How to find vanishing points from vanishing lines?

Question

I'm trying to find vanishing points of vanishing lines to estimate a depth map for a 2D image.

First, I detected the vanishing lines of the 2D image using hough transform. Here is my code in Matlab:

Img =imread('landscape.bmp'); %read the 2D image
%Convert the image to Grayscale
I=rgb2gray(Img);

%Edge Detection
Ie=edge(I,'sobel');

%Hough Transform
[H,theta,rho] = hough(Ie);

% Finding the Hough peaks (number of peaks is set to 5)
P = houghpeaks(H,5,'threshold',ceil(0.2*max(H(:))));
x = theta(P(:,2));
y = rho(P(:,1));

%Vanishing lines
lines = houghlines(I,theta,rho,P,'FillGap',170,'MinLength',350);
[rows, columns] = size(Ie);
figure, imshow(~Ie)
hold on
xy_1 = zeros([2,2]);
for k = 1:length(lines)
   xy = [lines(k).point1; lines(k).point2];
   % Get the equation of the line
   x1 = xy(1,1);
   y1 = xy(1,2);
   x2 = xy(2,1);
   y2 = xy(2,2);
   slope = (y2-y1)/(x2-x1);
   xLeft = 1; % x is on the left edge
   yLeft = slope * (xLeft - x1) + y1;
   xRight = columns; % x is on the reight edge.
   yRight = slope * (xRight - x1) + y1;
   plot([xLeft, xRight], [yLeft, yRight], 'LineWidth',1,'Color','blue');

   %intersection of two lines (the current line and the previous one)
   slopee = @(line) (line(2,2) - line(1,2))/(line(2,1) - line(1,1));
   m1 = slopee(xy_1);
   m2 = slopee(xy);
   intercept = @(line,m) line(1,2) - m*line(1,1);
   b1 = intercept(xy_1,m1);
   b2 = intercept(xy,m2);
   xintersect = (b2-b1)/(m1-m2);
   yintersect = m1*xintersect + b1;
   plot(xintersect,yintersect,'m*','markersize',8, 'Color', 'red')
   xy_1 = xy;

   % Plot original points on the lines .
   plot(xy(1,1),xy(1,2),'x','markersize',8,'Color','yellow'); 
   plot(xy(2,1),xy(2,2),'x','markersize',8,'Color','green');    
end

Now I need to find the vanishing point to be able to estimate the depth map. The vanishing point is chosen as the intersection point with the greatest number of intersections around it.

My question, in other words, is how can I find the intersection of a number of lines (vanishing lines) in Matlab? I guess one way to do it is to find the point whose sum of squared distances from all lines is minimal, but not sure how to do that in Matlab?

Any help would be appreciated.

Edit: I tried to find the intersection of the lines, but I could only find the intersection of each a line and the line after it. I don't know how to find the intersection of all the lines?

Here is an example of a picture I am using: https://www.dropbox.com/s/mbdt6v60ug1nymb/landscape.bmp?dl=0

I am posting a link because I don't have enough reputations to post an image.

Answer 1

A simplistic approach:

You should be able to create an array with all the intersection points between the lines.

Pseudo code:

for i = 1:length(lines)-1
  for j = i+1:length(lines)
     //add intersection of lines i and j 

If you have all the intersections, you could simply take the average.

OR, take the approach written up here:

https://math.stackexchange.com/questions/61719/finding-the-intersection-point-of-many-lines-in-3d-point-closest-to-all-lines

3d can be simplified to 2d :)