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My idea is that all vertical lines should not cross a diagonal one. Therefore I created a method, which returns all points of a diagonal line:

vector<Point> diagonalLines(Mat src) {
   vector<Point> hitPoint;

   Scalar mu, sigma;
   meanStdDev(src, mu, sigma);

   Canny(src, dst, mu.val[0] - sigma.val[0], mu.val[0] + sigma.val[0], 3, false);

   cvtColor(dst, cdst, CV_GRAY2BGR);
   HoughLinesP(dst, lines, 1, CV_PI / 180, 100, 50, 10);

   sort(lines.begin(), lines.end(), vec4iSortByX());

   for (size_t i = 1; i < lines.size(); i++) {
       Vec4i current = lines[i];
       Point pt1 = Point(current[0], current[1]);
       Point pt2 = Point(current[2], current[3]);

       double angle = atan2(pt2.y - pt1.y, pt2.x - pt1.x) * 180.0 / CV_PI;
       if (angle != -90 && angle != 90) {
           hitPoint.push_back(pt1);
           hitPoint.push_back(pt2);
       }
   }
   return hitPoint;
}

After that there's a method to detect all essential vertial lines:

void verticalLines(Mat src) {
   double maxLineGap = 200.0;
   vector<Vec4i> elemLinesCur, elemLinesPrev;                                           

   Scalar mu, sigma;
   meanStdDev(src, mu, sigma);

   Canny(src, dst, mu.val[0] - sigma.val[0], mu.val[0] + sigma.val[0], 3, false);

   cvtColor(dst, cdst, CV_GRAY2BGR);
   HoughLinesP(dst, lines, 1, CV_PI / 2, 100, 50, maxLineGap);                              

   sort(lines.begin(), lines.end(), vec4iSortByX());                                    
   Vec4i current, previous;
   Point pt1, pt2, ppt1, ppt2;

   for (size_t i = 1; i < lines.size(); i++) {

       current = lines[i];
       pt1 = Point(current[0], current[1]);
       pt2 = Point(current[2], current[3]);

       for (int k = 0; k < maxLineGap; k++) {
           for (auto d : diagonalLines(src)) 
               if (pt1 == d) 
                   maxLineGap--;
       }
       //do some stuff
}

Yes, they similiar. So in for (int k = 0; k < maxLineGap; k++) { //... I'm trying to say if any point of a vertical line hits a point of diagonal line, make the maxLineGap shorter, so it goes exactly to the diagonal line and not above of it. It should be something like a intersection and between the intersection points the line should be drawen.

As you could image, this approach doesn't work and the computing time is long. What am I doing wrong?