C
OMPUTE
I
NTERVALS
K { for k
= 0 to dimK L
k
= ∅; for j
= 0 to m − 1 { d = R
EMOVE
P
IVOT
R
OWS
σ
j
; if d
= ∅ Mark σ
j
; else {
i = maxindex d; k = dim σ
j
; Store j and d in T
[i]; L
k
= L
k
∪ {deg σ
i
, deg σ
j
} }
} for j
= 0 to m − 1 { if
σ
j
is marked and T [ j] is empty {
k = dim σ
j
; L
k
= L
k
∪ {deg σ
j
, ∞} }
} }
Fig. 7.15. Algorithm C
OMPUTE
I
NTERVALS
processes a complex of m simplices. It stores the sets of
P-intervals in dimension k in L
k
.
chain R
EMOVE
P
IVOT
R
OWS
σ { k
= dim σ; d = ∂
k
σ; Remove unmarked terms in d;
while d = ∅ {
i = maxindex d;
if T [i] is empty, break;
Let q be the coefficient of σ
i
in T [i];
d = d − q
−1
T [i];
} return d;
} Fig. 7.16. Algorithm R
EMOVE
P
IVOT
R
OWS
first eliminates rows not marked not cor- responding to the basis for
Z
k −1
and then eliminates terms in pivot rows.
pivot. If T [i] is nonempty, a pivot already exists in that row, and we use the in-
verse of its coefficient to eliminate the row from our chain. Otherwise, we have found a pivot and our chain is a pivot column. For our example filtration in Fig-
ure 7.14, the marked 0-simplices {a, b, c, d} and 1-simplices {ad, ac} generate
the P-intervals L
= {0, ∞, 0, 1, 1, 1, 1, 2} and L
1
= {2, 4, 3, 5}, re- spectively.
7.3.4 Discussion
From our derivation, it is clear that the algorithm has the same running time as Gaussian elimination over fields. That is, it takes O
m
3
in the worst case, where m is the number of simplices in the filtration. The algorithm is very
simple, however, and represents the matrices efficiently. Having derived the algorithm from the reduction scheme, we find the algorithm to have the same
structure as the persistence algorithm for Z
2
coefficients. It is different in two aspects:
1. It does its own marking, so it is independent of the Delfinado- Edelsbrunner algorithm. Therefore, the algorithm is no longer restricted
to subcomplexes of a triangulation of S
3
, but can compute over arbi- trary complexes in any dimension.
2. It allows for arbitrary fields as coefficients. This allows us to detect low-order torsion by computing over different rings.
Most significantly, the approach in this section places the persistence algorithm within the classical framework of algebraic topology.
7.4 Algorithm for PIDs
The correspondence we established in Section 6.1.5 eliminated any hope for a simple classification of persistent groups over rings that are not fields. Nev-
ertheless, we may still be interested in their computation. In this section, we give an algorithm to compute the persistent homology groups H
i ,p
k
of a filtered complex K for a fixed i and p. The algorithm we provide computes persistent
homology over any PID D of coefficients by utilizing a reduction algorithm over that ring.
To compute the persistent group, we need to obtain a description of the nu- merator and denominator of the quotient group in Equation 6.1. We already
know how to characterize the numerator. We simply reduce the standard ma- trix representation M
i k
of ∂
i k
using the reduction algorithm. The denominator,
B
i ,p
k
=
B
i +p
k
∩
Z
i k
, plays the role of the boundary group in Equation 6.1. There- fore, instead of reducing matrix M
i k
+1
, we need to reduce an alternate matrix M
i ,p
k +1
that describes this boundary group. We obtain this matrix as follows: 1 We reduce matrix M
i k
to its normal form and obtain a basis {z
j
} for
Z
i k
, using fact ii in Section 7.3.1. We may merge this computation with
that of the numerator. 2 We reduce matrix M
i +p
k +1
to its normal form and obtain a basis {b
l
} for
B
i +p
k
using fact iii in Section 7.3.1.
3 Let N = [{b
l
} {z
j
}] = [B Z], that is, the columns of matrix N consist of the basis elements from the bases we just computed, and B and Z are
the respective submatrices defined by the bases. We next reduce N to normal form to find a basis
{u
q
} for its null-space. As before, we obtain this basis using fact ii. Each u
q
= [α
q
ζ
q
], where α
q
, ζ
q
are vectors of coefficients of
{b
l
}, {z
j
}, respectively. Note that Nu
q
= Bα
q
+ Zζ
q
= 0 by definition. In other words, element B
α
q
= −Zζ
q
belongs to the span of both bases. Therefore, both
{Bα
q
} and {Zζ
q
} are bases for
B
i ,p
k
=
B
i +p
k
∩
Z
i k
. We form a matrix M
i ,p
k +1
from either. We now reduce M
i ,p
k +1
to normal form and read off the torsion coefficients and the rank of
B
i ,p
k
. It is clear from the procedure that we are computing the persistent groups correctly, giving us the following.
Theorem 7.6 For coefficients in any PID, persistent homology groups are computable in the order of time and space of computing homology groups.
8
Topological Simplification
In Chapter 6, we motivated the definition of persistence by the need for intel- ligent methods for topological simplification. In this chapter, we look at algo-
rithms for simplifying a space topologically, using persistence as a measure. We begin by reviewing prior work and formalizing a notion of topological sim-
plification within the framework of filtrations in Section 8.1. We then look at a simple algorithm for computing persistent Betti numbers, which motivates
the reordering algorithms for simplification in Section 8.2. There are conflicts, however, between the goals established for simplification. We formalize these
conflicts, and discuss their resolution or diminution in Section 8.3. To view the entire persistent history of a filtration, we develop color maps in Section 8.4.
We end this chapter with visualizations of simplified complexes.
8.1 Motivation
Topological issues arise in surface reconstruction and mesh optimization. Sur- face reconstruction is, by itself, a topological question, but it is often addressed
with geometric methods. Consequently, fast ad-hoc heuristics for surface re- construction usually give rise to defective surfaces, requiring hole-filling or
filtering as a post-processing step Curless and Levoy, 1996; Turk and Levoy, 1994. Furthermore, surface modification methods such as decimation, refine-
ment, thickening, and smoothing may cause changes in the surface’s topology. We gave an example of this connection in the discussion in Section 1.2.3 in
relation to surface decimation.
8.1.1 Prior Work
Topological questions have been mostly marginalized in the past. In the com- puter graphics community, for example, where appearance is the paramount
148
issue, the topological changes caused by a geometric simplification algorithm are often touted as a feature of the algorithm Garland and Heckbert, 1997;
Hoppe et al., 1993; Popovi´c and Hoppe, 1997; Schroeder et al., 1992. Dey et al. 1999 describe a topology-preserving decimation operation that disal-
lows topological changes all together. In general, however, geometrical con- cerns override topological ones, and there is little control or understanding of
the resulting topological changes.
There has been little work, moreover, in the area of topological simplifica- tion. Rossignac and Borrel 1993 use a global grid and simplify the topology
within grid elements. He et al. 1996 use low-pass filters for volume grid data sets. Their work does not apply, however, to polygonal objects, unless they
are voxelized. El-Sana and Varshney 1998 approach simplification using α-
shape inspired ideas and convolution. Wood and Guskov 2001 eliminate small tunnels by growing regions on a surface. None of the work considers the
problem using a theoretical foundation or a well-defined topological measure.
8.1.2 Approach and Goals
In this book, I advocate the approach of using persistence within the framework of filtrations. The topological complexity of a filtration is reflected in its Betti
numbers. Consequently, I consider topological simplification to be a process that decreases a space’s Betti numbers. If we view a filtration as a history of
a growing complex, simplification is a process that does not allow short-lived cycles to ever exist. Simply put, a cycle cannot be born unless it has a long
life, and persistence controls the prerequisite life-time for existence. There are two goals in the simplification process:
1. elimination of nonpersistent cycles, 2. and maintenance of the filtration.
As stated, it is not clear whether any conflicts exist between achieving the above two goals.
The simplification process reorders the simplices in the filtration to elimi- nate nonpersistent cycles. It is the entire history of a growing complex that is
being simplified, however, and not a single complex. Some may argue, there- fore, that no simplification has taken place: The same simplices exist as before
in the filtration, but in a new order. This argument is based on notions from geometric simplification, where simplices are removed and new ones are in-
troduced in a single complex. The argument is not valid, however, as the two simplification processes are not analogous. The filtrations in this book exist
in a geometric context, and the order of simplices has meaning. For example,
