Your suggestions are correct. I have tried to realize some of them before.
see my PR Optimization ximgproc::thinning by sturkmen72 · Pull Request #3801 · opencv/opencv_contrib (github.com)
You can use the following code to compare the performance of old and new code.
note: I was aware that using ROIs could make it even faster (see arnaud-ramey/voronoi: VoronoiSkeleton is a C++ class made for the fast computing of Voronoi diagrams of monochrome images. It contains different implementations of thinning algorithms. (github.com)). but I didn’t think about it because I think sometimes the slow result with less code is better.
but still open to development for new contributors
also see The THinning evaluation BEnchmark
#include <opencv2/core.hpp>
#include <opencv2/highgui.hpp>
#include <opencv2/imgproc.hpp>
#include <iostream>
using namespace std;
using namespace cv;
enum ThinningTypes{
THINNING_ZHANGSUEN = 0, // Thinning technique of Zhang-Suen
THINNING_GUOHALL = 1 // Thinning technique of Guo-Hall
};
namespace cv {
namespace current {
// look up table - there is one entry for each of the 2^8=256 possible
// combinations of 8 binary neighbors.
static uint8_t lut_zhang_iter0[] = {
1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1,
0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1,
0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1,
0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1,
1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1,
1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0,
1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1,
1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1};
static uint8_t lut_zhang_iter1[] = {
1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1,
0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1,
0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1,
0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1,
0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0,
1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1,
0, 1, 1, 1};
static uint8_t lut_guo_iter0[] = {
1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1,
0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0,
0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1,
0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1,
0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1,
0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1,
0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0,
1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1};
static uint8_t lut_guo_iter1[] = {
1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1,
1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0,
1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1,
1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0,
1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1,
1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1,
1, 1, 1, 1};
// Applies a thinning iteration to a binary image
static void thinningIteration(Mat img, int iter, int thinningType){
Mat marker = Mat::zeros(img.size(), CV_8UC1);
int rows = img.rows;
int cols = img.cols;
marker.col(0).setTo(1);
marker.col(cols - 1).setTo(1);
marker.row(0).setTo(1);
marker.row(rows - 1).setTo(1);
if(thinningType == THINNING_ZHANGSUEN){
marker.forEach<uchar>([=](uchar& value, const int postion[]) {
int i = postion[0];
int j = postion[1];
if (i == 0 || j == 0 || i == rows - 1 || j == cols - 1)
return;
auto ptr = img.ptr(i, j); // p1
// p9 p2 p3
// p8 p1 p4
// p7 p6 p5
uchar p2 = ptr[-cols];
uchar p3 = ptr[-cols + 1];
uchar p4 = ptr[1];
uchar p5 = ptr[cols + 1];
uchar p6 = ptr[cols];
uchar p7 = ptr[cols - 1];
uchar p8 = ptr[-1];
uchar p9 = ptr[-cols - 1];
int neighbors = p9 | (p2 << 1) | (p3 << 2) | (p4 << 3) | (p5 << 4) | (p6 << 5) | (p7 << 6) | (p8 << 7);
if (iter == 0)
value = lut_zhang_iter0[neighbors];
else
value = lut_zhang_iter1[neighbors];
//int A = (p2 == 0 && p3 == 1) + (p3 == 0 && p4 == 1) +
// (p4 == 0 && p5 == 1) + (p5 == 0 && p6 == 1) +
// (p6 == 0 && p7 == 1) + (p7 == 0 && p8 == 1) +
// (p8 == 0 && p9 == 1) + (p9 == 0 && p2 == 1);
//int B = p2 + p3 + p4 + p5 + p6 + p7 + p8 + p9;
//int m1 = iter == 0 ? (p2 * p4 * p6) : (p2 * p4 * p8);
//int m2 = iter == 0 ? (p4 * p6 * p8) : (p2 * p6 * p8);
//if (A == 1 && (B >= 2 && B <= 6) && m1 == 0 && m2 == 0) value = 0;
// else value = 1;
});
}
if(thinningType == THINNING_GUOHALL){
marker.forEach<uchar>([=](uchar& value, const int postion[]) {
int i = postion[0];
int j = postion[1];
if (i == 0 || j == 0 || i == rows - 1 || j == cols - 1)
return;
auto ptr = img.ptr(i, j); // p1
// p9 p2 p3
// p8 p1 p4
// p7 p6 p5
uchar p2 = ptr[-cols];
uchar p3 = ptr[-cols + 1];
uchar p4 = ptr[1];
uchar p5 = ptr[cols + 1];
uchar p6 = ptr[cols];
uchar p7 = ptr[cols - 1];
uchar p8 = ptr[-1];
uchar p9 = ptr[-cols - 1];
int neighbors = p9 | (p2 << 1) | (p3 << 2) | (p4 << 3) | (p5 << 4) | (p6 << 5) | (p7 << 6) | (p8 << 7);
if (iter == 0)
value = lut_guo_iter0[neighbors];
else
value = lut_guo_iter1[neighbors];
//int C = ((!p2) & (p3 | p4)) + ((!p4) & (p5 | p6)) +
// ((!p6) & (p7 | p8)) + ((!p8) & (p9 | p2));
//int N1 = (p9 | p2) + (p3 | p4) + (p5 | p6) + (p7 | p8);
//int N2 = (p2 | p3) + (p4 | p5) + (p6 | p7) + (p8 | p9);
//int N = N1 < N2 ? N1 : N2;
//int m = iter == 0 ? ((p6 | p7 | (!p9)) & p8) : ((p2 | p3 | (!p5)) & p4);
//if ((C == 1) && ((N >= 2) && ((N <= 3)) & (m == 0))) value = 0;
// else value = 1;
});
}
img &= marker;
}
// Apply the thinning procedure to a given image
void thinning(InputArray input, OutputArray output, int thinningType){
Mat processed = input.getMat().clone();
CV_CheckTypeEQ(processed.type(), CV_8UC1, "");
// Enforce the range of the input image to be in between 0 - 255
processed /= 255;
Mat prev = processed.clone();
Mat diff;
do {
thinningIteration(processed, 0, thinningType);
thinningIteration(processed, 1, thinningType);
absdiff(processed, prev, diff);
if (!hasNonZero(diff)) break;
processed.copyTo(prev);
}
while (true);
processed *= 255;
output.assign(processed);
}
} //namespace current
} //namespace cv
namespace cv {
namespace improved {
// look up table - there is one entry for each of the 2^8=256 possible
// combinations of 8 binary neighbors.
static uint8_t lut_zhang_iter0[] = {
1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1,
0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1,
0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1,
0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1,
1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1,
1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0,
1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1,
1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1};
static uint8_t lut_zhang_iter1[] = {
1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1,
0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1,
0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1,
0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1,
0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0,
1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1,
0, 1, 1, 1};
static uint8_t lut_guo_iter0[] = {
1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1,
0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0,
0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1,
0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1,
0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1,
0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1,
0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0,
1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1};
static uint8_t lut_guo_iter1[] = {
1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1,
1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0,
1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1,
1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0,
1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1,
1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1,
1, 1, 1, 1};
// Applies a thinning iteration to a binary image
static void thinningIteration(Mat &img, Mat &marker, const uint8_t* const lut, bool &changed) {
int rows = img.rows;
int cols = img.cols;
// Parallelized iteration over pixels excluding the boundary
parallel_for_(Range(1, rows - 1), [&](const Range& range) {
for (int i = range.start; i < range.end; i++) {
const uchar* imgRow = img.ptr(i);
uchar* markerRow = marker.ptr(i);
for (int j = 1; j < cols - 1; j++) {
if (imgRow[j]) {
uchar p2 = imgRow[j - cols] != 0;
uchar p3 = imgRow[j - cols + 1] != 0;
uchar p4 = imgRow[j + 1] != 0;
uchar p5 = imgRow[j + cols + 1] != 0;
uchar p6 = imgRow[j + cols] != 0;
uchar p7 = imgRow[j + cols - 1] != 0;
uchar p8 = imgRow[j - 1] != 0;
uchar p9 = imgRow[j - cols - 1] != 0;
int neighbors = p9 | (p2 << 1) | (p3 << 2) | (p4 << 3) | (p5 << 4) | (p6 << 5) | (p7 << 6) | (p8 << 7);
uchar lut_value = lut[neighbors];
if (lut_value == 0)
{
markerRow[j] = lut_value;
changed = true;
}
}
}
}
});
img &= marker;
}
// Apply the thinning procedure to a given image
void thinning(InputArray input, OutputArray output, int thinningType){
Mat input_ = input.getMat();
CV_Assert(!input_.empty());
CV_CheckTypeEQ(input_.type(), CV_8UC1, "");
Mat processed = input_ / 255;
Mat prev = processed.clone();
Mat marker;
Mat marker_inner = processed(Rect(1, 1, processed.cols - 2, processed.rows - 2));
copyMakeBorder(marker_inner, marker, 1, 1, 1, 1, BORDER_ISOLATED | BORDER_CONSTANT, Scalar(255));
const auto lutIter0 = (thinningType == THINNING_GUOHALL) ? lut_guo_iter0 : lut_zhang_iter0;
const auto lutIter1 = (thinningType == THINNING_GUOHALL) ? lut_guo_iter1 : lut_zhang_iter1;
do {
bool changed0 = false;
bool changed1 = false;
thinningIteration(processed, marker, lutIter0, changed0);
thinningIteration(processed, marker, lutIter1, changed1);
if (changed0 | changed1)
processed.copyTo(prev);
else
break;
} while (true);
output.assign(processed * 255);
}
} //namespace improved
} //namespace cv
void test1(Mat& src, Mat& dst, int thinningType = THINNING_ZHANGSUEN)
{
TickMeter tm;
tm.start();
current::thinning(src, dst, thinningType);
tm.stop();
cout << thinningType << " current (white pixels are mostly)" << endl << tm << endl;
tm.reset();
tm.start();
dst = ~src;
current::thinning(dst, dst, thinningType);
tm.stop();
cout << thinningType << " current (white pixels are in the minority)" << endl << tm << endl << endl;
}
void test2(Mat& src, Mat& dst, int thinningType = THINNING_ZHANGSUEN)
{
TickMeter tm;
tm.start();
improved::thinning(src, dst, thinningType);
tm.stop();
cout << thinningType << " improved (white pixels are mostly)" << endl << tm << endl;
tm.reset();
tm.start();
dst = ~src;
improved::thinning(dst, dst, thinningType);
tm.stop();
cout << " improved (white pixels are in the minority)" << endl << tm << endl << endl;
}
int main(int argc, const char** argv)
{
string filename = "08.png";
if (argc > 1)
filename = argv[1];
Mat src = imread(filename, IMREAD_GRAYSCALE);
Mat dst = src.clone();
for (int i = 0; i < 5; i++)
{
test1(src, dst);
test2(src, dst);
test1(src, dst, THINNING_GUOHALL);
test2(src, dst, THINNING_GUOHALL);
}
}