robert ghristuniversity of pennsylvaniadepts. of mathematics &

electrical/systems engineering

euler calculus & data

motivation

tools

euler calculus

χ = Σ (-1)k # {k-cells} k

χ = 2

χ = 7

χ = 3

χ = 2

χ = 3

euler calculus

χ = Σ (-1)k rank Hk k

sheaves

lemma: [classical]

χ(AuB) = χ(A)+ χ(B) – χ(A B)

u

χ(AuB) = χ(A)+ χ(B) – χ(A B)

u

∫ h dχ

geometry

probability

topology

networks

kashiwaramacpherson

schapiraviro

blaschkehadwigerrotachen

adlertaylor

resu

lts

axiomatic approach to tameness in the work on o-minimal structures

consider the sheaf of constructible functions

CF(X) = Z-valued functions whose level sets are locally finite and “tame”

collections {Sn}n=1,2,... of boolean algebras of sets in Rn closed under projections, products,...

all functions in CF(X) are of the form h = Σci1Ui for Ui definable

elements of {Sn}n=1,2,... are called “definable” or “tame” sets

all definable sets are triangulable & have a well-defined euler characteristic

all functions in CF(X) are integrable with respect to Euler characteristic

tool

s

explicit definition:

euler integral

∫ h dχ = ∫ (Σ ci1Ui) dχ = Σ(∫ ci1Ui

)dχ = Σci χ(Ui)

integration

[schapira, 1980’s; via kashiwara, macpherson, 1970’s]

the induced pushforward on sheaves of constructible functions is the correct way to understand dχ

F*

in the case where Y is a point, CF(Y)=Z, and the pushforward is a homomorphism from CF(X) to Z which respects all the gluings implicit in sheaves...

X Y

CF(X) CF(Y)

F

X pt

CF(X) CF(pt)=Z∫ dχ

corollary: [schapira, viro; 1980’s] fubini theorem

F*

X Y

CF(X) CF(Y)

Fpt

CF(pt)=Z∫ dχ

sheaf-theoretic constructions also give naturalconvolution operators, duality, integral transforms, ...

integration

a network of “minimal” sensors returns target counts without IDshow many targets are there?

= 0 = 1 = 2 = 3 = 4

problem

problem

theorem: [BG] assuming target supports with uniform χ(Ui)=N

# targets = (1/N) ∫X h dχ

trivial proof:

∫ h dχ = ∫ (Σ1Ui) dχ = Σ(∫ 1Ui

dχ) = Σ χ(Ui) = N # i

let W = “target space” = space where finite # of targets live

let X = “sensor space” = space which parameterizes sensors

target i is detected on a target support Ui in X

sensor field on X returns h(x) = #{ i : x lies in Ui }

amazingly, one needs no convexity, no leray (“good cover”) condition, etc.this is a purely topological result.

h:X→Z

2

N ≠ 0

counting

for h in CF(X), integrals with respect to dχ are computable via

∫ h dχ = Σ s χ({ h=s }) s=0

∞

= Σ χ({ h>s })-χ({ h<-s }) s=0

∞

= Σ h(V)χ(v) V

level set

upper excursion set

weighted euler index

“chambers” of h components of level sets

computation

h>3 : χ = 2

h>2 : χ = 3

h>1 : χ = 3

h>0 : χ = -1

net integral = 2+3+3-1 = 7

= Σ χ {h(x)>s}s=0

∞∫ h dχ

example

some applications in

minimal sensing

17

the resulting targetimpacts are stillnullhomotopic (no echoing)

3 booms…

whuh?

2 booms…

consider a sensor modality which counts each wavefronts andincrements an internal counter: used to count # events

accurate event counts obtained via ad hoc network of acoustic sensorswith no clocks, no synchronization, and no localization

waves

consider sensors which count passing vehicles and increment an internal counter

acoustic sensors embedded in roads…

such target impacts may not be contractible…

theorem: [BG] if sensors read h = the total number of time intervals in which

some vehicle is nearby, then # vehicles = ∫ h dχ

wheels

supports are the projected image of a contractible subset in space-time

recall:

∫X h(x) dχ(x) = ∫Y F*h(y) dχ(y)

F*h(y) = ∫F-1

(y) h(x) dχ(x)

let X = domain x time ; let Y = domain ; let F = temporal projection map then F*h(y) = total # of (compact) time intervals on which some vehicle is at/near point w

= sensor reading at y

F*

X Y

CF(X) CF(Y)

Fpt

Z∫ dχ

wheels

numerical integration

theorem: [BG] if the function h:R2→N is sampled over a network in a way that correctly samples the connectivity of upper and lower excursion sets, then the exact value of the euler integral of h is

Σ( #comp{ h≥s } - #comp{ h<s } + 1)s=1

∞

this is a simple application of alexander duality…

= Σ χ{ h ≥ s } s=1

∞

∫ h dχ = Σ b0 {h ≥ s } – b1{h ≥ s } s=1

∞

this works in ad hoc setting : clustering gives fast computation

= Σ b0{h ≥ s } – b0{h < s } + 1s=1

∞

~= Σ b0{h ≥ s } – b0{h < s }s=1

∞ χ = Σ (-1)k dim Hk

k

bk

ad hoc networks

eucharis

eucharis

eucharis

eucharis

eucharis

eucharis

eucharis

get real…

it’s helpful to have a well-defined integration theory for R-valued integrands:Def(X) = R-valued functions whose graphs are “tame” (definable in o-minimal)

unfortunately, ∫ _ dχ ● & ∫ _ dχ● are no longer homomorphisms Def(X)→R

take a riemann-sum approach

∫ h dχ● = lim 1/n∫ floor(nh) dχ ∫ h dχ● = lim 1/n∫ ceil(nh) dχ

however, ∫ _ dχ ● & ∫ _ dχ● have an interpretation in o-minimal category

if h is affine on an open k-simplex, then

∫ h dχ● = (-1)k inf (h) ∫ h dχ● = (-1)k sup (h)

h

lemma

real-valued integrands

I*, I* : Def(X)→CF(X)

intuition: the two measures correspond to the stratified morse indices ofthe graph of h in Def(X) with respect to two graph axis directions…

∫ h dχ∙ = Σ (-1)n-μ(p) h(p)

crit(h)

= Σ (-1)μ(p) h(p) crit(h)

μ =

mor

se in

dex

∫ h dχ∙

corollary: [BG] if h : X → R is morse on an n-manifold, then∫ h dχ∙ = ∫ h I*h dχ

theorem: [BG] for h in Def(X)

real-valued integrands

∫ h dχ∙ = ∫ h I*h dχ

corollary: [BG] if h is univariate, then ∫ h dχ∙ = totvar(h)/2 = - ∫ h dχ∙

∫ h dχ● = ∫R χ{h≥s} - χ{h<-s} ds ∫ h dχ● = ∫R χ{h>s} - χ{h≤-s} ds

∫ h dχ● = limε→0+∫R s χ{s ≤ h < s+ε} ds ∫ h dχ● = limε→0+∫R s χ{s < h ≤ s+ε} ds

Lebesgue

Morse

∫ h dχ● = Σ (-1)n-μ(p) h(p)

crit(h)

∫ h dχ● = Σ (-1)μ(p) h(p) crit(h)

∫ h dχ● = - ∫ - h dχ● (Dh)(x) = limε→0+∫ h 1B(ε,x) dχ

Duality

D(Dh) = h

∫X h dχ●(x) = ∫Y ∫ {F(x)=y} h(x) dχ

●(x)dχ

●(y)

Fubini

F:X→Y with h∙F=h

real-valued integrands

consider the following relative problem:

given h on the complement of a hole D,

estimate ∫ h dχ over the entire domain

reminder: f < g does not imply that ∫ f dχ < ∫ g dχ ...in this case the opposite occurs…

theorem: [BG] for h:R2→Z a sum of indicator functions over homotopically trivial supports, none of which lies entirely within a contractible hole D, then

∫R2 h dχ ≤ ∫R

2 h dχ ≤ ∫R2 h dχ

h = fill in D with maximum of h on ∂D h = fill in D with minimum of h on ∂D

D

incomplete data

but what to choose in between upper and lower bounds?

claim: a harmonic extension over a hole is a “best guess”...

the proof is surprisingly easy using morse theory:

theorem: [BG] For h:R2→Z a sum of indicator functions over homotopically trivial supports, none of which lies entirely within a contractible hole D, then

for f any “harmonic” extension of h over D (weighted average of h rel ∂D)

the integral over D is the heights of the maxima minus the heights of the saddles

a “harmonic” extension has no local maxima or minima within D... # saddles in D - # maxima on ∂D = χ(D)=1

∫R2 h dχ ≤ ∫R

2 f dχ ≤ ∫R2 h dχ

incomplete data

in practice, harmonic extensions lead to non-integer target counts

this is an “expected” target count

∫ h dχ = 1+1-c

weights for the laplacian can be chosen based on confidence of data

points toward a general theory of expected integrals

expected values

integral transforms

W

X

S

sensing relations

∫X h dχ = N ∫W 1T dχ = N #T

h = integral transform of 1T with kernel S

fourier transform

radon transform

bessel transform

eucharis

eucharis

eucharis

eucharis

eucharis

eucharis

eucharis

how to correct “side lobes” and energy loss in integral transforms?

open questions

what is the appropriate integration theory for multi-modal and logical-valued data?

how to efficiently compute integral transforms given discrete (sparse) data?

…and, well, numerical analysis in general

topological network topology

closing credits…

research sponsored by

professional support

a.j. friend, stanford

university of pennsylvaniaa. mitchell

darpa (stomp program)national science foundation

office of naval research

primary collaborator yuliy baryshnikov, bell labs

java code david lipsky, uillinois, urbana

naveen kasthuri, penn

work in progress with michael robinson, pennmatthew wright, penn