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An Algebraic Topological Method for Feature Identification.

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We develop a mathematical framework for describing local features of a geometric object| such as the edges of a square or the apex of a cone|in terms of algebraic topological invariants. The main tool is the construction of a tangent complex for an arbitrary geometrical object, generalising the usual tangent bundle of a manifold. This framework can be used to develop algorithms for automatic feature location. We give several examples of applying such algorithms to geometric objects represented by point-cloud data sets.
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An Algebraic Topological Method for Feature Identification
Erik Carlsson, Gunnar Carlssonand Vin de Silva
August 12, 2003
Abstract
We develop a mathematical framework for describing local features of a geometric object—
such as the edges of a square or the apex of a cone—in terms of algebraic topological invariants.
The main tool is the construction of a tangent complex for an arbitrary geometrical object,
generalising the usual tangent bundle of a manifold. This framework can be used to develop
algorithms for automatic feature location. We give several examples of applying such algorithms
to geometric objects represented by point-cloud data sets.
1 Introduction
In attempting to recognize geometric objects, it is often very useful to first recognize iden-
tifiable features of the object in question. For example, in correctly identifying a square a
natural first step is to locate the corners; this information is enough to determine which
square we are dealing with. Similarly, if the object in question is a convex polyhedron, then
the vertices and edges of the polyhedron are the most important features to identify. In the
case of a cone, one looks for the cone point. It is an interesting problem theoretically and
computationally to construct automatic methods for locating such features.
In order to develop such methods, it is first necessary to make mathematical sense of the
notion of “feature”. A reasonable starting point, based on the examples above, is to define
features as singular points of geometric curves, surfaces, etc. Accordingly, in this paper we
set ourselves the task of developing automatic methods for locating singular points on a
curve, surface, or higher dimensional geometric object.
A desirable feature of such methods is that they should be robust to deformation, to a
certain degree. For example, in optical character recognition, it is important that variously
deformed versions of a given character should be identified as being equivalent and having
equivalent features. The methods we develop here ought to be able to recognize the apex,
T-junctions and leg-ends of an upper-case letter “A”, even if that letter has been sheared,
or bent, or compressed in the vertical or horizontal direction. Another situation where
robustness is important occurs when an object is viewed in two different coordinate systems.
The locus given by {(r, θ) : 1 r2,0θπ/2}looks like a rectangle in (r, θ)-space,
but it looks like a sector of an annulus when viewed in rectangular coordinates. We develop
methods which detect properties of this locus which are invariant under such coordinate
changes.
A typical method (see [3] or [5]), for dealing with such questions is to develop templates,
equipped with parameters, with the hope that the figure in question will be very close to a
Supported in part by NSF DMS-0101364
Supported in part by NSF DMS-0101364
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template model, for some choice of the parameter values. For example, in the case of the
letter “A” above, one might have a template consisting of a standard letter “A”, together
with two parameters describing vertical and horizontal compression of the letter. This family
of templates may be adequate for a particular class of documents, but it would not be
adequate in documents where a “sheared” letter is permitted. Of course, a new parameter
can be added which describes the shear. To cover an even larger class of documents, perhaps
containing instances of “A” where some of the line segments defining it are in fact curves,
yet more parameters are necessary. Clearly this can become unwieldy quite quickly.
By contrast, our approach uses algebraic topology to locate and identify relevant features
of objects without requiring the choice of templates, or of parametrized families of defor-
mations. Our method permits us to conclude the existence of a singular point, without
having to match it with any particular model of the particular singularity. For example, a
sharp bend (“corner”) in a curve can be recognized without having to match that region of
the curve to any particular pair of lines locally. The idea is to identify algebraic topological
invariants which can recognize a singular point, and which are by their nature deformation
invariant; instead of trying to match with a larger and larger family of templates.
These invariants automatically distinguish between different kinds of singular points. For
example, if our underlying point set is a cube, the set of singular points consists of all the
edges on the cube; this includes the vertices, which are common to multiple edges. However,
we may wish to isolate the vertices directly. This can be done by adjusting a single parameter
in the search. We use a topological invariant referred to as the first Betti number β1in this
case, and setting β1>1 finds all singular points, while setting β1>3 will find only the
vertices.
A key consideration is in what form the geometric objects are presented. For instance,
if they are presented using finite systems of algebraic equations and inequalities, then it is
typically feasible to determine the collection of singular points explicitly. In this paper, we
will instead deal with point cloud data, i.e finite but large sets of points sampled from a
geometric object in Euclidean space. Dealing with spaces presented in this form produces
computational challenges for us, since one must determine how to “estimate” the topological
invariants from a geometric object using only a finite sample from it.
1.1 Overview of the method
We now give an informal description of our method. An initial observation is that many
singular points are topologically standard. This means that there is a continuous, but not
smooth, change of coordinates which transforms the surface locally into a smooth model.
Since topological invariants are insensitive to such coordinate changes, this means that we
cannot apply topological invariants directly to the spaces in question to detect these features.
We are instead forced to consider constructions on the surface, which are sensitive to the
local smooth structure, and which produce spaces which can be distinguished by topological
methods. In this paper, we will develop an extension of the concept of the tangent bundle
to a smooth submanifold of Rn, which applies to more general subsets. We will refer to
this construction as the tangent complex T(X) of a subset XRn; the tangent complex
is a subset of X×Rn. It is closely related to the notion of tangent cone used in geometric
measure theory (see [4]).
In many examples, which are topologically standard, T(X) nevertheless produces a space
2
which is topologically distinct from Tapplied to a smooth submanifold. This will ultimately
permit us to detect singular points by finding regions in which the tangent complex is
homotopically non-standard. Here are some examples of how the construction behaves;
we will give a formal definition in the body of the paper.
Example 1.1 When Xis a smooth submanifold of Rn,T(X)is the usual tangent unit sphere
bundle of X.
Contrast the following two examples.
Example 1.2 (Straight line.) If X1=R× {0}is the x-axis in R2, then T(X1)is the union
of two components R× {0} × {e1}and R× {0} × {−e1}. Here e1denotes the standard basis
vector (1,0) R2.
Example 1.3 (L-shaped line.) Let X2=R+×{0} ∪ {0} × R+, where R+denotes the set
of nonnegative reals {x:x0}. In this case, T(X2)is a disconnected union of four rays,
given by R+× {0} × {e1},R+× {0} × {−e1},{0} × R+× {e2}, and {0} × R+× {−e2}.
The sets in Examples 1.2 and 1.3 are topologically equivalent to the real line, but their
tangent complexes fall into two and four connected components respectively. Thus we dis-
tinguish X1and X2by simple topological invariants of T(X1) and T(X2), though the spaces
themselves are topologically indistinguishable. In fact for any smooth curve CRn, the
tangent complex T(C) is topologically equivalent to T(X1). In contrast, for a piecewise
smooth curve with ktangent discontinuities, the tangent complex has 2k+ 2 connected
components. Example 1.3 simply illustrates the case k= 1.
In these simple examples the existence of a corner, or the number of corners, can be
derived from the number of connected components of the tangent complex. This may be
computed exactly or up to some tolerance using a clustering algorithm ([7], pp. 453-480).
In higher-dimensional cases it is may not be enough to count connected components, as we
see next.
Example 1.4 (One wall.) Let X3be the set {0R2in R3. Then the tangent complex T(X3)
is connected and has the homotopy type of a circle.
Example 1.5 (Two walls meeting at a corner.) Let X4be the subset of R3given by
X4=X2×R
=R+× {0} × R∪ {0} × R+×R
In this case the tangent complex is connected, but has the homotopy type of a bouquet of three
circles.
Here the presence of singular points in X4along the subset {0} × {0} × Rcan be detected
using one-dimensional homology, which detects loops.
In this paper, we use these ideas as the basis for an algorithm to locate the singular set.
To give an idea of how the algorithm works, we consider the case of a curve in the plane.
The object is to locate any singular points. We suppose that we are dealing with a bounded
part of the curve contained in a square window, as in Figure 1.
The first step in the algorithm is to compute the homology of the tangent complex for the
part of the curve contained in the window. If the homology agrees with the standard model
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Figure 1: A curve with a singular point
Figure 2: A divide-and-conquer strategy for locating the singular point
of a single smooth curve, then we stop looking for singular points. In this case the tangent
complex has four connected components (as in Example 1.3), which is non-standard.
The next step is to divide the window into four smaller windows and repeat the homology
calculation in each window (Figure 2, left panel). In this case, one of the windows is empty
and two of the windows contain a standard curve, and hence have standard homology. As
indicated by the shading, we discard these three windows and apply the algorithm recursively
on the single remaining non-standard window. Two further iterations of this process are
shown in last two panels of Figure 2. The result is a nested sequence of windows converging
on the singular point. If there are several singular points, then the process will have several
active branches converging separately to the different singular points.
When implementing this algorithm in practice, we need to take account of the fact that
we are dealing with point cloud data. This presents two challenges:
How do we recover homology from a space represented as point cloud data?
How do we reconstruct a discrete tangent complex from point cloud data, when it
depends on limiting information concerning the underlying space?
In this paper we have taken a straightforward approach to the first question. Given
a point cloud space we build a simplicial complex approximation called the Rips complex
which depends on a choice of length scale and which has a vertex for every data point
considered. Given a simplicial complex, the homology calculation is straightforward linear
algebra. The Rips complex is simple to implement but not particularly efficient; it suffices
for the examples given here. A more sophisticated approach is the the synthetic Delaunay
triangulation developed in [1] .
We reconstruct the tangent complex by using local principal components analysis§at a
small number of base points in the complex to obtain a an approximation to the tangent
space at these points; then we sample the unit spheres in these tangent spaces uniformly to
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(a) (b) (c)
Figure 3: Example spaces with easily-computed homology
obtain a point cloud in Rn×Sn1. The resulting point cloud space is amenable to the Rips
complex construction, and the homology of the tangent complex can be recovered reliably
given sufficient data.
2 Homological Preliminaries
In this section, we will discuss the properties of homology groups we will need. The reader
is encouraged to consult a standard text such as [6] or [8] for a more detailed exposition of
these ideas.
Homology is a technique for assigning, to every topological space Xand nonnegative
integer n, a vector spaces Hn(X). We will deal exclusively with “mod 2 homology”, in
which case these are vector spaces over the finite field F2={0,1}. The dimension of this
vector space is referred to as the n-th Betti number of Xwith mod 2 coefficients, and will
be written βn(X). In an informal sense, the n-th Betti number of Xmeasures the number
of n-dimensional holes in the space X.
Example 2.1 Suppose that X=S1is the unit circle in the plane. Then H1(X)
=F2, so
β1(X) = 1. This represents the one dimensional hole “in the middle of the circle”.
Example 2.2 Suppose that Xis a bouquet of two circles, as shown in Figure 3(b). In this
case, β1(X) = 2, representing two distinct one dimensional holes.
Example 2.3 Suppose that X=S2, the unit sphere in 3-space. Then β2(X) = 1, measuring
the two dimensional hole in the sphere. More generally, we have that βi(Sn) = 0 when i6= 0, n
and βi(Sn) = 1 for i= 0, n.
Example 2.4 Suppose that Xconsists of kdistinct points. Then β0(X) = k. In general,
β0measures the number of path components of X.
The homology groups have the following properties.
Hnis functorial, i.e. every continuous map f:XYinduces a linear transformation
Hn(f): Hn(X)Hn(Y) for all n.
Hnis homotopy invariant, i.e. if two maps f, g:XYare homotopic, then the induced
linear transformations Hn(f) and Hn(g) are equal. This is an extremely important
property of these linear transformations. We say two spaces Xand Yare homotopy
equivalent if there are maps f:XYand g:YXso that fg is homotopic to idY
and gf is homotopic to idX. The homotopy property for Hnimplies that if Xand Y
are homotopy equivalent, then Hn(X) and Hn(Y) are isomorphic, and in particular
βn(X) = βn(Y).
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The phrase “can be deformed into” is loosely synonymous with “is homotopy equivalent to”,
and conveys roughly the right idea.
Example 2.5 The circle in Figure 3(a) and the annulus in Figure 3(b) are homotopy equiv-
alent and so have the same Betti numbers.
When a space is broken up as the union of different pieces, the homology can be com-
puted from the homology of the pieces and all possible overlaps of these pieces, using
Mayer–Vietoris techniques ([6], [8]).
When a space is described as a simplicial complex, the computation of homology re-
duces to straightforward linear algebra over the field F2. A simplicial complex is a subspace
of Rnexpressed as a union of simplices which overlap in faces, i.e. the intersection of any
pair of simplices is a face of each of the two simplices. Such a space is determined up to
homeomorphism by simple combinatorial data.
Definition 2.6 By an abstract simplicial complex, we will mean a pair (V, Σ), where Vis
a finite set whose objects are referred to as vertices, and where Σis a collection of subsets
of V, so that if σΣ, and στ, then τΣ. The elements of Σare referred to as faces.
If a face τΣconsists of exactly k+ 1 elements of Vthen we say that τ={v0, v1, . . . , vk}
is a k-simplex of Σwith vertices v0, v1,...,vk.
Any simplicial complex Sdetermines an abstract simplicial complex as follows. Let Vbe
the set of vertices of S, and let Σ consist of those sets of vertices τ={v0, v1, . . . , vk}which
span a simplex in S. Conversely, we can recover the topological type of Sfrom the abstract
simplicial complex by taking a simplex for each face of Σ and gluing these simplices together
appropriately.
The homology of a simplicial complex Sis computed from the abstract simplicial complex
associated to it. The idea is to set up a chain complex, which is a sequence of vector spaces
and linear maps between them:
00
C0
1
C1 · · · Ck1
k
Ck · · ·
Each Ckis a vector space over the field F2with a basis vector ¯τfor each k-simplex τΣ.
The linear map kis known as the boundary operator and is defined as follows. First
choose an ordering of the vertex set V. Writing τ={v0, v1,...,vk}with the vertices listed
in increasing order, we define the j-th face of τto be the (k1)-simplex τjobtained by
deleting the vertex vjfrom the list. Then kis defined to be the linear map defined by
k¯τ=
k
X
j=0
(1)j¯τj
on basis vectors ¯τ, and extended by linearity to all of Ck. [Note: the (1)jterms shown here
are necessary in general, but in our case they happen to be redundant since we are working
over F2.]
If στis a (k2)-simplex, then it is a face of a (k1)-face of τin exactly two different
ways. Using this observation it can be shown that k1k= 0 for all k. In other words, the
boundary of a boundary is always zero. Let ZkCkdenote the null space of the operator k,
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(a) (b) (c)
Figure 4: Chains and cycles
(a) (b)
Figure 5: A boundary cycle
and let BkCkdenote the image of the operator k+1. It follows that BkZk, and we
define the k-th homology group Hkto be the quotient vector space Zk/Bk. The structure
of Hkcan therefore be expressed in terms of matrix calculations over the field F2.
2.1 Chains, cycles and boundaries
It may be helpful to give some examples of how the definition Hk=Zk/Bkworks in practice.
We introduce the language of chains,cycles and boundaries.
Ak-chain is an element of the F2vector space Ckderived from a simplicial complex S.
There is a coefficient, 0 or 1, for each k-simplex of S; thus we can regard a k-chain simply as
a set of k-simplices, by picking out those simplices with coefficient 1. A k-cycle is an element
of Zk; in other words a k-chain whose boundary is zero (empty). Finally a k-boundary is a
k-chain which is the boundary of some (k+ 1)-chain. Every k-boundary is automatically a
k-cycle; this is equivalent to the assertion = 0. The homology Hkis defined to be the
space of k-cycles modulo all the uninteresting k-cycles that be created cheaply by taking the
boundary of some (k+ 1)-chain.
In Figure 4(a), a typical 1-chain is shown highlighted in red. It is not a 1-cycle, since
its boundary 0-chain is nonempty (Figure 4(b)). Figure 4(c) shows a 1-cycle. This is not
the boundary of any 2-chain, so it corresponds to a genuine non-zero element of H1. On
the other hand, the cycle in Figure 5(a) is the boundary of the 2-cycle shown in Figure ref-
fig:1boundary(b), and so it is zero in homology.
3 Point Cloud Data
We have seen that homology is readily computable for spaces which are equipped with a
triangulation, i.e. a homeomorphism to a simplicial complex. The geometric objects we
will deal with will rarely come equipped with such a structure. In fact, we will be trying
to recover topological information about a geometric object from point cloud data obtained
from the space, by which we mean a finite set of points sampled from the object. In order
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to make calculations, this means that we must somehow construct a simplicial complex from
the point cloud data, which we believe approximates the space in question.
The idea is as follows. Let Xbe a topological space, and suppose we have a finite covering
U={Uα}αAof Xindexed by a set A.
Definition 3.1 The Cech complex of U,C(U), is the simplicial complex whose vertex set
is A, and where a subset {α0, α1,...,αk}is a simplex if and only if
Uα0Uα1...Uαk6=
It is frequently the case that the Cech complex of the covering Uis homotopy equivalent
to X, and therefore has homology isomorphic to that of X. For example, if all sets of the
form
Uα0Uα1...Uαk
are either empty or contractible, then C(U) is homotopy equivalent to X. For any Rieman-
nian manifold M, there is an so that if {x1,...,xN}has the property that the balls B(xi)
cover M, then the Cech complex of the covering {B(x1),...,B(xN)}is homotopy equivalent
to M.
If Sis a finite subset of a metric space, we write C(S) to mean C(B), where Bis the
collection of metric balls {B(s) : sS}. In the case of Euclidean data there is the following
approximation theorem.
Theorem 3.2 If SRnis a finite set of points in Euclidean space, then C(S)is homotopy
equivalent to the space:
S=[
sS
B(s)
When Sis sampled from a space XRn, it may well be the case that the union of balls S
covers and is homotopy equivalent to X. If so, then this theorem implies that C(S) has the
same homology as X.
There is a second complex we can construct to approximate the homotopy type of a space
which is equipped with a metric.
Definition 3.3 Suppose that Xis a metric space, with metric d. For any finite subset S
of X, and any  > 0, we define the -Rips complex of the subset Sto be the abstract simplicial
complex whose vertex set is S, and where a subset {s0, s1,...,sk}is a simplex if and only if
d(xi, xj)for all i, j so that 0i, j k. We write R(S)for this complex.
Suppose again that Xis a metric space, and that Sis a finite subset so that
[
sS
B(s) = X
We have an evident inclusion C/2(S)R(S): the vertex sets of the two complexes are
the same, and it follows from the triangle inequality that if B/2(s1)B/2(s2)6=, then
d(s1, s2). If we are dealing with points in Rn, there is also an inclusion R(S)C(S),
as one can readily check. This comparability suggests that both complexes can be useful in
approximating homotopy types.
8
(a) (b)
Figure 6: ‘Holes’ due to uneven sampling lead to incorrect homology.
Remark. The two complexes have different useful properties. The Cech complex is theoret-
ically amenable in that there are results (such as Theorem 3.2 above) which establish that
under certain conditions the homotopy type of the Cech complex of a covering of Xis the
same as that of X. However the Cech complex is computationally more involved, since one
needs to determine for every collection of metric balls whether they have a common intersec-
tion. This is a slightly awkward calculation even in Euclidean space. The Rips complex, on
the other hand, does not have such good theoretical properties, but is computationally more
convenient, since one only needs to identify the 1-simplices (edges), which then determine
the rest of the complex.
3.1 Uneven sampling, and persistent homology
In spite of the theorems alluded to above, in practice it is unusual for the Cech complex to
exactly recover the homotopy type of the underlying space X. The usual problem is that
our sampling from the geometric object may not be adequate.
To see how this happens, consider Figure 6. Here we suppose that we have obtained
point cloud data by sampling from an annulus, which has the homotopy type of a circle.
However, the sampling is not completely uniform. The blue shaded region in (a) represents
the cloud of sampled points, with the white holes representing subregions where there are no
sample points. Each of the holes which is entirely contained in the shaded region will create
a new generator in homology, so when we compute the homology of the Cech complex, for
a suitable small value of , we find that rank H1(C) = 4 instead of the desired rank of 1.
The simple solution—make bigger–is not always as helpful as it seems; see Figure 6(b).
Here we have thickened the data cloud (green region), to represent the effect of choosing a
larger value 0> . Although we have successfully closed the three small holes in (a), a new
hole has formed and this time rank H1(C0) = 2, again not the desired value.
This phenomenon suggests that instead of computing homology for a Cech or Rips com-
plex for a single value of , we instead compute homology for several values of , and consider
the image of the homomorphism
Hi(R(X)) Hi(R0(X))
for  < 0. This construction is known as persistent homology becuse it picks out those
homology classes already existing in Cwhich persist when we move to the larger complex C0.
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Equivalently, we consider the k-cycles of Cmodulo those which can be expressed as
boundaries of (k+ 1)-chains in C0. In the example of Figure 6, a 1-cycle encircling any
of the three small holes in Cbecomes the boundary of a 2-chain when we move to C0.
On the other hand, the newly-created hole in C0does not correspond to any 1-cycle in C
itself. Thus the persistent homology with respect to , 0detects only a single nontrivial
1-dimensional homology class, coming from the obvious cycle which encircles the annulus.
This is the approach we adopt. We will select different length scales for our complexes,
which we believe will be of the right scale to capture the features we are interested in, and
so that any spurious classes vanish under passage to the longer length scale.
Note: The idea of considering homology for Cech complexes of varying length scales and
defining persistent homology groups was introduced by H. Edelsbrunner in [2]. An effective
algorithm for simultaneously computing all the persistent homology groups over an interval
range of values for , 0is given in [2].
4 The Tangent Complex
In this section, we will consider subsets Xof Euclidean space Rn, which in many cases
are contractible, but which nevertheless carry features which we would intuitively regard as
qualitative. The idea of this section is that it is possible to make a construction on X, whose
homotopy type is sensitive to non-smooth features in X.
Definition 4.1 Let XRn. We define the open tangent complex to X,T0(X)to be the
subset of X×Sn1defined by
T0(X) = (x, v) : lim
t0+
d(x+tv, X )
t= 0
where d(ξ, X)denotes infxXd(ξ , x). We define the closed tangent complex T(X)to be the
closure of T0(X)in X×Sn1.
Note first that T(X) comes equipped with a projection p:T(X)X. For any xX, we
will denote by Tx(X) the fiber at x, i.e. p1(x). There is also the projection q:T(X)Sn1.
We have the following two useful propositions concerning this construction.
Proposition 4.2 Suppose that xXis a smooth point of X, i.e. so that there is a
neighborhood Uof xin Rn, and a smooth function f:URm, so that
UX=f1(0)
Df(ξ)has rank mfor every ξin U
Then Tx(X)
=Snm1.
Example 4.3 Let Lbe a line in the xy-plane, given by the equation ξ·(xx0) = 0, for
vectors ξand x0. Then we have q(T(L)) = η}, where ηis a unit vector perpendicular
to ξ, and
T(L)
=L× {±η}
More generally, Let WRnbe the hyperplane determined by the equation ξ·(xx0) = 0,
where ξand x0are n-vectors. Then T(W)
=W×S(ξ), where S(ξ)denotes the unit sphere
10
in the plane of vectors perpendicular to ξ. This result holds with Wreplaced by any halfplane
or quadrant in W.
It is typically easy to work directly with the definition of the tangent complex in the case
of one-dimensional objects in the plane.
Example 4.4 Consider the example in the introduction, with XR2,X=R+× {0} ∪
{0} × R+.
We evaluate the fibers Tx(X) directly. For any smooth point x, the fiber will consist
of two distinct points, i.e. a zero dimensional sphere S0. For points along the x-axis, the
two points will be (x, (1,0)) and (x, (1,0)), and along the y-axis, they will be (x, (0,1))
and (x, (0,1)). At the origin, though, the fiber T(0,0) (X) consists of four points, namely
((0,0),(±1,0)) and ((0,0),(0,±1)). We can easily verify that the tangent complex is actually
the union of two pieces, one from the tangent complex of R+×0 and the other from the
tangent complex of 0 ×R+:
T(R+× {0}) = (R+× {0})× {±e1}
and
T({0} × R+) = ({0} × R+)× {±e2}
Thus T(X) is equal to:
(R+× {0})×{±e1} ∪ ({0} × R+)×e2}
It is easy to see that this space is a disjoint union of four distinct half lines.
Example 4.5 Let XR3be the boundary of the positive octant, i.e.
X={(x, y, z) : x, y, z 0,and one of x,y,z is equal to zero}
In this case, Xis a union of three pieces, namely the intersections of Xwith the three co-
ordinate planes. Denote the intersection of Xwith the xy-plane by Xxy , and let Xyz and Xxz
be the other intersections. Each of these intersections is a quadrant in the corresponding
coordinate plane. From the previous example, we find that T(Xxy)
=Xxy ×S1
xy, where S1
xy
denotes the unit circle in the xy-plane. There are similar descriptions for each of the other
coordinate planes. If we now examine the fibers of the projection T(X)X, we find the
following.
For any smooth point v(i.e. any point of Xwhich does not lie on a coordinate axis),
the fiber Tv(X) is a circle.
For any point vwhich lies on a coordinate axis, but which is not equal to the “cone
point” (0,0,0), Tv(X) is the union of two circles which overlap at a a pair of antipodal
points.
For the cone point, we have T(0,0,0) (X) is homeomorphic to the union of three circles
which pairwise overlap at pairs of antipodal points.
In order to analyze some higher dimensional examples, we will give a result which analyzes
the effect of taking the product of a set in Rnwith a copy of R. We first recall the notion of
the join.
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Definition 4.6 Let XSn1Rn. By the join of Xwith Sk1Rk, we will mean all
points (x, v)in Sn+k1Rn+kso that x
kxkXwhenever x6= 0.
The join has an intrinsic meaning in terms of Xwithout reference to the embedding. The
join of Xand Yis denoted by XY, and is defined to be the quotient X×Y×[0,1]/', where
'is the equivalence relation generated by the equivalences (x, y, 0) '(x0, y, 0) for all x, x0,
and (x, y, 1) '(x, y0,1) for all y, y0. The join of any space Xwith Skis homeomorphic to
the (k+ 1)-fold suspension of the space. In particular, we have SnSm
=Sn+m+1.
Proposition 4.7 Let XRn, and let Y=X×RRn+1. Then the fibre T(x,t)(Y)is equal
to the join of the fiber Tx(X)with S0R. Informally we say that T(x,t)(Y)is the fiberwise
join of T(X)with the 0-sphere.
To illustrate the application of this idea, suppose that Xis obtained by folding a plane
in R3along a line in the plane. For example, consider the set X={(x, y, z)|x0, y
0,and x= 0 or y= 0}. This set is the product of the set Yin R2given by Y={(x, y)|x
0, y 0,and x= 0 or y= 0}and R. We analyzed the tangent complex for Yin this set
in Example 4.4 above, and found that the fiber T(0,0)(Y) consisted of four distinct points.
Proposition 4.7 now tells us that the fiber T(0,0,0)(X) is the join of these four distinct points
with the S0.
Note that this fiber is homeomorphic to the union of two circles along two points, as in
In general, it is possible to give an explicit description of the fibers Tx(X) in the case
when xis a conelike singular point.
12
Definition 4.8 For any subset Lof Sn1Rn, we define the cone on L,cL, to be the set
cL ={rv|r[0,1],and vL}. Let XRn. We say xXis a conelike point in X
if there is a neighborhood Ucontaining xin Rn, with boundary ∂U , so that there is a map
f:U Dn, which is smooth and has a smooth inverse, so that f(XU) = c(f(X∂U )). In
other words, the singularity is locally diffeomorphic to the cone on the space f(X∂U ).
Remark: Conelike singularities are common. For instance, if Xis an algebraic variety,
and xis an isolated singular point, then xis conelike in the above described sense.
It is possible to analyze the fiber Tx(X) in the case of a conelike singularity. Since the
topological type of the tangent complex is unchanged by smooth changes of coordinates, it
is enough to study the case of cL, where LSn1Rn.Lis a subset of Rn, and as such we
may study its tangent complex T(L). For each xL, we have the fiber Tx(L)Rn×Sn1. If
we let q:Rn×Sn1Sn1denote the projection, we obtain the subset q(Tx(L)) Sn1. In
order to describe T(cL), we coordinatize the cone cL via coordinates (t, λ), where 0 t1,
and λL, with all points with t= 0 being identified with the single cone point. Here tis
the parameter describing the line segment from a point xLto the cone point.
Proposition 4.9 T(cL)is described as follows.
For t > 0,T(t,λ)(cL)is the join of Tλ(L)with S0, so is homeomorphic to the suspension
of Tλ(L).
Let pdenote the cone point, i.e. the origin. Then
q(Tp(cL)) = [
λL
S0q(Tλ(L))
Example 4.10 Consider the cone singularity, which occurs at the point (0,0,0) of the sub-
set Xof R3defined by
x2+y2=z2and z0
In this case the fiber Tv(X) consists of a circle for all points vaway from the origin, since
these points are all smooth. However, the fiber at the origin is given by
T(0,0,0)(X)
={ξS2such that ξ·(0,0,1) 1/2}
This space is homeomorphic to an annulus S1×[0,1].
5 Homology detection of singular points
In this section, we will show that in many cases, homology groups can be used to detect and
distinguish between singular points. Let XRnbe a subset. What we will show is that for
many choices of Xand xX, the Betti numbers βkwill provide useful information about
the nature of the point x.
13
Example 5.1 Suppose that xis a smooth point in X, i.e. a point for which there is a
neighborhood UXof x, so that Uis diffeomorphic to a Euclidean disc Dkfor some k.
Then βj(Tx(X)) = 0 for j6= 0, k 1, and βj(Tx(X)) = 1 for j= 0, k 1
Example 5.2 Suppose that Xis a union of llines in R2, intersecting in a single point p.
Then β0(Tp(X)) = 2l, and βi(Tp(X)) = 0 otherwise.
Example 5.3 We consider the case where Xis the surface of a polyhedron. There are now
three distinct possibilities for a point PX, namely
1. Pis in the interior of a face of X
2. Plies on an edge of X, but is not a vertex.
3. Pis a vertex
It turns out that in all cases, β0(TP(X)) = 1, and that βi(TP(X)) = 0 for i2. We examine
the behavior of β1
1. Pis in the interior of a face of the polyhedron. In this case, Pis a smooth point of X,
so TP(X)is a circle. This tells us that β1(TP(X)) = 1.
2. In this case, a local smooth model for the space Xnear xis as the product of a line with
the space YR2which is the union of the non-negative xand y-axes. It now follows
from Proposition 4.7 that TP(X)is the join of S0with T(0,0) (Y), which is the union
of two circles with intersection a pair of distinct points. It is now readily verified that
β1(TP(X)) = 3.
3. In this case, we must count the number Nof faces containing P, or equivalently the
number of edges containing P.TP(X)is a union of Ncircles, with each pair of circles
intersecting in a pair of distinct points, and where all of the pairs of points are disjoint.
One finds that β1(TP(X)) = 1 + PN1
i=0 2i= 1 + N(N1). Note that in this case
N3.
Observe that all the different cases are distinguished by the value of β1on TP(X).
6 Locating singular points
In the last section, we have shown how to use homology to determine whether or not a given
point is a singular point, and what type it is. An important question, though, is whether
one can use homological methods to locate singular points without prior knowledge of where
they might be. The key idea is the following.
14
Proposition 6.1 Let XSn1Rn, and as before let CX Rndenote the cone on X.
Let pdenote the cone point. Then the inclusion Tp(CX)T(CX)is a homotopy equiv-
alence, and hence induces an isomorphism on homology. More generally, let CRXdenote
{zCX :kzk ≤ R}. Then Tp(CRX)T(CRX)is also a homotopy equivalence.
Proof. There is a smooth deformation retraction of CX into the single point p. It is covered
by a deformation retraction of T(CX) into Tp(CX)
This means that if we have found a conelike neighborhood of a conelike singular point,
we can compute the homology of the fiber over the singular point. This fact suggests the
existence of an algorithm for location of singular points in that portion a set Xwhich is
contained in a rectangular subset URn, consisting of the following steps.
1. Compute H(T(X)). If the homology is that of a smooth subset, i.e. H(T(X)) '
H(Sk) for some k, then we assume that the rectangular region in question does not
contain any singular points, and we remove this rectangular region from consideration.
2. Divide the rectangular region into a family of smaller rectangular regions {Uα}αA, say
by bisecting or trisecting in each of the coordinate directions.
3. Apply step 1 to each of the smaller windows, retaining only those rectangular regions Uα
in which H(T(XUα)) is not that of a sphere.
4. Repeat step 3 until one arrives at a sufficiently good approximation to the singular set.
Remark. The assumption that the “homological standardness” of the intersection XUα
implies that there are no singular points of Xin Uαis not a rigorous one. It is surely possible
to construct situations where H(T(XUα)) is isomorphic to H(Sk) for some k, but where
XUαdoes contain singular points. However, one generally expects that homological
complexity of Tx(X) will carry into homological complexity of T(XUα). If one suspects
that one has missed a singular point, though, one can subdivide the region more finely, and
begin at a finer level of subdivision.
Remark. As we have described the algorithm above, it is designed to search for all possible
singular points. However, it is possible to modify it to search for singular points of a particular
type. For instance, if one is searching for the vertices of a cube, and is not interested in the
edges, one can use the calculations in Example 5.3 to see that if one’s criterion for retaining
a rectangular region is that β1(T(XUα)) 7, one will locate the vertices.
7 Point cloud approximation to T X
In order to apply the ideas described above to point cloud data, an attractive option is to find
a method for associating to a set of point cloud data D ⊆ Rnwhich is obtained by sampling
from a geometric object Xa new set of point cloud data T(D) which one believes is what
one might obtain by sampling a finite set of points from T(X). There are many subtle and
interesting issues regarding such constructions, and many natural ways in which one might
proceed. One problem with all these methods is that they construct very large complexes.
We plan to discuss these issues in a systematic way in a future paper, but for the present we
will restrict ourselves to an ad hoc construction of a simplicial complex which is well related
15
to the tangent complex T(X), and for which the algorithm described above successfully
locates the singular set in a number of examples. The goal throughout the construction is
to make sure that not only is the vertex set as small as possible, but that the collections
of simplices should also be as small as possible. Therefore, in addition to choosing a small
vertex set, we use a criterion described below to “prune” edges. Our construction proceeds
as follows.
We suppose that we know the dimension of the original subset X, say l. The construction
begins by selecting a set B={β1, β2,...,βN}of base points from D. In order to maximize
coverage of the space by these points, one chooses them in a way which is biased in favor
of large interpoint distances. Specifically, a relatively large set Ris sampled from D, then
the sequence of points {βi}is chosen from Rin such a way that βiis the furthest point
in Rfrom the collection {β1, β2,...,βi1}. The number of Nbase points is set in advance.
At each base point β, we find the knearest neighbors {βi, β2, . . . , βk}to βin the set D,
where kis a parameter we choose beforehand. We then perform local principal component
analysis [9] to obtain the best linear subspace approximation to Dnear β, and we write Lβ
for this subspace. For us, this means that we form the n×kmatrix Awhose columns are
the differences {β1β, β2β, . . . βkβ}, then construct the covariance matrix C=AAT.
We then diagonalize this matrix, and let Lβbe the span of the eigenvectors corresponding
to the llargest eigenvalues. If the set of llargest eigenvalues doesn’t “stand out”, we assume
that there is not a natural best fitting l-dimensional linear subspace, and we omit the base
point β. Our criterion for “standing out” is as follows. We let λ1λ2 · · · λldenote the
llargest eigenvalues of the matrix C, and our criterion for inclusion is that λl1should be
less than a fixed threshold, which is a parameter in the algorithm. We also choose parameters
δ,ρ, and a parameter ν. We next build a small simplicial complex whose vertex set is Bby
considering the Rips complex on Bfor a suitable value of ν, and then removing edges in a
way which is biased in favor of short edges and against 2-simplices with small angles. This
is done as follows. For each edge e={β1, β2}, we let L(e) denote the length of e. For any
other edge e0={β1, β0}, which contains β1as a vertex, we define σ(e, e0) to be the length of
the vector
β2β1
kβ2β1kβ0β1
kβ0β1k
and similarly for edges e00 ={β0, β2}. We let θ(e) denote the minimum value of σ(e, e0), as
e0varies over all edges which share a vertex with e. We now assign to the edge ethe score
L(e)
θ(e)1.5. We let Sdenote the subcomplex of the Rips complex obtained by removing all edges
whose score is greater than a certain threshhold. This threshhold is also a parameter in the
algorithm. We have constructed a small complex modelling the base space, i.e. the original
data set. In order to build a complex Tfor the tangent complex, we proceed as follows. For
each β∈ B, we now sample a fixed number tof points {vβ
1, vβ
2,...,vβ
t}uniformly from the unit
sphere in Lβ. The vertex set of Tis the set {(β, vβ
i)}β∈B,1it. We define a graph structure
on this set as follows. For every pair of points {(β, vβ
i),{(β, vβ
j)}, we insert this potential edge
if and only if d(vi, vj)δ. For βand β0which are adjacent in S, we define a bipartite graph
structure on the set V(β, β0) = {vβ
1, vβ
2,...,vβ
t} ∪ {vβ0
1, vβ0
2,...,vβ0
t}as the intersection of two
bipartite graph structures Γ1and Γ2on V(β , β0). A pair {(β, vβ
i),(β0, vβ0
j)}spans an edge in
Γ1if and only if d(vβ
i, vβ0
j)pδ2+ρ2. In Γ2, we say {(β, vβ
i),(β0, vβ0
j)}spans an edge if and
16
only if vβ
iis among the mclosest points to vβ0
jin the set {vβ
1, vβ
2, . . . , vβ
t}and vβ0
jis among
the mclosest points to vβ
iin {vβ0
1, vβ0
2,...,vβ0
t}.mis again a parameter. We insert the edge
{(β, vβ
i),(β0, vβ0
j)}if and only if it is an edge both in Γ1and Γ2. If β, β0∈ B are not adjacent
in the complex S, we do not insert any edges of the form {(β , vβ),(β0, vβ0)}. This completes
the construction of the complex T. The rationale for this complicated construction is that
it in practice succeeds in removing small loops which otherwise distort the calculation.
8 Sample Results
We show the results of running our algorithm on various example point sets. The reader
will notice that in some cases, the singular set we obtain is “chunky”, i.e. that we have only
obtained a neighborhood of the singular set. This performance can certainly be improved
with more sampling. The purpose of this paper is show the validity of the concept, rather
than to demonstrate a fully optimized algorithm.
Figure 7: This figure shows the results of applying the algorithm to a curve with intersections in the plane.
Increasing redness indicates longer survival under the algorithm, and so the “reddest points” are those found
by the algorithm to be singular points. In this case, the algorithm searches for small sets for which the
tangent complex has more than two components, i.e for which β0>2. In this example, 5000 points were
used, and the algorithm had a running time of c:a 10 seconds.
References
[1] Carlsson, Gunnar and de Silva, Vin, Synthetic Delaunay triangulations, (in preparation).
[2] Edelsbrunner, Herbert, Letscher, David and Zomorodian, Afra, Topological persistence
and simplification, Discrete Comput. Geom. 28 (2002), 511-533.
[3] Fan, Ting-Jun, Describing and Recognizing 3D Objects Using Surface Properties,
Springer Verlag, Berlin–New York, 1990.
17
Figure 8: These figures show the results of applying the algorithm to two curved surfaces which meet
transversely in a curve, which becomes the singular locus of the union of the two surfaces. This is obtained
by searching for small sets for which the tangent complex has β1>1. In both cases, point clouds of 20,000
points were used, with a running time of c:a 2 minutes.
[4] Federer, Herbert, Geometric measure theory, Die Grundlehren der mathematischen Wis-
senschaften, Band 153, Springer-Verlag New York Inc., New York 1969.
[5] Fisher, Robert B., From Surfaces to Objects: Computer Vision and Three-Dimensional
Scene Analysis, John Wiley and Sons, New York, 1989.
[6] Greenberg, Marvin J. and Harper, John R., Algebraic topology. A first course, Mathe-
matics Lecture Note Series, 58 . Benjamin/Cummings Publishing Co., Inc., Advanced
Book Program, Reading, Mass., 1981.
[7] Hastie, Trevor, Tibshirani, Robert, and Friedman, Jerome. The elements of statistical
learning. Data mining, inference, and prediction, Springer Series in Statistics. Springer-
Verlag, New York, 2001.
[8] Hatcher, Allen, Algebraic topology, Cambridge University Press, Cambridge, 2002.
[9] Jolliffe, I.T., Principal component analysis, Springer Series in Statistics. Springer-Verlag,
New York, 1986.
18
Figure 9: This figure shows the results for a search for a vertex in a portion of the surface of a cube. In
this case, the search is for small sets for which the tangent complex has β1>3. Sample size: 20,000 points,
running time: c:a 1.5 minutes.
00.2 0.4 0.6 0.8 100.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 10: This figure shows the results for a search for the edges in a three simplex. Sample size: 10,000
points, running time: c:a 2 minutes.
Figure 11: Results of a search for the edges in an icosahedron. Sample size: 20,000 points, running time: c:a
3 minutes.
19
Figure 12: A two dimensional projection of the results of searching for the vertices in a 4-simplex. Sample
size: 80,000 points, running time: c:a 5 minutes.
20
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We present the implementation results of a shape segmentation technique and an associated shape matching method whose input is a point sample from the shape. The sample is allowed to be noisy in the sense that they may scatter around the boundary of the shape instead of lying exactly on it. The algorithm is simple and mostly combinatorial in that it builds a single data structure, the Delaunay triangulation of the point set, and groups the tetrahedra to form the segments. A small set of weighted points are derived from the segments which are used as signatures to match shapes. Experimental results establish the effectiveness of the method in practice.
Book
1 Introduction.- 1.1 The Input.- 1.2 Issues in Shape Description.- 1.2.1 Criteria for shape description.- 1.2.2 Choosing segmented surface descriptions.- 1.3 Issues of Recognition.- 1.3.1 Description of models.- 1.3.2 Matching primitives and algorithms.- 1.4 Questions for the Research.- 1.5 The Contribution of the Research.- 1.6 Organization of the Book.- 2 Survey of Previous Work.- 2.1 Survey of Shape Descriptions.- 2.1.1 Volume descriptions.- 2.1.2 Curve/line descriptions.- 2.1.3 Surface descriptions.- 2.1.4 Summary.- 2.2 Survey of Recognition Systems.- 2.2.1 3DPO.- 2.2.2 Nevatia and Binford.- 2.2.3 ACRONYM.- 2.2.4 Extended Gaussian Image (EGI).- 2.2.5 Oshima and Shirai.- 2.2.6 Grimson and Lozano-Perez.- 2.2.7 Faugeras and Hebert.- 2.2.8 Bhanu.- 2.2.9 Ikeuchi.- 2.2.10 Summary.- 3 Surface Segmentation and Description.- 3.1 Curvature Properties and Surface Discontinuities.- 3.2 Detecting Surface Features.- 3.2.1 Method 1: using directional curvatures and scale-space tracking.- 3.2.2 Method 2: using principal curvatures at a single scale.- 3.2.3 Method 3: using anisotropic filtering.- 3.3 Space Grouping.- 3.4 Spatial Linking.- 3.5 Segmentation into Surface Patches.- 3.6 Surface Fitting.- 3.7 Object Inference.- 3.7.1 Labeling boundaries.- 3.7.2 Occlusion and connectivity.- 3.7.3 Inferring and describing objects.- 3.8 Representing Objects by Attributed Graphs.- 3.8.1 Node attributes.- 3.8.2 Link attributes.- 4 Object Recognition.- 4.1 Representation of Models.- 4.2 Overview of the Matching Process.- 4.3 Module 1: Screener.- 4.4 Module 2: Graph Matcher.- 4.4.1 Compatibility between nodes of the model view and scene graph.- 4.4.2 Compatibility between two pairs of matching nodes.- 4.4.3 Computing the geometric transform.- 4.4.4 Modifications based on the geometric transform.- 4.4.5 Measuring the goodness of a match.- 4.5 Module 3: Analyzer.- 4.5.1 Splitting objects.- 4.5.2 Merging objects.- 4.6 Summary.- 5 Experimental Results.- 5.1 The Models.- 5.2 A Detailed Case Study.- 5.2.1 Search nodes expanded in recognition.- 5.3 Results for Other Scenes.- 5.4 Parallel Versus Sequential Search.- 5.5 Unknown Objects.- 5.6 Occlusion.- 6 Discussion and Conclusion.- 6.1 Discussion.- 6.1.1 Problems of segmentation.- 6.1.2 Problems of approximation.- 6.2 Contribution.- 6.3 Future Research.- 6.3.1 From surface to volume.- 6.3.2 Applications.- A Directional Curvatures.- B Surface Curvature.- C Approximation by Quadric Surfaces.
Article
We consider coverage problems in sensor networks with minimal sens-ing capabilities. In particular, we demonstrate that a stationary collection of sensor nodes with no localization can verify coverage in a bounded domain of unknown topological type, so long as the boundary is not too pinched. The only sensing ca-pabilities required by the nodes are a binary form of distance estimation between nodes and a binary proximity sensor for the boundary. The methods we introduce come from persistent homology theory.