Matthew WrightInstitute for Mathematics and its Applications

University of Minnesota

Applied Topology in BędlewoJuly 24, 2013

How can we assign a notion of size to functions?

Lebesgue integral

Anything else?

Euler Characteristic

Let 𝑋 be a finite simplicial complex containing 𝐴𝑖open simplices of dimension 𝑖.

𝐴0 = number of vertices of 𝐴

𝐴1 = number of edges of 𝐴

𝐴2 = number of faces of 𝐴

etc.

Then the Euler Characteristic of 𝐴 is:

𝜒 𝐴 =

𝑖

−1 𝑖𝐴𝑖combinatorial

v

Key Property

For sets 𝐴 and 𝐵,

𝜒 𝐴 ∪ 𝐵 = 𝜒 𝐴 + 𝜒 𝐵 − 𝜒 𝐴 ∩ 𝐵 .

This property is called additivity, or the inclusion-exclusion principle.

𝐴 𝐵𝐴 ∩ 𝐵

Euler Integral

Let 𝐴 be a “tame” set in ℝ𝑛, and let 𝟏𝐴 be the function with value 1 on set 𝐴 and 0 otherwise.

The Euler Integral of 𝟏𝐴 is:

ℝ𝑛𝟏𝐴 𝑑𝜒 = 𝜒(𝐴)

For a “tame” function 𝑓:ℝ𝑛 → ℤ, with finite range,

ℝ𝑛𝑓 𝑑𝜒 =

𝑐

𝑐 𝜒{𝑓 = 𝑐} .

set on which 𝑓 = 𝑐

Example

𝑓(𝑥)

1

2

3

𝑥

Consider 𝑓: ℝ → ℤ:

= 1 ⋅ 0

+ 2 ⋅ (−1)

+ 3 ⋅ 2

= 4

← 𝑐 = 1

← 𝑐 = 2

← 𝑐 = 3

ℝ𝑛𝑓 𝑑𝜒 =

𝑐

𝑐 𝜒{𝑓 = 𝑐}

Euler integral of 𝑓

Continuous Functions

How can we extend the Euler integral to a continuous function 𝑓: ℝ → ℝ?

Idea: Approximate 𝑓 by step functions.

𝑓

1

𝑥

2

3Make the step size smaller.

Consider the limit of the Euler integrals of the approximations as the step size goes to zero:

lim𝑚→∞

1

𝑚 𝑚𝑓 𝑑𝜒

1

2∙ 2𝑓

𝑓

Does it matter if we use lower or upper approximations?

To extend the Euler integral to a function 𝑓: ℝ𝑛 → ℝ, define two integrals:

These limits exist, but are not equal in general.

Lower integral:

Upper integral:

Continuous Functions

𝑓 𝑑𝜒 = lim𝑚→∞

1

𝑚 𝑚𝑓 𝑑𝜒

𝑓 𝑑𝜒 = lim𝑚→∞

1

𝑚 𝑚𝑓 𝑑𝜒

Application

Euler Integration is useful in sensor networks:

• Networks of cell phones or computers

• Traffic sensor networks

• Surveillance and radar networks

LocalData

GlobalData

How can we assign a notion of size to functions?

Lebesgue integralEuler integral

Anything else?

The intrinsic volumes are the 𝑛 + 1 Euclidean-invariant valuations on subsets of ℝ𝑛, denoted 𝜇0, … , 𝜇𝑛.

Intrinsic Volumes

𝜇0: Euler characteristic

1 0

𝜇1: “length”

𝜇𝑛−1: ½(surface area)

𝜇𝑛: (Lebesgue) volume

𝑉 = 𝑙𝑤ℎ

Let 𝐾 be an 𝑛-dimensional closed box with side lengths 𝑥1, 𝑥2, … , 𝑥𝑛. The 𝑖th intrinsic volume of 𝐾 is 𝑒𝑖(𝑥1, 𝑥2, … , 𝑥𝑛), the elementary symmetric polynomial of degree 𝑖 on 𝑛 variables.

𝜇0 𝐾 = 𝑒0 𝑥1, … , 𝑥𝑛 = 1

𝜇1 𝐾 = 𝑒1 𝑥1, … , 𝑥𝑛= 𝑥1 + 𝑥2 +⋯+ 𝑥𝑛

𝜇2 𝐾 = 𝑒2(𝑥1, … , 𝑥𝑛)

= 𝑥1𝑥2 + 𝑥1𝑥3 +⋯+ 𝑥𝑛−1𝑥𝑛⋮

𝜇𝑛 𝐾 = 𝑒𝑛 𝑥1, … , 𝑥𝑛 = 𝑥1𝑥2⋯𝑥𝑛

𝑥1𝑥2

𝑥3

Example

For a “tame” set 𝐾 ⊂ ℝ, the 𝑘th intrinsic volume can be defined:

Hadwiger’s Formula

Intrinsic Volume Definition

𝐴𝑛,𝑛−𝑘 is the affine Grassmanian of (𝑛 − 𝑘)–dimensional planes in ℝ𝑛, and 𝜆 is Harr measure on 𝐴𝑛,𝑛−𝑘 with appropriate normalization.

𝜇𝑘 𝐾 = 𝐴𝑛,𝑛−𝑘

𝜒 𝐾 ∩ 𝑃 𝑑𝜆(𝑃)

Tube Formula

Steiner Formula: For compact convex 𝐾 ⊂ ℝ𝑛 and 𝑟 > 0,

𝜇𝑛(tube 𝐾, 𝑟 ) =

𝑗=0

𝑛

𝜔𝑛−𝑗 𝜇𝑗(𝐾)𝑟𝑛−𝑗

𝐾

tube(𝐾, 𝑟)

𝑟

The volume of a tube around 𝐾 is a

polynomial in 𝑟, whose coefficients

involve intrinsic volumes of 𝐾.

volume of unit (𝑛 − 𝑗)-ball intrinsic volume

Let 𝑓 ∶ ℝ𝑛 → ℤ have finite range. Integration of 𝑓 with respect to 𝜇𝑘 is straightforward:

ℝ𝑛𝑓 𝑑𝜇𝑘 =

𝑐

𝑐 𝜇𝑘{𝑓 = 𝑐}

Integration of 𝑓 ∶ ℝ𝑛 → ℝ is more complicated:

Lower integral:

Upper integral:

Hadwiger Integral

ℝ𝑛𝑓 𝑑𝜇𝑘 = lim

𝑚→∞

1

𝑚 ℝ𝑛𝑚𝑓 𝑑𝜇𝑘

ℝ𝑛𝑓 𝑑𝜇𝑘 = lim

𝑚→∞

1

𝑚 ℝ𝑛𝑚𝑓 𝑑𝜇𝑘

set on which 𝑓 = 𝑐

𝑓 𝑑𝜇𝑘 = 𝜇𝑘 𝑓 ≥ 𝑠 𝑑𝑠 = 𝑓 𝑑𝜒 𝑑𝛾

𝑓

level sets slices

s = 0

∞

𝐴𝑛,𝑛−𝑘 𝑃 ∩ 𝑋

𝑓

Let 𝑋 ⊆ ℝ𝑛 be compact and 𝑓 ∶ 𝑋 → ℝ+ bounded.

Hadwiger Integral

X

Example

𝑋

Let 𝑓 𝑥, 𝑦 = 4 − 𝑥2 − 𝑦2 on 𝑋 = 𝑥, 𝑦 | 𝑥2 − 𝑦2 ≤ 4 .

𝑋

𝑓 𝑑𝜇0 = 0

4

1 𝑑𝑠 = 4

𝑋

𝑓 𝑑𝜇1 = 0

4

𝜋 4 − 𝑠 𝑑𝑠 =16𝜋

3

𝑋

𝑓 𝑑𝜇2 = 0

4

𝜋(4 − 𝑠) 𝑑𝑠 = 8𝜋

𝑠

Excursion set 𝑓 ≥ 𝑠 is a circle

of radius 4 − 𝑠.

𝑓 Hadwiger Integrals:

Valuations on Functions

A valuation on functions is an additive map

𝑣 ∶ {“tame” functions on ℝ𝑛} → ℝ.

For a valuation on functions, additivity means

𝑣(𝑓 ∨ 𝑔) + 𝑣(𝑓 ∧ 𝑔) = 𝑣(𝑓 ) + 𝑣(𝑔),

or equivalently,𝑣(𝑓 ) = 𝑣(𝑓 ⋅ 𝟏𝐴) + 𝑣(𝑓 ⋅ 𝟏𝐴𝑐)

for any subset 𝐴 and its complement 𝐴𝑐.

pointwise max pointwise min

A valuation on functions is an additive map

𝑣 ∶ {“tame” functions on ℝ𝑛} → ℝ.

Valuation 𝑣 is:

• Euclidean-invariant if 𝑣(𝑓 ) = 𝑣(𝑓(𝜑)) for any Euclidean motion 𝜑 of ℝ𝑛.

• continuous if a “small” change in 𝑓corresponds to a “small” change in 𝑣(𝑓)(a precise definition of continuity requires a discussion of the flat topology on functions).

Valuations on Functions

Any Euclidean-invariant, continuous valuation 𝑣on “tame” functions can be written

𝑣 𝑓 =

𝑘=0

𝑛

ℝ𝑛𝑐𝑘 𝑓 𝑑𝜇𝑘

for some increasing functions 𝑐𝑘: ℝ → ℝ.

That is, any valuation on functions can be written as a sum of Hadwiger integrals.

Hadwiger’s Theorem for Functions(Baryshnikov, Ghrist, Wright)

How can we assign a notion of size to functions?

Lebesgue integralEuler integral

Hadwiger Integral

Any valuation on functions can be written in terms of Hadwiger integrals.

Surveillance

Suppose function 𝑓 counts the number of objects at each point in a domain.

Hadwiger integrals provide data about the set of objects:

𝑓 𝑑𝜇0 gives a count

𝑓 𝑑𝜇1 gives a “length”

𝑓 𝑑𝜇2 gives an “area”

etc.

2

2

2

2

23

3

3

1

1

1

1

1

1

0

0

00

2

𝑓

Cell Dynamics

As the cell structure changes by a certain process that minimizes energy, cell volumes change according to:𝑑𝜇𝑛𝑑𝑡𝐶 = −2𝜋𝑀 𝜇𝑛−2 𝐶𝑛 −

1

6𝜇𝑛−2(𝐶𝑛−2)

𝑛-dimensional structure (𝑛 − 2)-dimensional structure

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, Hadwigerintegrals may be useful to return information about an image.

Applications may also include color or hyperspectral images, or images on higher-dimensional domains.

Percolation

Functional approach: Define a permeability function in a solid material.

Hadwiger integrals may be useful in such a functional approach to percolation theory.

ℝ3

Question: Can liquid flow through a porous material from top to bottom?

SurveillanceLet 𝑓: 𝑇 → ℤ count objects locally in a domain 𝑇 ⊆ ℝ2.

What if part of 𝑇 is not observable?

Idea: Model the function with a random field. Estimate the global count via the expected Euler integral.

𝑇 𝑓 𝑑𝜇0 = 5

2

2

2

2

23

3

3

1

1

1

1

1

1

0

0

00

2

𝑓 Then the Euler integral gives the global count:

?

?

?

Random Field

Intuitively: A random field is a function whose value at any point in its domain is a random variable.

Formally: Let Ω,ℱ, ℙ be a probability space and 𝑇 a topological space. A measurable mapping 𝑓: Ω → ℝ𝑇 (the space of all real-valued functions on 𝑇) is called a real-valued random field.

Note: 𝑓(𝜔) is a function, (𝑓 𝜔 )(𝑡) is its value at 𝑡.

Shorthand: Let 𝑓𝑡 = (𝑓 𝜔 )(𝑡).

Expected Hadwiger Integral

Theorem: Let 𝑓 ∶ 𝑇 → ℝ𝑘 be a 𝑘-dimensional Gaussian field satisfying the conditions of the Gaussian Kinematic Formula. Let 𝐹 ∶ ℝ𝑘 → ℝ be a piecewise 𝐶2 function. Let 𝑔 = 𝐹 ∘ 𝑓, so 𝑔 ∶ 𝑇 → ℝ is a Gaussian-related field. Then the expected lower Hadwiger integral of 𝑔 is:

𝔼 𝑇

𝑔 𝑑𝜇𝑖 = 𝜇𝑖 𝑇 𝔼 𝑔 +

𝑗=1

dim 𝑇 −𝑖𝑖 + 𝑗𝑗2𝜋 −𝑗/2𝜇𝑖+𝑗 𝑇

ℝ

ℳ𝑗𝛾{𝐹 ≥ 𝑢} 𝑑𝑢

and similarly for the expected upper Hadwiger integral.

Computational DifficultiesComputing expected Hadwiger integrals of random fields is difficult in general.

𝔼 𝑇

𝑔 𝑑𝜇𝑖 = 𝜇𝑖 𝑇 𝔼 𝑔 +

𝑗=1

dim 𝑇 −𝑖𝑖 + 𝑗𝑗2𝜋 −𝑗/2𝜇𝑖+𝑗 𝑇

ℝ

ℳ𝑗𝛾{𝐹 ≥ 𝑢} 𝑑𝑢

intrinsic volumes: tricky, but possible to compute

Gaussian Minkowski functionals: very difficult to compute, except in special cases

Challenge: Non-Linearity

Consider the following Euler integrals:

𝑥 𝑑𝜒 = 1[0, 1]

(1 − 𝑥) 𝑑𝜒 = 1 1 𝑑𝜒 = 1[0, 1] [0, 1]

1

𝑥

𝑦 = 𝑥1

𝑥

𝑦 = 1 − 𝑥1

𝑥

𝑦 = 1

Upper and lower Hadwiger integrals are not linear in general.

Challenge: ContinuityA change in a function 𝑓 on a small set (in the Lebesgue) sense can result in a large change in the Hadwiger integrals of 𝑓.

𝑓 𝑑𝜒 = 1

𝑓

2

𝑥

1𝑔

2

𝑥

1

𝑔 𝑑𝜒 = 2

Working with Hadwiger integrals requires different intuition than working with Lebesgue integrals.

Similar examples exist for higher-

dimensional Hadwigerintegrals.

Challenge: ApproximationsHow can we approximate the Hadwiger integrals of a function sampled at discrete points?

Hadwiger integrals of interpolations of 𝑓 might diverge, even when the approximations converge pointwise to 𝑓.

𝑓: 0,1 2 → ℝ triangulated approximations of 𝑓

Summary

• The intrinsic volumes provide notions of size for sets, generalizing both Euler characteristic and Lebesguemeasure.

• Analogously, the Hadwiger integrals provide notions of size for real-valued functions.

• Hadwiger integrals are useful in applications such as surveillance, sensor networks, cell dynamics, and image processing.

• Hadwiger integrals bring theoretical and computational challenges, and provide many open questions for future study.

References

• Yuliy Baryshnikov and Robert Ghrist. “Target Enumeration via Euler Characteristic Integration.” SIAM J. Appl. Math. 70(3), 2009, 825–844.

• Yuliy Baryshnikov and Robert Ghrist. “Definable Euler integration.” Proc. Nat. Acad. Sci. 107(21), 2010, 9525-9530.

• Yuliy Baryshnikov, Robert Ghrist, and Matthew Wright. “Hadwiger’s Theorem for Definable Functions.” Advances in Mathematics. Vol. 245 (2013) p. 573-586.

• Omer Bobrowski and Matthew Strom Borman. “Euler Integration of Gaussian Random Fields and Persistent Homology.” Journal of Topology and Analysis, 4(1), 2012.

• S. H. Shanuel. “What is the Length of a Potato?” Lecture Notes in Mathematics. Springer, 1986, 118 – 126.

• Matthew Wright. “Hadwiger Integration of Definable Functions.” Publicly accessible Penn Dissertations. Paper 391. http://repository.upenn.edu/edissertations/391.