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robert ghrist university of pennsylvania depts. of mathematics & electrical/systems engineering euler calculus & data

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motivation

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tools

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euler calculus

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χ = Σ (-1) k # { k-cells } k χ = 2 χ = 7 χ = 3 χ = 2 χ = 3 euler calculus χ = Σ (-1) k rank H k k

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sheaves

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lemma: [classical] χ (AuB) = χ (A)+ χ (B) – χ (A B) u

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u ∫ h d χ geometry probability topology networks kashiwara macpherson schapira viro blaschke hadwiger rota chen adler taylor

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results 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 {S n } n=1,2,... of boolean algebras of sets in R n closed under projections, products,... all functions in CF(X) are of the form h = Σ c i 1 U i for U i definable elements of {S n } 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 tools explicit definition: euler integral ∫ h d χ = ∫ ( Σ c i 1 U i ) d χ = Σ ( ∫ c i 1 U i ) d χ = Σ c i χ (U i ) integration

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[schapira, 1980’s; via kashiwara, macpherson, 1970’s] the induced pushforward on sheaves of constructible functions is the correct way to understand d χ F*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... XY CF(X)CF(Y) F X pt CF(X)CF( pt )=Z ∫ d χ corollary: [schapira, viro; 1980’s] fubini theorem F*F* XY CF(X)CF(Y) F pt CF( pt )=Z ∫ d χ sheaf-theoretic constructions also give natural convolution operators, duality, integral transforms,... integration

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a network of “minimal” sensors returns target counts without IDs how many targets are there? = 0= 1= 2= 3= 4 problem

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theorem: [BG] assuming target supports with uniform χ (U i )=N # targets = ( 1/N ) ∫ X h d χ trivial proof: ∫ h d χ = ∫ ( Σ 1 U i ) d χ = Σ ( ∫ 1 U i d χ ) = Σ χ ( U i ) = 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 U i in X sensor field on X returns h(x) = #{ i : x lies in U i } amazingly, one needs no convexity, no leray (“good cover”) condition, etc. this is a purely topological result. h:X → Z 2 N ≠ 0 counting

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for h in CF(X), integrals with respect to d χ are computable via ∫ h d χ = 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

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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 d χ example

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some applications in minimal sensing

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17 the resulting target impacts are still nullhomotopic (no echoing) 3 booms… whuh? 2 booms… consider a sensor modality which counts each wavefronts and increments an internal counter: used to count # events accurate event counts obtained via ad hoc network of acoustic sensors with no clocks, no synchronization, and no localization waves

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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 d χ wheels

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supports are the projected image of a contractible subset in space-time recall: ∫ X h(x) d χ (x) = ∫ Y F * h(y) d χ (y) F * h(y) = ∫ F -1 (y) h(x) d χ (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*F* XY CF(X)CF(Y) F pt Z ∫ d χ wheels

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numerical integration

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theorem: [BG] if the function h:R 2 → 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~~
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eucharis

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get real…

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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, ∫ _ d χ ● & ∫ _ d χ ● are no longer homomorphisms Def(X)→R take a riemann-sum approach ∫ h d χ ● = lim 1/n ∫ floor(nh) d χ ∫ h d χ ● = lim 1/n ∫ ceil(nh) d χ however, ∫ _ d χ ● & ∫ _ d χ ● have an interpretation in o-minimal category if h is affine on an open k-simplex, then ∫ h d χ ● = (-1) k inf (h) ∫ h d χ ● = (-1) k sup (h) h lemma real-valued integrands

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I *, I * : Def(X)→CF(X) intuition: the two measures correspond to the stratified morse indices of the graph of h in Def(X) with respect to two graph axis directions… ∫ h d χ ∙ = Σ (-1) n- μ (p) h(p) crit(h) = Σ (-1) μ (p) h(p) crit(h) μ = morse index ∫ h d χ ∙ corollary: [BG] if h : X → R is morse on an n-manifold, then ∫ h d χ ∙ = ∫ h I * h d χ theorem: [BG] for h in Def(X) real-valued integrands ∫ h d χ ∙ = ∫ h I * h d χ corollary: [BG] if h is univariate, then ∫ h d χ ∙ = totvar(h)/2 = - ∫ h d χ ∙

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Blaschke, Hadwiger,... ∫ h d χ ● = ∫ R χ {h≥s} - χ {h<-s} ds ∫ h d χ ● = ∫ R χ {h>s} - χ {h≤-s} ds ∫ h d χ ● = lim ε→0+ ∫ R s χ {s ≤ h < s+ε} ds ∫ h d χ ● = lim ε→0+ ∫ R s χ {s < h ≤ s+ε} ds Lebesgue Morse ∫ h d χ ● = Σ (-1) n- μ (p) h(p) crit(h) ∫ h d χ ● = Σ (-1) μ (p) h(p) crit(h) ∫ h d χ ● = - ∫ - h d χ ● (Dh)(x) = lim ε→0+ ∫ h 1 B(ε,x) d χ Duality D(Dh) = h ∫ X h d χ ● (x) = ∫ Y ∫ {F(x)=y} h(x) d χ ● (x)d χ ● (y) Fubini F:X→Y with h∙F=h real-valued integrands

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consider the following relative problem: given h on the complement of a hole D, estimate ∫ h d χ over the entire domain reminder: f < g does not imply that ∫ f d χ < ∫ g d χ...in this case the opposite occurs… theorem: [BG] for h:R 2 → Z a sum of indicator functions over homotopically trivial supports, none of which lies entirely within a contractible hole D, then ∫ R 2 h d χ ≤ ∫ R 2 h d χ ≤ ∫ R 2 h d χ h = fill in D with maximum of h on ∂ D h = fill in D with minimum of h on ∂ D D incomplete data

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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:R 2 → 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 ∫ R 2 h d χ ≤ ∫ R 2 f d χ ≤ ∫ R 2 h d χ incomplete data

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in practice, harmonic extensions lead to non-integer target counts this is an “expected” target count ∫ h d χ = 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

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integral transforms

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W X S sensing relations ∫ X h d χ = N ∫ W 1 T d χ = N #T h = integral transform of 1 T with kernel S

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fourier transform

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radon transform

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bessel transform

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eucharis

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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

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topological network topology

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closing credits… research sponsored by professional support a.j. friend, stanford university of pennsylvania a. mitchell darpa (stomp program) national science foundation office of naval research primary collaboratoryuliy baryshnikov, bell labs java codedavid lipsky, uillinois, urbana naveen kasthuri, penn work in progress withmichael robinson, penn matthew wright, penn

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