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copied!<p>Using Mathematica 8, I created a morphological graph from the list of white pixels in the image. It is working fine on your first image:</p> <p><img src="https://i.stack.imgur.com/rAW9V.png" alt="enter image description here"></p> <p><img src="https://i.stack.imgur.com/RiNTM.png" alt="enter image description here"></p> <p>Create the morphological graph:</p> <pre><code>graph = MorphologicalGraph[binaryimage]; </code></pre> <p>Then you can query the graph properties that are of interest to you.</p> <p>This gives the names of the vertex in the graph:</p> <pre><code>vertex = VertexList[graph] </code></pre> <p>The list of the edges:</p> <pre><code>EdgeList[graph] </code></pre> <p>And that gives the positions of the vertex:</p> <pre><code>pos = PropertyValue[{graph, #}, VertexCoordinates] & /@ vertex </code></pre> <p>This is what the results look like for the first image:</p> <pre><code>In[21]:= vertex = VertexList[graph] Out[21]= {1, 3, 2, 4, 5, 6, 7, 9, 8, 10} In[22]:= EdgeList[graph] Out[22]= {1 \[UndirectedEdge] 3, 2 \[UndirectedEdge] 4, 3 \[UndirectedEdge] 4, 3 \[UndirectedEdge] 5, 4 \[UndirectedEdge] 6, 6 \[UndirectedEdge] 7, 6 \[UndirectedEdge] 9, 8 \[UndirectedEdge] 9, 9 \[UndirectedEdge] 10} In[26]:= pos = PropertyValue[{graph, #}, VertexCoordinates] & /@ vertex Out[26]= {{54.5, 191.5}, {98.5, 149.5}, {42.5, 185.5}, {91.5, 138.5}, {132.5, 119.5}, {157.5, 72.5}, {168.5, 65.5}, {125.5, 52.5}, {114.5, 53.5}, {120.5, 29.5}} </code></pre> <p>Given the documentation, <a href="http://reference.wolfram.com/mathematica/ref/MorphologicalGraph.html" rel="noreferrer">http://reference.wolfram.com/mathematica/ref/MorphologicalGraph.html</a>, the command MorphologicalGraph first computes the skeleton by morphological thinning:</p> <pre><code>skeleton = Thinning[binaryimage, Method -> "Morphological"] </code></pre> <p>Then the vertex are detected; they are the branch points and the end points:</p> <pre><code>verteximage = ImageAdd[ MorphologicalTransform[skeleton, "SkeletonEndPoints"], MorphologicalTransform[skeleton, "SkeletonBranchPoints"]] </code></pre> <p><img src="https://i.stack.imgur.com/KbITD.png" alt="enter image description here"></p> <p>And then the vertex are linked after analysis of their connectivity.</p> <p>For example, one could start by breaking the structure around the vertex and then look for the connected components, revealing the edges of the graph:</p> <pre><code>comp = MorphologicalComponents[ ImageSubtract[ skeleton, Dilation[vertices, CrossMatrix[1]]]]; Colorize[comp] </code></pre> <p><img src="https://i.stack.imgur.com/E6B9K.png" alt="enter image description here"></p> <p>The devil is in the details, but that sounds like a solid starting point if you wish to develop your own implementation.</p>

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