{"id":997,"date":"2014-03-27T15:55:06","date_gmt":"2014-03-27T19:55:06","guid":{"rendered":"https:\/\/blogs.mathworks.com\/steve\/?p=997"},"modified":"2019-11-01T11:01:34","modified_gmt":"2019-11-01T15:01:34","slug":"comparing-the-geometries-of-bwboundaries-and-poly2mask","status":"publish","type":"post","link":"https:\/\/blogs.mathworks.com\/steve\/2014\/03\/27\/comparing-the-geometries-of-bwboundaries-and-poly2mask\/","title":{"rendered":"Comparing the geometries of bwboundaries and poly2mask"},"content":{"rendered":"\r\n

MATLAB user Meshooo<\/a> asked a question on MATLAB Answers<\/a> about a problem with the createMask<\/tt> function associated with impoly<\/tt>. Meshooo observed a discrepancy between the output of bwboundaries<\/tt> and the mask created by createMask<\/tt>.<\/p>

I want to describe the issue in more general terms here as a conflict between the geometry of the bwboundaries<\/tt> function and the geometry of the poly2mask<\/tt> function (which is used by createMask<\/tt>).<\/p>

Here's a simple example that illustrates the discrepancy. Start by creating a small binary image.<\/p>

BW = [ ...<\/span>\r\n    0 0 0 0 0 0 0 0 0\r\n    0 0 0 0 0 0 0 0 0\r\n    0 0 0 0 1 1 1 0 0\r\n    0 0 1 1 1 1 1 0 0\r\n    0 0 1 1 1 1 1 0 0\r\n    0 0 1 1 1 1 1 0 0\r\n    0 0 1 1 1 1 1 0 0\r\n    0 0 1 1 0 0 0 0 0\r\n    0 0 0 0 0 0 0 0 0\r\n    0 0 0 0 0 0 0 0 0 ];\r\n<\/pre>
Call bwboundaries<\/tt>, which traces the perimeter pixels of all the objects (and holes) in the image.<\/p>
B = bwboundaries(BW);\r\n<\/pre>
There's only one boundary in this case. Extract and plot it.<\/p>
B = B{1};\r\nBx = B(:,2);\r\nBy = B(:,1);\r\nplot(Bx,By)\r\naxis ij<\/span>\r\naxis equal<\/span>\r\naxis([.5 9.5 .5 10.5])\r\n<\/pre>\"\" 
If we now pass Bx<\/tt> and By<\/tt> to poly2mask<\/tt>, we don't get exactly the same binary mask image that we started with.<\/p>
BW2 = poly2mask(Bx,By,10,9)\r\n<\/pre>
\r\nBW2 =\r\n\r\n     0     0     0     0     0     0     0     0     0\r\n     0     0     0     0     0     0     0     0     0\r\n     0     0     0     0     0     0     0     0     0\r\n     0     0     0     0     1     1     1     0     0\r\n     0     0     0     1     1     1     1     0     0\r\n     0     0     0     1     1     1     1     0     0\r\n     0     0     0     1     1     1     1     0     0\r\n     0     0     0     1     0     0     0     0     0\r\n     0     0     0     0     0     0     0     0     0\r\n     0     0     0     0     0     0     0     0     0\r\n\r\n<\/pre>
To understand the reason for the discrepancy, it helps to understand that bwboundaries<\/tt> treats the foreground pixels of the input image as points in space. Here's a plot to illustrate:<\/p>
plot(Bx,By)\r\naxis ij<\/span>\r\naxis equal<\/span>\r\naxis([.5 9.5 .5 10.5])\r\nhold on<\/span>\r\n[yy,xx] = find(BW);\r\nplot(xx,yy,'*'<\/span>)\r\nhold off<\/span>\r\nlegend('Boundary polygon'<\/span>,'Foreground pixels'<\/span>)\r\n<\/pre>\"\" 
Now let's do a plot that shows the pixels as squares of unit area. I'll include some code to overlay the pixel edges as gray lines.<\/p>
imshow(BW,'InitialMagnification'<\/span>,'fit'<\/span>)\r\nhold on<\/span>\r\n\r\nx = [.5 9.5];\r\nfor<\/span> k = .5:10.5\r\n    y = [k k];\r\n    plot(x,y,'Color'<\/span>,[.7 .7 .7]);\r\nend<\/span>\r\n\r\ny = [.5 10.5];\r\nfor<\/span> k = .5:9.5\r\n    x = [k k];\r\n    plot(x,y,'Color'<\/span>,[.7 .7 .7]);\r\nend<\/span>\r\n\r\nplot(Bx,By,'r'<\/span>)\r\n\r\nplot(xx,yy,'*'<\/span>)\r\nhold off<\/span>\r\n<\/pre>\"\" 
Now you can see that the polygon produced by bwboundaries<\/tt> does not completely contain the pixels along the border of the object. In fact, most of those pixels are only half inside the polygon (or less).<\/p>
That's the clue needed to explain the discrepancy with poly2mask<\/tt>. That function treats images pixels not as points, but as squares having unit area. Its algorithm is carefully designed to treat partially covered pixels in a geometrically consistent way. I wrote several blog posts (POLY2MASK and ROIPOLY Part 1<\/a>, Part 2<\/a>, and Part 3<\/a>) about this back in 2006. Here's a diagram from Part 3 that illustrates a bit of the algorithm for handling partially covered pixels.<\/p>
\"\" <\/p>
It turns out that there is a way to get boundary polygons from bwboundaries<\/tt> that are consistent with the poly2mask<\/tt> geometry. The idea is to upsample the binary image so that the polygon produced by bwboundaries<\/tt> is outside the pixel centers instead of running directly through the centers.<\/p>
BW3 = imresize(BW,3,'nearest'<\/span>);\r\nB3 = bwboundaries(BW3);\r\nB3 = B3{1};\r\n<\/pre>
Now shift and scale the polygon coordinates back into the coordinate system of the original image.<\/p>
Bx = (B3(:,2) + 1)\/3;\r\nBy = (B3(:,1) + 1)\/3;\r\n\r\nimshow(BW,'InitialMagnification'<\/span>,'fit'<\/span>)\r\nhold on<\/span>\r\n\r\nx = [.5 9.5];\r\nfor<\/span> k = .5:10.5\r\n    y = [k k];\r\n    plot(x,y,'Color'<\/span>,[.7 .7 .7]);\r\nend<\/span>\r\n\r\ny = [.5 10.5];\r\nfor<\/span> k = .5:9.5\r\n    x = [k k];\r\n    plot(x,y,'Color'<\/span>,[.7 .7 .7]);\r\nend<\/span>\r\n\r\nplot(Bx,By,'r'<\/span>)\r\n\r\nplot(xx,yy,'*'<\/span>)\r\nhold off<\/span>\r\n<\/pre>\"\" 
You can see that the pixel centers are now clearly inside the modified polygon (Bx,By). That means that when we try poly2mask<\/tt> again, we'll get the same mask as the original image.<\/p>
BWout = poly2mask(Bx,By,10,9)\r\n<\/pre>
\r\nBWout =\r\n\r\n     0     0     0     0     0     0     0     0     0\r\n     0     0     0     0     0     0     0     0     0\r\n     0     0     0     0     1     1     1     0     0\r\n     0     0     1     1     1     1     1     0     0\r\n     0     0     1     1     1     1     1     0     0\r\n     0     0     1     1     1     1     1     0     0\r\n     0     0     1     1     1     1     1     0     0\r\n     0     0     1     1     0     0     0     0     0\r\n     0     0     0     0     0     0     0     0     0\r\n     0     0     0     0     0     0     0     0     0\r\n\r\n<\/pre>
isequal(BW,BWout)\r\n<\/pre>
\r\nans =\r\n\r\n     1\r\n\r\n<\/pre>
To everyone in the Northern Hemisphere: Happy Spring!<\/p>