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EULER INTEGRATION OF GAUSSIAN RANDOM FIELDSAND PERSISTENT HOMOLOGYOMER BOBROWSKI & MATTHEW STROM BORMAN
Presented by Roei Zohar
THE EULER INTEGRAL- REASONING
As we know, the Euler characteristic is an additive operator oncompact sets:
which reminds us of a measure
That is why it seems reasonable to define an integral with respect to this “measure”.
BABABA
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
We continue to derive the following Morse like expression for the integral:
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
Under this metric the L-K curvatures are computed in the GKF, and in it the manifold M is bounded
0 Where
,
xfExm
tfsfEtmtfsmsfEtsC TT
When taking we receive and:
Now we are interested in computing the Euler integral of a Gaussian random field:
Let M be a stratified space and let 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.
0i **0
DmMDfE jj
M
j
j
2 dim
12
1
Mg :
Theorem: Let M be a compact d-dimensional stratified space, and let be a k-dimensional Gaussian random field satisfying the GKF conditions. For piecewise c² functionlet setting , we have
kMf :
kG :.fGg
uGDu ,1
mean.constant a has Where
2 1
2
g(t)g
duDmMgEMdgE ujj
d
j
j
M
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:
dxxxgxfgf
ex
xdx
dxxH
GGsignHMgEMdgE
x
n
nn
n
j
jj
j
d
j
j
M
,
2
1
2
)(,1
22/1
1
2
1
1
2
And if G is strictly monotone:
d
j j
j
j
M
j
j
j
d
j
j
M
GHMdgE
GHMdgE
02
20
2
, :Decreasing
2
,1 :Increasing
WEIGHTED SUM OF CRITICAL VALUES
Taking in Theorem and using Proposition yields the following compact formula
xxHxG 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 , a 1-
dimensional measure of M and not the
volume , as one might have expected. Remark: if we scale the metric by , then
scales by
M1
Md
Md
k
INTRODUCTION : PERSISTENT HOMOLOGY
Have a single topological space, X
Get a chain complex
For k=0, 1, 2, …
compute Hk(X)Hk=Zk/Bk
Ck(X) C1(X) C0(X)Ck-1(X) 0∂ ∂∂ ∂ ∂ ∂
The Usual Homology
INTRODUCTION : PERSISTENT HOMOLOGY
Persistent homology is a way of tracking how the homology of a sequence of spaces changes
Given a filtration of spaces such that
if s < t, the persistent homology of , ,consists of families of homology classes that ‘persist’ through time.
uuXts XX
*PH
INTRODUCTION : PERSISTENT HOMOLOGY
o Explicitly an element of is a family of homology classesfor
o The map induced by the inclusion , maps
kPH t tkt XHbat where,],[
tksk XHXH ts XX
. to ts
Given a tame function
.in elements of sdeath time andbirth tocorespond will valuescritical
hen thefunction t Morse a is iffor theory,Morse oftion generaliza a
asseen becan :function tamea ofhomology persistent The
.,
., spaces of filtration associatedan is there,:
*
1**
1
fPH
f
Xf
ufPHfPH
ufXf
INTRODUCTION : PERSISTENT HOMOLOGY
X1 X2 X3 …Xn
a b a b
cd
a b
cd
a b
cd
a b
cd
a b
cd
a, b c, d, ab,bc cd, ad ac abc acd
t=0 t=1 t=2 t=3 t=4 t=5
A filtration of spaces (Simplicial complexes example):
THE EXPECTED EULER CHARACTERISTIC OF THE PERSISTENT HOMOLOGY OF AGAUSSIAN 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
Computing 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 gE
DEFINITIONS
A “tame” function :
finite. always is ,f of ypehomotopy t theif is
sticcharacteriEuler finite a with X space ltopologica
compact aon X : ffunction continuousA
1- u
tame