{"id":392,"date":"2011-09-30T07:00:12","date_gmt":"2011-09-30T11:00:12","guid":{"rendered":"https:\/\/blogs.mathworks.com\/steve\/2011\/09\/30\/binary-image-convex-hull\/"},"modified":"2019-10-29T16:56:06","modified_gmt":"2019-10-29T20:56:06","slug":"binary-image-convex-hull","status":"publish","type":"post","link":"https:\/\/blogs.mathworks.com\/steve\/2011\/09\/30\/binary-image-convex-hull\/","title":{"rendered":"Binary image convex hull"},"content":{"rendered":"
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I've been intending to mention a new function bwconvhull<\/tt><\/a> that was introduced in the Image Processing Toolbox last spring in the R2011a release. Now that R2011b is out, I figure I\r\n better go ahead and do it!\r\n <\/p>\r\n

bwconvhull<\/tt> computes the \"convex hull of a binary image.\" Now I have to admit that this terminology is a little loose, so I'd better\r\n clarify. The convex hull of a set of 2-D points is the smallest convex polygon that contains the entire set. Here's an example\r\n from the MATLAB documentation for convhull<\/tt><\/a>:\r\n <\/p>

xx = -1:.05:1; yy = abs(sqrt(xx));\r\n[x,y] = pol2cart(xx,yy);\r\nk = convhull(x,y);\r\nplot(x(k),y(k),'r-'<\/span>,x,y,'b+'<\/span>)<\/pre> 
The polygon in red is the convex hull of the set of points shown in blue. So convhull<\/tt> takes a set of points and returns a polygon, whereas bwconvhull<\/tt> takes a binary image and returns another binary image. What's up with that? It means simply that bwconvhull<\/tt> computes the convex hull of all the foreground pixels in the input image, and then it produces an output binary image with\r\n      all the pixels inside the convex hull set to white. It's a little easier to show than to say, so here's what it looks like:\r\n   <\/p>
bw = imread('text.png'<\/span>);\r\nimshow(bw)<\/pre> 
bw2 = bwconvhull(bw);\r\nimshow(bw2);<\/pre> 
Let's overlay the foreground pixel locations from the input image (using blue dots) and the convex hull computed by convhull<\/tt> (using a thick red line).\r\n   <\/p>
[y, x] = find(bw);\r\nk = convhull(x, y);\r\nhold on<\/span>\r\nplot(x, y, 'b.'<\/span>)\r\nplot(x(k), y(k), 'r'<\/span>, 'LineWidth'<\/span>, 4)\r\nhold off<\/span><\/pre> 
An important variation supported by bwconvhull<\/tt> is to compute the convex hulls of the individual objects in the input image. Here's how you do that:\r\n   <\/p>
bw3 = bwconvhull(bw, 'objects'<\/span>);\r\nimshow(bw3)<\/pre> 
Something called the convex deficiency<\/i> is sometimes used in shape recognition applications. Loosely speaking, the convex deficiency of a shape is the convex hull\r\n      of the shape minus the shape. Here's an example using the letter \"T\" from the text image above.\r\n   <\/p>
bwt = bw(7:24, 4:18);\r\nimshow(bwt, 'InitialMagnification'<\/span>, 'fit'<\/span>)<\/pre> 
bwtc = bwconvhull(bw_t);\r\nimshow(bwtc, 'InitialMagnification'<\/span>, 'fit'<\/span>)\r\ntitle('Convex hull image'<\/span>)<\/pre> 
bwtcd = bwtc & ~bwt;\r\nimshow(bwtcd, 'InitialMagnification'<\/span>, 'fit'<\/span>)\r\ntitle('Convex deficiency image'<\/span>)<\/pre> 
The convex deficiency of the letter \"T\" has two connected components. This kind of measurement can be useful for recognizing\r\n      shapes.\r\n   <\/p>