec3f>(row, col)[k] = sums[k] / static_cast<float>(count);
            }
        }
    }

    // compute mean for column-wise
    for (int col = 0; col < cols; ++col) {
        // initialize sums for column
        vector<float> sums(channels, 0.0f);
        int count = 0;
        for (int row = 0; row < radius; ++row) {
            if (row < rows) {
                count++;
                for (int k = 0; k < channels; ++k) {
                    sums[k] += row_result.at<Vec3f>(row, col)[k];
                }
            }
        }
        // process column
        for (int row = 0; row < rows; ++row) {
            int up = row - radius - 1;
            int down = row + radius;
            if (up >= 0) {
                count--;
                for (int k = 0; k < channels; ++k) {
                    sums[k] -= row_result.at<Vec3f>(up, col)[k];
                }
            }
            if (down < rows) {
                count++;
                for (int k = 0; k < channels; ++k) {
                    sums[k] += row_result.at<Vec3f>(down, col)[k];
                }
            }
            for (int k = 0; k < channels; ++k) {
                result.at<Vec3f>(row, col)[k] = sums[k] / static_cast<float>(count);
            }
        }
    }
    result = cv::max(cv::min(result, 1.0), 0.0);
    time_end = clock();
    cout << "box-filter-3 time cost: " << time_end - time_beg << endl;

    return result;
}