How to get a line segment rectangle?

Question

I want to find a weight matrix for the algebraic reconstruction method. To do this, I need to find the intersection of the line with the grid. I can find the direct intersection of the line with the line, but I need to store the grid number of the crossed line segment, depending on the number. Therefore, suppose that if the first square of the grid does not intersect with the grid, then put zero in the first element of the weight matrix.

Here is the code I tried to cross the line:

ak = 3:6 aka = 3:6 x = zeros(size(aka)) y = zeros(size(ak)) for k = 1:length(ak) line([ak(1) ak(end)], [aka(k) aka(k)],'color','r') end % Vertical grid for k = 1:length(aka) line([ak(k) ak(k)], [aka(1) aka(end)],'color','r') end hold on; X =[0 15.5] Y = [2.5 8.5] m = (Y(2)-Y(1))/(X(2)-X(1)) ; c = 2.5 ; plot(X,Y) axis([0 10 0 10]) axis square % plotting y intercept for i = 1:4 y(i) = m * ak(i) + c if y(i)<2 || y(i)>6 y(i) = 0 end end % plotting x intercept for i = 1:4 x(i) = (y(i) - c)/m if x(i)<2 || x(i)>6 x(i) = 0 end end z = [x' y'] 

I have a line defined by the parameters m, h , where y = m*x + h This line passes through the grid (i.e. in pixels).

For each square (a, b) grid (that is, the square [a, a+1]x[b, b+1] ) I want to determine whether this line intersects this square or not, and if so, what is the length of the segment in squared so that I can build the matrix of weights, which is necessary for the method of algebraic reconstruction.

Answer 1

Here's a good way to intersect a line with a grid of rectangles and get the lengths of each of the intersection segments: I used the line intersection from the pseudocode in the third answer from this link

 % create some line form the equation y=mx+h m = 0.5; h = 0.2; x = -2:0.01:2; y = m*x+h; % create a grid on the range [-1,1] [X,Y] = meshgrid(linspace(-1,1,10),linspace(-1,1,10)); % create a quad mesh on this range fvc = surf2patch(X,Y,zeros(size(X))); % extract topology v = fvc.vertices(:,[1,2]); f = fvc.faces; % plot the grid and the line patch(fvc,'EdgeColor','g','FaceColor','w'); hold on; plot(x,y); % use line line intersection from the link DC = [f(:,[1,2]);f(:,[2,3]);f(:,[3,4]);f(:,[4,1])]; D = v(DC(:,1),:); C = v(DC(:,2),:); A = repmat([x(1),y(1)],size(DC,1),1); B = repmat([x(end),y(end)],size(DC,1),1); E = AB; F = DC; P = [-E(:,2),E(:,1)]; h = dot(AC,P,2)./dot(F,P,2); % calc intersections idx = (0<=h & h<=1); intersections = C(idx,:)+F(idx,:).*repmat(h(idx),1,2); intersections = uniquetol(intersections,1e-8,'ByRows',true); % sort by x axis values [~,ii] = sort(intersections(:,1)); intersections = intersections(ii,:); scatter(intersections(:,1),intersections(:,2)); % get segments lengths directions = diff(intersections); lengths = sqrt(sum(directions.^2,2)); directions = directions./repmat(sqrt(sum(directions.^2,2)),1,2); directions = directions.*repmat(lengths,1,2); quiver(intersections(1:end-1,1),intersections(1:end-1,2),directions(:,1),directions(:,2),'AutoScale','off','Color','k'); 

This is the result (the lengths of the arrows in the image are the lengths of the segment)