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23. Topological Data Analysis

23. Topological Data Analysis

MatrixToTopologicalSpace(A,n)

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.

ReadImageAsTopologicalSpace("file.png",n) ReadImageAsTopologicalSpace("file.png",[m,n])

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].

ReadImageAsMatrix("file.png")

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.

WriteTopologicalSpaceAsImage(T,"filename","ext")

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.

ViewTopologicalSpace(T) ViewTopologicalSpace(T,"mozilla")

Inputs a topological space T, and optionally a command such as "mozilla" for viewing image files. A black/white image is displayed.

Bettinumbers(T,n) Bettinumbers(T)

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.

PathComponent(T,n)

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.

SingularChainComplex(T)

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.

ContractTopologicalSpace(T)

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.

BoundaryTopologicalSpace(T)

Inputs a topological space T and returns its boundars as a topological space.

BoundarySingularities(T)

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.

ThickenedTopologicalSpace(T) ThickenedTopologicalSpace(T,n)

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.

If a positive integer n is input as a second argument then the thickening process is repeated n times.

ComplementTopologicalSpace(T)

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.

ConcatenatedTopologicalSpace(L)

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)].


 


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