EULER INTEGRATION OF
GAUSSIAN RANDOM FIELDS
AND PERSISTENT HOMOLOGY
OMER BOBROWSKI & MATTHEW STROM BORMAN
Presented by Roei Zohar
THE EULER INTEGRAL- REASONING

As we know, the Euler
characteristic is an
additive operator on
compact sets:
  A  B     A    B     A  B 
which reminds us of a
measure

That is why it seems
reasonable to define
an integral with
respect to this
“measure”.
THE EULER INTEGRAL- DRAWBACK
The main problem with this kind of integration is
that the Euler characteristic is only finitely
additive.
 This is why under some conditions it can be
defined for “constructible functions”

as

But we can’t go on from here approximating other
functions using CF functions
EULER INTEGRAL - EXTENSIONS

We shall define another form of integration that
will be more useful for calculations:
For a tame function the limits are well defined,
but generally not equal.
 This definition enable us to use the following
useful proposition

EULER INTEGRAL - EXTENSIONS
The proof appears in [3]
We will only use the upper Euler integral

EULER INTEGRAL - EXTENSIONS

We continue to derive the following Morse like
expression for the integral:
FURTHER HELPFUL DEVELOPMENT
GAUSSIAN RANDOM FIELDS AND THE
GAUSSIAN KINEMATIC FORMULA

Now we turn to show the GK formula which is an
explicit expression for the mean value of all
Lipschitz-Killing curvatures of excursion sets for
zero mean, constant value variance, c², Gaussian
random fields. We shall not go into details, you
can take a look in [2].

The metric here, under certain conditions is
C s , t   E
Where

  f  s   m  s    f t   m t     E  f  s  f t  
T
T
m  x   E  f  x   0
Under this metric the L-K curvatures are
computed in the GKF, and in it the manifold M is
bounded

When taking i  0 we receive  0 *    *  and:
E

   f  D    
1
dim M
j 1
 j
 2  2

j
M  m j  D 
Now we are interested in computing the Euler
integral of a Gaussian random field:
Let M be a stratified space and let g : M   be a
Gaussian or Gaussian related random field.
We are interested in computing the expected
value of the Euler integral of the field g over M.

Theorem: Let M be a compact d-dimensional
stratified space, and let f : M   k be a kdimensional Gaussian random field satisfying the
GKF conditions. For piecewise c² function G :  k  
let g  G  f .
setting D u  G  1    , u  , we have


E   g d       M
M

Where
j
 E g    j 1  2   2  j  M  m j  D u du
g  g(t) has a constant
d

mean.
The proof:
The difficulty in evaluating the expression above
lies in computing the Minkowski functionals
 In the article few cases where they have been
computed are presented, which allows us to
simplify , I will mention one of them
Real Valued Fields:



E   g  d       M  E g  
M

H n  x    1   x 
n
  x   2
f,g 
1 / 2 e
x
1
d
dx
n
n
 x 
2
2
 f  x g  x   x dx

  1   M 
d
j 1
j
j
H
, ( sign G  ) G 
j
j 1
j
2  2

And if G is strictly monotone:
Increasing
Decreasing


: E   g d    
M




: E   g d    
M

d
j0

 1
d
j0

j
j

j
M 
M 
H j,G
j
2  2
H j,G
j
2  2
WEIGHTED SUM OF CRITICAL VALUES

Taking G  x   H  x   x in Theorem
and using
Proposition
yields the following compact
formula
1
WEIGHTED SUM OF CRITICAL VALUES

The thing to note about the last result, is that the
expected value of a weighted sum of the critical
values scales like
 1 M
 , a 1-dimensional measure
of M and not the volume  d  M  , as one might
have expected.

Remark: if we scale the metric by  , then
 d  M  scales by  k
WE’LL NEED THIS ONE UP AHEAD
The proof relies on
INTRODUCTION : PERSISTENT HOMOLOGY
The Usual Homology
 Have
a single
topological space, X
 Get a chain complex
∂
∂
Ck(X)
 For
Ck-1(X)
k=0, 1, 2, …
compute Hk(X)
 Hk=Zk/Bk
∂
∂
C1(X)
∂
C0(X)
∂
0
INTRODUCTION : PERSISTENT HOMOLOGY


Persistent homology is a way of tracking how the
homology of a sequence of spaces changes
Given a filtration of spaces   X u u
such that X s  X t
if s < t, the persistent homology of  ,PH *   
,consists of families of homology classes that
‘persist’ through time.
INTRODUCTION : PERSISTENT HOMOLOGY
o Explicitly an element of PH    is a family of homology classes
k
   t  for t  [ a , b ], where  t  H k  X t 
o The map H k  X s   H k  X t  induced by the inclusion X s  X t , maps
 s to  t .
Given a tame function
f : X   , there is an associated
PH
*
f 
PH
The persistent
a generaliza
*
 f .
of spaces
 f   , u .
 f   , u .
1
1
*
homology
of a tame function
f : X   can be seen as
tion of Morse theory, for if f is a Morse function t
critical values will corespond
PH
filtration
to birth and death time
hen the
s of elements
in
BARCODE DEMONSTRATION (1)
BARCODE DEMONSTRATION (2)
INTRODUCTION : PERSISTENT HOMOLOGY
A filtration of spaces (Simplicial complexes example):
X1  X2  X3  …Xn
t=0
t=1
a
b
a
d
a, b
t=2
b
c
c, d, ab,bc
t=3
a
d
b
c
cd, ad
t=4
a
b
d
c
ac
t=5
a
b
d
c
abc
a
b
d
c
acd
GETTING TO THE POINT…
1.
2.
I’ll leave the proof of the second claim to you
THE EXPECTED EULER CHARACTERISTIC OF
THE PERSISTENT HOMOLOGY OF A
GAUSSIAN RANDOM FIELD

In light of the connection between the Euler
integral of a function and the Euler
characteristic of the function’s persistent
homology in place, we will now reinterpret our
computations about the expected Euler integral
of a Gaussian random field
1.
2.
The proof makes use of the following two expressions we saw before:

Computing E g  is not usually possible, but in
the case of a real random Gaussian field we can
get around it and it comes out that:
max
AN APPLICATION – TARGET ENUMERATION
THE END
DEFINITIONS

A “tame” function :
A continuous
topologica
function
f : X   on a compact
l space X with a finite
is tame if the homotopy t
Euler characteri stic
ype of f
-1
   , u  is
always finite.
REFERENCES
1.
Paper presented here, by Omer Bobrowski & Matthew Strom Borman
2.
R.J. Adler and J.E. Taylor. Random Fields and Geometry. Springer,
2007.
3.
Y . Baryshikov and R. Ghrist. Euler integration over definable
functions. In print, 2009.