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<div class="ChapSects"><a href="chap23.html#X7B7E077887694A9F">23. <span class="Heading"> Topological Data Analysis</span></a>
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<h3>23. <span class="Heading"> Topological Data Analysis</span></h3>
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<td class="tdleft"><code class="code"> MatrixToTopologicalSpace(A,n)</code></p>
<p>Inputs an integer matrix A and an integer n. It returns a 2-dimensional topological space corresponding to the black/white image determined by the threshold n and the values of the pixels in A.</td>
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<td class="tdleft"><code class="code"> ReadImageAsTopologicalSpace("file.png",n)</code> <code class="code"> ReadImageAsTopologicalSpace("file.png",[m,n])</code></p>
<p>Reads an image file ("file.png", "file.eps", "file.bmp" etc) and an integer n or pair [m,n] of integers between 0 and 765. It returns a topological space based on the black/white version of the image determined by the threshold n or threshold range [m,n].</td>
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<td class="tdleft"><code class="code"> ReadImageAsMatrix("file.png")</code></p>
<p>Reads an image file ("file.png", "file.eps", "file.bmp" etc) and returns an integer matrix whose entries are the sum of the RGB values of the pixels in the image.</td>
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<td class="tdleft"><code class="code"> WriteTopologicalSpaceAsImage(T,"filename","ext")</code></p>
<p>Inputs a 2-dimensional topological space T, and a filename followed by its extension (e.g. "myfile" followed by "png"). A black/white image is saved to the file.</td>
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<td class="tdleft"><code class="code"> ViewTopologicalSpace(T)</code> <code class="code"> ViewTopologicalSpace(T,"mozilla")</code></p>
<p>Inputs a topological space T, and optionally a command such as "mozilla" for viewing image files. A black/white image is displayed.</td>
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<td class="tdleft"><code class="code"> Bettinumbers(T,n)</code> <code class="code"> Bettinumbers(T)</code></p>
<p>Inputs a topological space T and a non-negative integer n. It returns the n-th betti number of T. If the integer n is not input then a list of all betti numbers is returned.</td>
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<td class="tdleft"><code class="code"> PathComponent(T,n)</code></p>
<p>Inputs a topological space T and an integer n in the rane 0, ..., Bettinumbers(T,0) . It returns the n-th path component of T as a topological space.</td>
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<td class="tdleft"><code class="code"> SingularChainComplex(T)</code></p>
<p>Inputs a topological space T and returns a (usually very large) integral chain complex that is homotopy equivalent to the singular chain complex of T.</td>
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<td class="tdleft"><code class="code"> ContractTopologicalSpace(T)</code></p>
<p>Inputs a topological space T of dimension d and removes d-dimensional cells from T without changing the homotopy type of T. When the function has been applied, no further d-cells can be removed from T without changing the homotopy type.</td>
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<td class="tdleft"><code class="code"> BoundaryTopologicalSpace(T)</code></p>
<p>Inputs a topological space T and returns its boundars as a topological space.</td>
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<td class="tdleft"><code class="code"> BoundarySingularities(T)</code></p>
<p>Inputs a topological space T and returns the subspace of points in the boundary where the boundary is not differentiable. (The method for deciding differentiability at a point is crude/discrete and prone to errors.) The zeroth betti number of the set of points is a measure of the number of "corners" in the boundary of T.</td>
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<td class="tdleft"><code class="code"> ThickenedTopologicalSpace(T)</code> <code class="code"> ThickenedTopologicalSpace(T,n)</code></p>
<p>Inputs a topological space T and returns a topological space S. If a euclidean point is in T then this point and all its perpendicularly neighbouring euclidean points are included in S.</p>
<p>If a positive integer n is input as a second argument then the thickening process is repeated n times.</td>
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<td class="tdleft"><code class="code"> ComplementTopologicalSpace(T)</code></p>
<p>Inputs a topological space T and returns a topological space S. A euclidean point is in S precisely when the point is not in T.</td>
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<td class="tdleft"><code class="code"> ConcatenatedTopologicalSpace(L)</code></p>
<p>Inputs a list L of topological spaces whose underlying arrays of numbers all have equal dimensions. It returns a topological space T got by juxtaposing the spaces L[1], L[2], ..., L[Length(L)].</td>
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