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An
image such as the CHA logo![]() can be viewed as a 2-dimensional (blue) space with white holes. The following commands show that this space has two connected components (i.e. betti number b0=2) and three 1-dimensional holes (i.e. betti number b1=3). |
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gap>
T:=ReadImageAsTopologicalSpace("cha.png",500);; Topological space of dimension 2. gap> ContractTopologicalSpace(T); gap> C:=SingularChainComplex(T); Chain complex of length 2 in characteristic 0 . gap> Homology(C,0); [ 0, 0 ] gap> Homology(C,1); [ 0, 0, 0 ] |
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General
homology algorithms are not the most efficient way to compute betti
numbers of image spaces as they take no advantage of the special
features of such spaces. The function
BettiNumbersOfMatrix(A) is a more efficient function. For example, this function can be used to show that the following image (borrowed from the CHOMP web pages!) ![]() has betti numbers b0=3
and b1=20.
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gap>
T:=ReadImageAsTopologicalSpace("bw_image.bmp",500);; gap> BettiNumbers(T,0); 3 gap> BettiNumbers(T,1); 20 |
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The
idea behind Topological Data Analysis is that one should be able to
gain a qualitative understanding of difficult data from its homological
properties. For 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. |
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gap>
T:=ReadImageAsTopologicalSpace("digital_photo.jpg",400);; gap> for i in [1..15] do > Print(BettiNumbers(T),"\n"); > T:=ThickenedTopologicalSpace(T);; > od; [ 206, 5070 ] [ 11, 10 ] [ 4, 4 ] [ 3, 3 ] [ 3, 3 ] [ 3, 4 ] [ 3, 3 ] [ 3, 3 ] [ 3, 3 ] [ 3, 3 ] [ 3, 3 ] [ 3, 3 ] [ 3, 3 ] [ 3, 3 ] [ 3, 3 ] |
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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 3: ![]() ![]() Space 5: ![]() By considering the betti numbers of the "inverted spaces" 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 spaces 2, 3 or 5. |
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gap>
T:=ReadImageAsTopologicalSpace("digital_photo.jpg",400);; gap> for i in [1..8] do > T:=ThickenedMatrix(T); > od; gap> T1:=ReadImageAsTopologicalSpace("space1.jpg",400);; gap> T2:=ReadImageAsTopologicalSpace("space2.jpg",400);; gap> T3:=ReadImageAsTopologicalSpace("space3.jpg",400);; gap> T4:=ReadImageAsTopologicalSpace("space4.jpg",400);; gap> T5:=ReadImageAsTopologicalSpace("space5.jpg",400);; gap> BettiNumbers(ComplementTopologicalSpace(T)); [ 3, 2 ] gap> BettiNumbers(ComplementTopologicalSpace(T1)); [ 3, 2 ] gap> BettiNumbers(ComplementTopologicalSpace(T2)); [ 4, 3 ] gap> BettiNumbers(ComplementTopologicalSpace(T3)); [ 4, 2 ] gap> BettiNumbers(ComplementTopologicalSpace(T4)); [ 3, 2 ] gap> BettiNumbers(ComplementTopologicalSpace(T5)); [ 4, 3 ] |
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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. |
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gap>
T1:=ReadImageAsTopologicalSpace("space1.jpg",400);; gap> BettiNumbers(T1,0); 3 gap> BettiNumbers(PathComponent(T1,1)); [ 1, 3 ] gap> BettiNumbers(PathComponent(T1,2)); [ 1, 0 ] gap> BettiNumbers(PathComponent(T1,3)); [ 1, 0 ] gap> T4:=ReadImageAsTopologicalSpace("space4.jpg",400);; gap> BettiNumbers(PathComponent(T4,1)); [ 1, 2 ] gap> BettiNumbers(PathComponent(T4,2)); [ 1, 1 ] gap> BettiNumbers(PathComponent(T4,3)); [ 1, 0 ] |
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The
2-dimensional data cloud![]() seems to be sampled from a connected space with a 1-dimensional hole. The following computations agree with this observation. |
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gap>
T:=ReadImageAsTopologicalSpace("sample_from_circle.gif",400);; gap> T:=ComplementTopologicalSpace(T);; ##These commands should reduce noise. gap> T:=ThickenedTopologicalSpace(T);; ## gap> T:=ComplementTopologicalSpace(T);; ## gap> for i in [1..50] do > Print(BettiNumbers(T),"\n"); > T:=ThickenedTopologicalSpace(T);; > od; [ 924, 0 ] [ 602, 29 ] [ 174, 153 ] [ 75, 181 ] [ 30, 107 ] [ 18, 44 ] [ 13, 31 ] [ 9, 18 ] [ 6, 10 ] [ 4, 5 ] [ 4, 3 ] [ 2, 3 ] [ 1, 1 ] [ 1, 1 ] [ 1, 2 ] [ 1, 1 ] [ 1, 1 ] [ 1, 1 ] [ 1, 1 ] [ 1, 1 ] [ 1, 1 ] [ 1, 1 ] [ 1, 1 ] [ 1, 1 ] [ 1, 1 ] [ 1, 1 ] [ 1, 1 ] [ 1, 1 ] [ 1, 1 ] [ 1, 1 ] [ 1, 1 ] [ 1, 1 ] [ 1, 1 ] [ 1, 1 ] [ 1, 1 ] [ 1, 1 ] [ 1, 0 ] [ 1, 0 ] [ 1, 0 ] [ 1, 0 ] [ 1, 0 ] [ 1, 0 ] [ 1, 0 ] [ 1, 0 ] [ 1, 0 ] [ 1, 0 ] [ 1, 0 ] [ 1, 0 ] [ 1, 0 ] [ 1, 0 ] |
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One
approach to calculating the homology of a space X is to simplify the
calculation by finding a smaller homotopy equivalent subspace Y and then to calculate the homology of Y. The command ContractTopologicalSpace(X) provides a method for finding Y. The following commands illustrate this. |
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gap>
T:=ReadImageAsTopologicalSpace("example.eps",400);; gap> ViewTopologicalSpace(T); #T is the following space. ![]() gap> ContractTopologicalSpace(T);; gap> ViewTopologicalSpace(T); #Now T is reduced to the following homotopy. ![]() |
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The
following commands find the boundary of the space T. |
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gap>
T:=ReadImageAsTopologicalSpace("example.eps",400);; gap> B:=BoundaryTopologicalSpace(T);; gap> ViewTopologicalSpace(B); ![]() |
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A
"feature" of a shape could be defined as a singularity in the boundary
of the shape (i.e. a point where the boundary is not
differentiable). According to this definition the following shapes
have respectively 6, 10 and 0 features. ![]() ![]() ![]() The number of features of each of these shapes can be computed using the following commands. |
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gap>
T:=ReadImageAsTopologicalSpace("shape1.jpg",400);; gap> S:=BoundarySingularities(T);; gap> BettiNumbers(S,0); 6 gap> T:=ReadImageAsTopologicalSpace("shape2.jpg",400);; gap> S:=BoundarySingularities(T);; gap> BettiNumbers(S,0); 10 gap> T:=ReadImageAsTopologicalSpace("shape3.jpg",400);; gap> S:=BoundarySingularities(T);; gap> BettiNumbers(S,0); 0 |
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The
hope is that Topological Data Analysis can be used to analyze images
such as the following Computed Tomography scan.![]() Increasing the threshold in steps
of 10, and computing the betti numbers each time, produces the
following results.
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gap>
for n in [1..70] do > T:=ReadImageAsTopologicalSpace("ctprostate.jpg",n*10);; > Print(BettiNumbers(T),"\n"); > od; [ 1, 0 ] [ 1, 2 ] [ 7, 0 ] [ 5, 0 ] [ 10, 0 ] [ 9, 1 ] [ 21, 1 ] [ 122, 0 ] [ 835, 7 ] [ 136, 383 ] [ 110, 169 ] [ 20, 95 ] [ 6, 20 ] [ 5, 2 ] [ 5, 1 ] [ 6, 0 ] [ 5, 0 ] [ 6, 1 ] [ 6, 0 ] [ 5, 1 ] [ 5, 6 ] [ 4, 5 ] [ 4, 11 ] [ 5, 7 ] [ 4, 3 ] [ 5, 3 ] [ 6, 3 ] [ 5, 5 ] [ 2, 8 ] [ 3, 10 ] [ 4, 23 ] [ 5, 60 ] [ 5, 102 ] [ 18, 117 ] [ 193, 129 ] [ 419, 115 ] [ 111, 342 ] [ 95, 207 ] [ 114, 201 ] [ 270, 208 ] [ 169, 306 ] [ 35, 558 ] [ 30, 134 ] [ 38, 82 ] [ 23, 44 ] [ 11, 32 ] [ 16, 22 ] [ 10, 20 ] [ 6, 16 ] [ 7, 13 ] [ 5, 17 ] [ 4, 14 ] [ 3, 23 ] [ 2, 25 ] [ 2, 21 ] [ 2, 17 ] [ 4, 16 ] [ 1, 18 ] [ 1, 20 ] [ 1, 26 ] [ 1, 25 ] [ 1, 26 ] [ 1, 23 ] [ 1, 21 ] [ 1, 22 ] [ 1, 25 ] [ 1, 25 ] [ 1, 23 ] [ 1, 23 ] [ 1, 26 ] |
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The
above suggests that the CT image is interesting in the threshold range
[340,390]. We can view it using this threshold. |
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gap>
T:=ReadImageAsTopologicalSpace(A,[340,390]);; gap> ViewTopologicalSpace(BoundaryTopologicalSpace(T)); ![]() |
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