Extracting segments from a list of 8-connected pixels

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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:

enter image description here

enter image description here

Create the morphological graph:

graph = MorphologicalGraph[binaryimage];

Then you can query the graph properties that are of interest to you.

This gives the names of the vertex in the graph:

vertex = VertexList[graph]

The list of the edges:

EdgeList[graph]

And that gives the positions of the vertex:

pos = PropertyValue[{graph, #}, VertexCoordinates] & /@ vertex

This is what the results look like for the first image:

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}}

Given the documentation, http://reference.wolfram.com/mathematica/ref/MorphologicalGraph.html, the command MorphologicalGraph first computes the skeleton by morphological thinning:

skeleton = Thinning[binaryimage, Method -> "Morphological"]

Then the vertex are detected; they are the branch points and the end points:

verteximage = ImageAdd[
                  MorphologicalTransform[skeleton, "SkeletonEndPoints"],   
                  MorphologicalTransform[skeleton, "SkeletonBranchPoints"]]

enter image description here

And then the vertex are linked after analysis of their connectivity.

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:

comp = MorphologicalComponents[
           ImageSubtract[
               skeleton, 
               Dilation[vertices, CrossMatrix[1]]]];
Colorize[comp]

enter image description here

The devil is in the details, but that sounds like a solid starting point if you wish to develop your own implementation.

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