institute for mathematics and its applications...
TRANSCRIPT
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.