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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 j0 1 d j0 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.