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The idea behind Topological Data Analysis is that one might be able to gain a useful qualitative understanding of difficult data from its homological properties. The main focus is on high-dimensional data, but below we describe two low-dimensional toy examples.
A 3-dimensional example
Twenty-five parallel slices of some 3-dimensional object are stored as colour images 01.png, 02.png, ..., 25.png in the directory pictures.tar.gz . The following commands compute a 3-dimensional pure cubical complex M from these images, and show that the complex M is contractible. (The function Bettinumbers(M) returns the ranks of the free parts of the homology groups of M in dimensions 0,1, .., Dimension(M).)
gap> M:=ReadImageSequenceAsPureCubicalComplex("pictures",400); Pure cubical complex of dimension 3. gap> Bettinumbers(M); [ 1, 0, 0, 0 ]
The next commands show that the boundary of M has the same homology as that of a 2-sphere.
gap> B:=BoundaryOfPureCubicalComplex(M); Pure cubical complex of dimension 3. gap> Bettinumbers(B); [ 1, 0, 1, 0 ]
These computations are consistent with the possibility that M is homeomorphic to a 3-dimensional ball. If one happened to know that M actually represents some polytope (which indeed in this case it does), then the following command computes the 1-skeleton of this polytope. The function SingularitiesOfPureCubicalComplex(M,radius,tolerance) returns the space of singular points on the boundary of M. A boundary point is "singular" if the sphere of given radius around the point is divided into two equal sized components (up to some tolerance) by the boundary of M.
gap> S:=SingularitiesOfPureCubicalComplex(M,3,15); Pure cubical complex of dimension 3. gap> Bettinumbers(S); [ 1, 7, 0, 0 ]
Since S has the homology of a wedge of seven circles, it follows that if M represents a 3-dimensional polytope, then the polytope must have eight facets. In order to investigate the combinatorial type of the facets we form the pure cubical complex D=B-S of nonsingular boundary points of M, and determine the adjacency relation between the path components in the pure cubical complex D. For example, the following commands show that there are eight path components in D (one for each facet), and that the first path component is adjacent to precisely four other path components.
gap> M:=ReadImageSequenceAsPureCubicalComplex("pictures",400);; gap> S:=SingularitiesOfPureCubicalComplex(M,3,15);; gap> B:=BoundaryOfPureCubicalComplex(M);; gap> D:=PureCubicalComplexDifference(B,S); Pure cubical complex of dimension 3. gap> Bettinumbers(D,0)); #Number of path componenets in D. 8 gap> P:=List([1..8],n->PathComponentOfPureCubicalComplex(D,n));; gap> for n in [2..8] do > U:=PureCubicalComplexUnion(P[1],P[n]); > U:=ThickenedPureCubicalComplex(U); > Print(Bettinumbers(U,1),"\n"); > od;
The first path component of D is thus either a quadrilateral or a triangle. To see that it is not a triangle we can compute the size of the common boundaries with its adjacent facets to see that in each case the common boundary is more than just a vertex.
gap> for n in [2..8] do > U:=ThickenedPureCubicalComplex(P[1]); > U:=ThickenedPureCubicalComplex(U); > V:=ThickenedPureCubicalComplex(P[n]); > V:=ThickenedPureCubicalComplex(V); > W:=PureCubicalComplexIntersection(U,V); > Print(Size(W),"\n"); > od; 128 95 95 50 0 0 0
A 2-dimensional example
The following commands investigate a digital photograph by calculating the betti numbers of successive thickenings of the image. The thickenings are intended to reduce the "noise" in the image and to realize the image's "true" betti numbers. Without actually viewing the photograph we can detect that there are probably three connected components and three 1-dimensional holes in it.
gap> T:=ReadImageAsPureCubicalComplex("digital_photo.jpg",400);; gap> for i in [1..15] do > Print(Bettinumbers(T),"\n"); > T:=ThickenedPureCubicalComplex(T);; > od; [ 206, 5070, 0 ] [ 11, 10, 0 ] [ 4, 4, 0 ] [ 3, 3, 0 ] [ 3, 3, 0 ] [ 3, 4, 0 ] [ 3, 3, 0 ] [ 3, 3, 0 ] [ 3, 3, 0 ] [ 3, 3, 0 ] [ 3, 3, 0 ] [ 3, 3, 0 ] [ 3, 3, 0 ] [ 3, 3, 0 ] [ 3, 3, 0 ]
There are quite a number of different "ambient isotopy types" of black/white images with betti numbers b0=3, b1=3. A few of these are: Space 1: Space 2: Space 3: Space 4: Space 5: By considering the betti numbers of the "inverted manifolds" obtained by inverting black and white, we can eliminate a few of these as possible ambient isotopy types for the digital photograph. For example, the following commands show that the photograph is not ambient isotopic to manifolds 2, 3 or 5.
gap> T:=ReadImageAsPureCubicalComplex("digital_photo.jpg",400);; gap> for i in [1..8] do > T:=ThickenedPureCubicalComplex(T); > od; gap> T1:=ReadImageAsPureCubicalComplex("space1.jpg",400);; gap> T2:=ReadImageAsPureCubicalComplex("space2.jpg",400);; gap> T3:=ReadImageAsPureCubicalComplex("space3.jpg",400);; gap> T4:=ReadImageAsPureCubicalComplex("space4.jpg",400);; gap> T5:=ReadImageAsPureCubicalComplex("space5.jpg",400);; gap> Bettinumbers(ComplementOfPureCubicalComplex(T)); [ 3, 2, 0 ] gap> Bettinumbers(ComplementOfPureCubicalComplex(T1)); [ 3, 2, 0 ] gap> Bettinumbers(ComplementOfPureCubicalComplex(T2)); [ 4, 3, 0 ] gap> Bettinumbers(ComplementOfPureCubicalComplex(T3)); [ 4, 2, 0 ] gap> Bettinumbers(ComplementOfPureCubicalComplex(T4)); [ 3, 2, 0 ] gap> Bettinumbers(ComplementOfPureCubicalComplex(T5)); [ 4, 3, 0 ]
Further distinctions can be made between Spaces 1-5 by considering individual path components. For example, the following additional commands show that Spaces 1 and 4 are not ambient isotopic.
gap> T1:=ReadImageAsPureCubicalCmplex("space1.jpg",400);; gap> Bettinumbers(T1)[1]; 3 gap> Bettinumbers(PathComponentOfPureCubicalComplex(T1,1)); [ 1, 3, 0 ] gap> Bettinumbers(PathComponentOfPureCubicalComplex(T1,2)); [ 1, 0, 0 ] gap> Bettinumbers(PathComponentOfPureCubicalComplex(T1,3)); [ 1, 0, 0 ] gap> T4:=ReadImageAsPureCubicalComplex("space4.jpg",400);; gap> Bettinumbers(PathComponentOfPureCubicalComplex(T4,1)); [ 1, 2, 0 ] gap> Bettinumbers(PathComponentOfPureCubicalComplex(T4,2)); [ 1, 1, 0 ] gap> Bettinumbers(PathComponentOfPureCubicalComplex(T4,3)); [ 1, 0, 0 ]
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