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Hadwiger Integration and Applications

Hadwiger Integration and Applications. Matthew Wright Institute for Mathematics and its Applications University of Minnesota November 22, 2013. A valuation is a notion of size. Let be a collection of subsets of . A valuation on is a function such that whenever .

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Hadwiger Integration and Applications

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  1. Hadwiger Integration and Applications Matthew Wright Institute for Mathematics and its Applications University of Minnesota November 22, 2013

  2. A valuationis a notion of size. Let be a collection of subsets of . A valuation on is a function such that whenever . additive property

  3. Valuation is Euclidean-invariant if is unchanged by rigid motions of . Valuation is continuous if a small change in results in a small change in . rigid motion

  4. Example: volume • volume : sets • additive, Euclidean-invariant, continuous Example: Euler characteristic Let be a finite simplicial complex containing opensimplices of dimension . number of vertices of number of edges of number of faces of etc. The Euler characteristic of is: v combinatorial

  5. The intrinsic volumes generalize both volume and Euler characteristic. -dimensional volume half of the -dimensional surface area The intrinsic volumes are continuous, Euclidean-invariant valuations on “tame” subsets of . gives a -dimensional notion of size “mean width” Euler characteristic

  6. Example: Let be an -dimensional closed box with side lengths . The thintrinsic volume of is , the elementary symmetric polynomial of degree on variables.

  7. Definition: For a “tame” set , the th intrinsic volume can be defined: Hadwiger’s Formula is the affine Grassmanian of –dimensional planes in , and is Harr measure on with appropriate normalization.

  8. Let be compact and convex. What is the area of a tube of radius around ? Euler charac-teristic of perimeter of area of tube area of The area of tube around is a polynomial in whose coefficients involve the intrinsic volumes of .

  9. In general: the volume of a tube around is a polynomial in , whose coefficients involve intrinsic volumes of . Steiner Formula: For compact convex and , volume of unit -ball intrinsic volume

  10. Hadwiger’s Theorem says that the intrinsic volumes form a basis for the vector space of valuations on . Hadwiger’s Theorem: If is a continuous, Euclidean-invariant valuation on “tame” subsets of , then for some constants .

  11. If you understand the intrinsic volumes, then you understand all notions of size for sets in . WHAT IS THE LENGTH OF A POTATO?

  12. Example: How big is this set? volume: 763 (cm3) or any linear combination of these area: 489 (cm2) length: 65 (cm) number: 3 (potatoes)

  13. What valuations exist on functions on ? A valuationon functions is an additive map “tame” functions on . In this case, additivemeans that . Equivalently, for any subset and its complement . pointwise max pointwise min

  14. Valuation is Euclidean-invariant if for any Euclidean motion of . Valuation is continuous if a small change in results in a small change in .

  15. Lebesgue integral is a map: {functions} . Example: Lebesgue integral Lebesgue integral is: Additive Euclidean-invariant Continuous

  16. Example: Euler integral Let be a “tame” set in , and let be the function with value 1 on set and 0 otherwise. The Euler Integral of is: For a “tame” function , with finite range, set on which

  17. Concrete example: consider 3 2 1 Euler integral of

  18. What is the Euler integral of a continuous function ? Idea: Approximate by step functions. Make the step size smaller. Consider the limit of the Euler integrals of the approximations as the step size goes to zero: 3 ∙ 2 1 Does it matter if we use lower or upper approximations?

  19. Lower Euler integral: Definition: For a function , we define two “dual” Euler integrals: Upper integral: These limits exist, but are not equal in general.

  20. Theorem: The lower and upper Euler integrals can be expressed in terms of Riemann integrals: lower: upper:

  21. Example: Consider : 3 2 The Euler integral depends only on the critical points of . 1

  22. In general: Suppose is a Morse function. Let be the set of critical points of , and let denote the Morse index of a critical point . Example: Morse index: 1 0 2 Then:

  23. The Euler integral is a valuation on functions. Additive: Euclidean-invariant: Euclidean motions of the domain of do not change . Continuous: A “small” change to results in a “small” change to .

  24. We can also define integrals with respect to the intrinsic volumes; we call these Hadwiger integrals. Let have finite range. Integration of with respect to is straightforward: Integration of is more complicated: Lower integral: Upper integral:

  25. Hadwiger integrals can be expressed in various ways: Let be compact and bounded. ∞ s = 0 X slices level sets ℝ ℝ

  26. Example: Let on . Excursion set is a circle of radius . ℝ Hadwiger Integrals:

  27. We obtain a functional version of the classic Hadwiger Theorem: Theorem: Any Euclidean-invariant, continuous valuation on “tame” functions can be written for some increasing functions . Baryshnikov, Ghrist, Wright That is, any valuation on functions can be written as a sum of Hadwiger integrals.

  28. How can we measure the size of a function? and

  29. Applications and Challenges

  30. Suppose we have a set of objects and a function that counts the number of objects at each point. 1 Hadwiger integrals provide data about the set of objects: gives a count gives a “length” gives an “area” etc. 3 1 2 3 4 0 2 2 1 0 2 1 0 2 4 1 1 1 3 3 2 3 2 2 2 1 1 1 0

  31. Example: Euler integral of compute: 1 3 1 6 2 3 4 0 3 2 2 2 1 0 2 1 0 2 1 4 1 1 1 3 3 7 2 3 4 2 2 sum 2 1 5 1 1 0 There are 7 objects!

  32. Proposition: For subsets of and counting function , the Hadwiger integral with respect to gives the sum of the intrinsic volumes . That is: Proof: In particular, if is constant for all , then:

  33. These sets each have Euler characteristic : Therefore: Application: Euler integration is useful for target enumeration (Baryshnikov and Ghrist).

  34. Can we count by sensors? ???

  35. Theorem: For a “tame” function , where denotes the zerothBetti number, the number of connected components of the set. For compact and nonempty: Proof: by homological definition of Euler characteristic by Alexander duality With , we obtain:

  36. Computation:

  37. How can we compute the other Hadwiger integrals in a network context? • This is challenging if because: • cannot be reduced to connectivity • we need metric information to compute • cannot be reduced to critical values of • computing given a discrete sampling of is difficult

  38. Challenge: A change in a function on a small set (in the Lebesgue) sense can result in a large change in the Hadwiger integrals of . Similar examples exist for higher-dimensional Hadwiger integrals. Working with Hadwiger integrals requires different intuition than working with Lebesgue integrals.

  39. Challenge: How can we approximate the Hadwiger integrals of a function sampled at discrete points? triangulated approximations of Hadwiger integrals of interpolations of might diverge, even when the approximations converge pointwise to .

  40. Challenge: Non-linearity of Hadwiger integrals of continuous functions. Consider the following Euler integrals: [0, 1] [0, 1] [0, 1]

  41. Application: image processing Intrinsic volumes are of utility in image processing. A greyscale image can be viewed as a real-valued function on a planar domain. With such a perspective, Hadwiger integrals may be useful to return information about an image. Applications may also include color or hyperspectralimages, or images on higher-dimensional domains. K. Schladitz, J. Ohser, and W. Nagel. “Measuring Intrinsic Volumes in Digital 3d Images.” Discrete Geometry for Computer Images. Springer, 2006.

  42. Application: cell dynamics MacPherson and Srolovitz: As the cell structure changes by a certain process that minimizes energy, cell volumes change according to: -dimensional structure -dimensional structure

  43. References • Yuliy Baryshnikov and Robert Ghrist. “Target Enumeration via Euler Characteristic Integration.” SIAM Journal on Applied Mathematics, vol. 70, no. 3 (2009), p. 825–844. • Yuliy Baryshnikov and Robert Ghrist. “Definable Euler integration.” Proceedings of the National Academy of Sciences. vol. 107, no. 21 (2010), p. 9525–9530. • Yuliy Baryshnikov, Robert Ghrist, and Matthew Wright. “Hadwiger’s Theorem for Definable Functions.” Advances in Mathematics. vol. 245 (2013), p. 573–586. • S. H. Shanuel. “What is the Length of a Potato?” Lecture Notes in Mathematics. Springer, 1986, p. 118–126. • Matthew Wright. “HadwigerIntegration of Definable Functions.” Publicly accessible Penn Dissertations.Paper 391. http://repository.upenn.edu/edissertations/391.

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