How can I detect the maximum-sized rectangle that I can draw onto the mask?

Question

I\'m making an image processing project and I have stuck in one the project\'s steps. Here is the situation;

This is my mask:

Answer 1

Here is a correct answer. You must use dynamic programming! Other methods of direct calculation (like cutting 1 pixel from each edge) might produce sub-optimal results. My method guarantees that it selects the largest possible rectangle that fits in the mask. I assume that the mask has 1 big convex white blob of any shape with black background around it.

I wrote 2 methods. findRect() which finds the best possible square (starting on x,y with length l). The second method LargestInscribedImage() is an example of how to find any rectangle (of any aspect ratio). The trick is to resize the mask image, find a square and resize it back. In my example the method finds the larges rectangle that can be fit in the mask having the same aspect ration as the mask image. For example if the mask image is of size 100x200 pixels than the algorithm will find the largest rectangle having aspect ratio 1:2.

% ----------------------------------------------------------
function LargestInscribedImage()
% ----------------------------------------------------------
    close all
    im = double(imread('aa.bmp')); % Balck and white image of your mask
    im = im(:,:,1);  % If it is colored RGB take only one of the channels
    b = imresize(im,[size(im,1) size(im,1)]); Make the mask square by resizing it by its aspect ratio.
    SC = 1;  % Put 2..4 to scale down the image an speed up the algorithm

    [x1,y1,l1] = findRect(b,SC);          % Lunch the dyn prog algorithm
    [x2,y2,l2] = findRect(rot90(b),SC);   % rotate the image by 90deg and solve
    % Rotate back: x2,y2 according to rot90
    tmp = x2;
    x2 = size(im,1)/SC-y2-l2;
    y2 = tmp;

% Select the best solution of the above (for the original image and for the rotated by 90degrees
        if (l1>=l2)
           corn = sqCorn(x1,y1,l1);
        else
           corn = sqCorn(x2,y2,l2);
        end

        b = imresize(b,1/SC);
    figure;imshow(b>0); hold on;
    plot(corn(1,:),corn(2,:),'O')

    corn = corn*SC;
    corn(1,:) = corn(1,:)*size(im,2)/size(im,1);
    figure;imshow(im); hold on;
    plot(corn(1,:),corn(2,:),'O')
end

function corn = sqCorn(x,y,l)
     corn = [x,y;x,y+l;x+l,y;x+l,y+l]';
end
% ----------------------------------------------------------
function [x,y,l] = findRect(b,SC)
b = imresize(b,1/SC);
res = zeros(size(b,1),size(b,2),3);
% initialize first col
for i = 1:1:size(b,1)
    if (b(i,1) > 0)
       res(i,1,:) = [i,1,0];
    end
end
% initialize first row
for i = 1:1:size(b,2)
    if (b(1,i) > 0)
       res(1,i,:) = [1,i,0];
    end
end

% DynProg
for i = 2:1:size(b,1)
for j = 2:1:size(b,2)
    isWhite = b(i,j) > 0;
    if (~isWhite)
       res(i,j,:)=res(i-1,j-1,:); % copy
    else
        if (b(i-1,j-1)>0)  % continuous line
           lineBeg    = [res(i-1,j-1,1),res(i-1,j-1,2)]; 
           lineLenght = res(i-1,j-1,3); 
           if ((b(lineBeg(1),j)>0)&&(b(i,lineBeg(2))>0))  % if second diag is good
              res(i,j,:) = [lineBeg,lineLenght+1];
           else
              res(i,j,:)=res(i-1,j-1,:); % copy since line has ended
           end
        else
           res(i,j,:) = [i,j,0];         % Line start
        end
    end

end
end

% check last col
[maxValCol,WhereCol] = max(res(:,end,3));
% check last row
[maxValRow,WhereRow] = max(res(end,:,3));

% Find max
x= 0; y = 0; l = 0;
if (maxValCol>maxValRow)
   y = res(WhereCol,end,1);
   x = res(WhereCol,end,2);
   l = maxValCol;
else
   y = res(end,WhereRow,1);
   x = res(end,WhereRow,2);
   l = maxValRow;
end

    corn = [x,y;x,y+l;x+l,y;x+l,y+l]';
%    figure;imshow(b>0); hold on;
 %   plot(corn(1,:),corn(2,:),'O')
return;
end