Introduction Definition Structure Question References

Relative Critical Sets: Structure and application

Dr. Jason Miller

Truman State University

8 January 2009

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

About the Talk

1 Introduction

2 Definition

3 What’s known

4 Question

5 References

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

The concept of d-dimensional relative critical set generalizes theconcept of (zero dimensional) critical point of a differentiablefunction.

Let U ⊂ Rn and f : U −→ R be a smooth function. Let x ∈ U.

Let H(f ) be the Hessian of f , λi ≤ λi+1 its eigenvalues and ei aunit eigenvector for λi so that {ei}

n

i=1 an orthonormal basis of Rn.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

The concept of d-dimensional relative critical set generalizes theconcept of (zero dimensional) critical point of a differentiablefunction.

Let U ⊂ Rn and f : U −→ R be a smooth function. Let x ∈ U.

Let H(f ) be the Hessian of f , λi ≤ λi+1 its eigenvalues and ei aunit eigenvector for λi so that {ei}

n

i=1 an orthonormal basis of Rn.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

The concept of d-dimensional relative critical set generalizes theconcept of (zero dimensional) critical point of a differentiablefunction.

Let U ⊂ Rn and f : U −→ R be a smooth function. Let x ∈ U.

Let H(f ) be the Hessian of f , λi ≤ λi+1 its eigenvalues and ei aunit eigenvector for λi so that {ei}

n

i=1 an orthonormal basis of Rn.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

Critical Set, v.1

The x is a critical point iff ∇f = 0 at x .

Alternatively...

Critical Set, v.2

The x is a critical point iff, at x , ∇f · ei = 0 for all i .

If we specify that λn < 0 at x , then x is a local maximum.

Structure

Generically, a function’s critical set is a set of isolated points.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

Critical Set, v.1

The x is a critical point iff ∇f = 0 at x .

Alternatively...

Critical Set, v.2

The x is a critical point iff, at x , ∇f · ei = 0 for all i .

If we specify that λn < 0 at x , then x is a local maximum.

Structure

Generically, a function’s critical set is a set of isolated points.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

Critical Set, v.1

The x is a critical point iff ∇f = 0 at x .

Alternatively...

Critical Set, v.2

The x is a critical point iff, at x , ∇f · ei = 0 for all i .

If we specify that λn < 0 at x , then x is a local maximum.

Structure

Generically, a function’s critical set is a set of isolated points.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

Critical Set, v.1

The x is a critical point iff ∇f = 0 at x .

Alternatively...

Critical Set, v.2

The x is a critical point iff, at x , ∇f · ei = 0 for all i .

If we specify that λn < 0 at x , then x is a local maximum.

Structure

Generically, a function’s critical set is a set of isolated points.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

Critical Set, v.1

The x is a critical point iff ∇f = 0 at x .

Alternatively...

Critical Set, v.2

The x is a critical point iff, at x , ∇f · ei = 0 for all i .

If we specify that λn < 0 at x , then x is a local maximum.

Structure

Generically, a function’s critical set is a set of isolated points.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

Let 0 < d < n.

0-dimensional Relative Critical Set

The x is a critical point iff, at x , ∇f · ei = 0 for all i .

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

Let 0 < d < n.

0-dimensional Relative Critical Set

The x is a critical point iff, at x , ∇f · ei = 0 for all i .

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

Let 0 < d < n.

d-dimensional Relative Critical Set

The x is a critical point iff, at x , ∇f · ei = 0 for i ≤ n − d .

If we specify that λn−d < 0 at x , the x is a point in the function’sd-dimensional height ridge.

Structure Question

What is the local generic structure of a function’s d-dimensionalridge in R

n (esp. near partial umbilics)?

The d = 1 dimensional height ridge has applications in imageanalysis, so knowing its generic structure is important.[PE+, PS, Ebe96]

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

Let 0 < d < n.

d-dimensional Relative Critical Set

The x is a critical point iff, at x , ∇f · ei = 0 for i ≤ n − d .

If we specify that λn−d < 0 at x , the x is a point in the function’sd-dimensional height ridge.

Structure Question

What is the local generic structure of a function’s d-dimensionalridge in R

n (esp. near partial umbilics)?

The d = 1 dimensional height ridge has applications in imageanalysis, so knowing its generic structure is important.[PE+, PS, Ebe96]

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

Let 0 < d < n.

d-dimensional Relative Critical Set

The x is a critical point iff, at x , ∇f · ei = 0 for i ≤ n − d .

If we specify that λn−d < 0 at x , the x is a point in the function’sd-dimensional height ridge.

Structure Question

What is the local generic structure of a function’s d-dimensionalridge in R

n (esp. near partial umbilics)?

The d = 1 dimensional height ridge has applications in imageanalysis, so knowing its generic structure is important.[PE+, PS, Ebe96]

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

Let 0 < d < n.

d-dimensional Relative Critical Set

The x is a critical point iff, at x , ∇f · ei = 0 for i ≤ n − d .

If we specify that λn−d < 0 at x , the x is a point in the function’sd-dimensional height ridge.

Structure Question

What is the local generic structure of a function’s d-dimensionalridge in R

n (esp. near partial umbilics)?

The d = 1 dimensional height ridge has applications in imageanalysis, so knowing its generic structure is important.[PE+, PS, Ebe96]

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

Theorem ([Dam99, Mil98])

Generically, the closure of the 1-dimensional ridge is

a discrete set of smooth embedded curves, that

has boundary points at partial umbilic points (λn−1 = λn) orat singular points (λn−1 = 0) of the Hessian.

Theorem ([Mil98])

Generically, the closure of the 2-dimensional ridge is

a discrete set of smooth embedded surfaces surfaces, that

has boundary curves at partial umbilic points (λn−2 = λn−1)or at singular points (λn−2 = 0) of the Hessian, and its

boundary is smooth except at a corner whereλn−2 = λn−1 = 0.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

Theorem ([Dam99, Mil98])

Generically, the closure of the 1-dimensional ridge is

a discrete set of smooth embedded curves, that

has boundary points at partial umbilic points (λn−1 = λn) orat singular points (λn−1 = 0) of the Hessian.

Theorem ([Mil98])

Generically, the closure of the 2-dimensional ridge is

a discrete set of smooth embedded surfaces surfaces, that

has boundary curves at partial umbilic points (λn−2 = λn−1)or at singular points (λn−2 = 0) of the Hessian, and its

boundary is smooth except at a corner whereλn−2 = λn−1 = 0.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

Theorem ([Dam99, Mil98])

Generically, the closure of the 1-dimensional ridge is

a discrete set of smooth embedded curves, that

has boundary points at partial umbilic points (λn−1 = λn) orat singular points (λn−1 = 0) of the Hessian.

Theorem ([Mil98])

Generically, the closure of the 2-dimensional ridge is

a discrete set of smooth embedded surfaces surfaces, that

has boundary curves at partial umbilic points (λn−2 = λn−1)or at singular points (λn−2 = 0) of the Hessian, and its

boundary is smooth except at a corner whereλn−2 = λn−1 = 0.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

Theorem ([Dam99, Mil98])

Generically, the closure of the 1-dimensional ridge is

a discrete set of smooth embedded curves, that

has boundary points at partial umbilic points (λn−1 = λn) orat singular points (λn−1 = 0) of the Hessian.

Theorem ([Mil98])

Generically, the closure of the 2-dimensional ridge is

a discrete set of smooth embedded surfaces surfaces, that

has boundary curves at partial umbilic points (λn−2 = λn−1)or at singular points (λn−2 = 0) of the Hessian, and its

boundary is smooth except at a corner whereλn−2 = λn−1 = 0.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

Theorem ([Dam99, Mil98])

Generically, the closure of the 1-dimensional ridge is

a discrete set of smooth embedded curves, that

has boundary points at partial umbilic points (λn−1 = λn) orat singular points (λn−1 = 0) of the Hessian.

Theorem ([Mil98])

Generically, the closure of the 2-dimensional ridge is

a discrete set of smooth embedded surfaces surfaces, that

has boundary curves at partial umbilic points (λn−2 = λn−1)or at singular points (λn−2 = 0) of the Hessian, and its

boundary is smooth except at a corner whereλn−2 = λn−1 = 0.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

Theorem ([Dam99, Mil98])

Generically, the closure of the 1-dimensional ridge is

a discrete set of smooth embedded curves, that

has boundary points at partial umbilic points (λn−1 = λn) orat singular points (λn−1 = 0) of the Hessian.

Theorem ([Mil98])

Generically, the closure of the 2-dimensional ridge is

a discrete set of smooth embedded surfaces surfaces, that

has boundary curves at partial umbilic points (λn−2 = λn−1)or at singular points (λn−2 = 0) of the Hessian, and its

boundary is smooth except at a corner whereλn−2 = λn−1 = 0.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

Theorem ([Dam99, Mil98])

Generically, the closure of the 1-dimensional ridge is

a discrete set of smooth embedded curves, that

has boundary points at partial umbilic points (λn−1 = λn) orat singular points (λn−1 = 0) of the Hessian.

Theorem ([Mil98])

Generically, the closure of the 2-dimensional ridge is

a discrete set of smooth embedded surfaces surfaces, that

has boundary curves at partial umbilic points (λn−2 = λn−1)or at singular points (λn−2 = 0) of the Hessian, and its

boundary is smooth except at a corner whereλn−2 = λn−1 = 0.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

This and related genericity result is established by

collecting closed submanifolds and stratified sets of jet spaceand then

using a set of mappings,

applying Thom’s Transversality Theorem to get the result.

Theorem (Thom’s Transversality Theorem)

For M and N smooth manifolds with Γ a submanifold of Jk(M,N),let

TΓ = {f ∈ C∞(M,N) | jk (f ) is transverse to Γ}.

Then TΓ is a residual subset of C∞(M,N) in the WhitneyC∞-topology. If Γ is closed, then TΓ is open.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

This and related genericity result is established by

collecting closed submanifolds and stratified sets of jet spaceand then

using a set of mappings,

applying Thom’s Transversality Theorem to get the result.

Theorem (Thom’s Transversality Theorem)

For M and N smooth manifolds with Γ a submanifold of Jk(M,N),let

TΓ = {f ∈ C∞(M,N) | jk (f ) is transverse to Γ}.

Then TΓ is a residual subset of C∞(M,N) in the WhitneyC∞-topology. If Γ is closed, then TΓ is open.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

This and related genericity result is established by

collecting closed submanifolds and stratified sets of jet spaceand then

using a set of mappings,

applying Thom’s Transversality Theorem to get the result.

Theorem (Thom’s Transversality Theorem)

For M and N smooth manifolds with Γ a submanifold of Jk(M,N),let

TΓ = {f ∈ C∞(M,N) | jk (f ) is transverse to Γ}.

Then TΓ is a residual subset of C∞(M,N) in the WhitneyC∞-topology. If Γ is closed, then TΓ is open.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

This and related genericity result is established by

collecting closed submanifolds and stratified sets of jet spaceand then

using a set of mappings,

applying Thom’s Transversality Theorem to get the result.

Theorem (Thom’s Transversality Theorem)

For M and N smooth manifolds with Γ a submanifold of Jk(M,N),let

TΓ = {f ∈ C∞(M,N) | jk (f ) is transverse to Γ}.

Then TΓ is a residual subset of C∞(M,N) in the WhitneyC∞-topology. If Γ is closed, then TΓ is open.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

This and related genericity result is established by

collecting closed submanifolds and stratified sets of jet spaceand then

using a set of mappings,

applying Thom’s Transversality Theorem to get the result.

Theorem (Thom’s Transversality Theorem)

For M and N smooth manifolds with Γ a submanifold of Jk(M,N),let

TΓ = {f ∈ C∞(M,N) | jk (f ) is transverse to Γ}.

Then TΓ is a residual subset of C∞(M,N) in the WhitneyC∞-topology. If Γ is closed, then TΓ is open.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

The boundary of the d-dimensional ridge inherits its geometryfrom that of the

geometry of the set of partial umbilic matrices (semialgebraic)

geometry of singular (algebraic)

as subsets of in S2R

n.

Theorem (The ”ℓ chose two” Test)

There is a closed semialgebraic set V (ℓ) ⊂ J2(n, 1) with theproperty that if d <

(

ℓ

2

)

and another transversality condition holds,then the closure of a d-dimensional ridge of f misses the partialumbilics of order ℓ.

Example

The 3-dimensional ridge fails this test for the partial umbilics oforder ℓ = 3.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

The boundary of the d-dimensional ridge inherits its geometryfrom that of the

geometry of the set of partial umbilic matrices (semialgebraic)

geometry of singular (algebraic)

as subsets of in S2R

n.

Theorem (The ”ℓ chose two” Test)

There is a closed semialgebraic set V (ℓ) ⊂ J2(n, 1) with theproperty that if d <

(

ℓ

2

)

and another transversality condition holds,then the closure of a d-dimensional ridge of f misses the partialumbilics of order ℓ.

Example

The 3-dimensional ridge fails this test for the partial umbilics oforder ℓ = 3.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

The boundary of the d-dimensional ridge inherits its geometryfrom that of the

geometry of the set of partial umbilic matrices (semialgebraic)

geometry of singular (algebraic)

as subsets of in S2R

n.

Theorem (The ”ℓ chose two” Test)

There is a closed semialgebraic set V (ℓ) ⊂ J2(n, 1) with theproperty that if d <

(

ℓ

2

)

and another transversality condition holds,then the closure of a d-dimensional ridge of f misses the partialumbilics of order ℓ.

Example

The 3-dimensional ridge fails this test for the partial umbilics oforder ℓ = 3.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

The boundary of the d-dimensional ridge inherits its geometryfrom that of the

geometry of the set of partial umbilic matrices (semialgebraic)

geometry of singular (algebraic)

as subsets of in S2R

n.

Theorem (The ”ℓ chose two” Test)

There is a closed semialgebraic set V (ℓ) ⊂ J2(n, 1) with theproperty that if d <

(

ℓ

2

)

and another transversality condition holds,then the closure of a d-dimensional ridge of f misses the partialumbilics of order ℓ.

Example

The 3-dimensional ridge fails this test for the partial umbilics oforder ℓ = 3.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

The boundary of the d-dimensional ridge inherits its geometryfrom that of the

geometry of the set of partial umbilic matrices (semialgebraic)

geometry of singular (algebraic)

as subsets of in S2R

n.

Theorem (The ”ℓ chose two” Test)

There is a closed semialgebraic set V (ℓ) ⊂ J2(n, 1) with theproperty that if d <

(

ℓ

2

)

and another transversality condition holds,then the closure of a d-dimensional ridge of f misses the partialumbilics of order ℓ.

Example

The 3-dimensional ridge fails this test for the partial umbilics oforder ℓ = 3.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

The boundary of the d-dimensional ridge inherits its geometryfrom that of the

geometry of the set of partial umbilic matrices (semialgebraic)

geometry of singular (algebraic)

as subsets of in S2R

n.

Theorem (The ”ℓ chose two” Test)

There is a closed semialgebraic set V (ℓ) ⊂ J2(n, 1) with theproperty that if d <

(

ℓ

2

)

and another transversality condition holds,then the closure of a d-dimensional ridge of f misses the partialumbilics of order ℓ.

Example

The 3-dimensional ridge fails this test for the partial umbilics oforder ℓ = 3.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

Part of the boundary of the 3-dimensional ridge will coincide withpartial umbilics of order 2 where λn−3 = λn−2

The ”ℓ chose two” Test implies the possibility that this part of theboundary also contains partial umbilics of order 3.

Question

Knowing the geometry of the set of umbilics of order 2 in a normalslice to the set of umbilics of order 3 will illuminate the boundarystructure of the 3-dimensional ridge. Analogous information forhigher umbilics could complete the structure theorem for ridges ofall dimension.

In [Arn72], Arnol’d remarks without proof that the structure is of acone over projective space.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

Part of the boundary of the 3-dimensional ridge will coincide withpartial umbilics of order 2 where λn−3 = λn−2

The ”ℓ chose two” Test implies the possibility that this part of theboundary also contains partial umbilics of order 3.

Question

Knowing the geometry of the set of umbilics of order 2 in a normalslice to the set of umbilics of order 3 will illuminate the boundarystructure of the 3-dimensional ridge. Analogous information forhigher umbilics could complete the structure theorem for ridges ofall dimension.

In [Arn72], Arnol’d remarks without proof that the structure is of acone over projective space.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

Part of the boundary of the 3-dimensional ridge will coincide withpartial umbilics of order 2 where λn−3 = λn−2

The ”ℓ chose two” Test implies the possibility that this part of theboundary also contains partial umbilics of order 3.

Question

Knowing the geometry of the set of umbilics of order 2 in a normalslice to the set of umbilics of order 3 will illuminate the boundarystructure of the 3-dimensional ridge. Analogous information forhigher umbilics could complete the structure theorem for ridges ofall dimension.

In [Arn72], Arnol’d remarks without proof that the structure is of acone over projective space.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

Part of the boundary of the 3-dimensional ridge will coincide withpartial umbilics of order 2 where λn−3 = λn−2

The ”ℓ chose two” Test implies the possibility that this part of theboundary also contains partial umbilics of order 3.

Question

Knowing the geometry of the set of umbilics of order 2 in a normalslice to the set of umbilics of order 3 will illuminate the boundarystructure of the 3-dimensional ridge. Analogous information forhigher umbilics could complete the structure theorem for ridges ofall dimension.

In [Arn72], Arnol’d remarks without proof that the structure is of acone over projective space.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

Part of the boundary of the 3-dimensional ridge will coincide withpartial umbilics of order 2 where λn−3 = λn−2

The ”ℓ chose two” Test implies the possibility that this part of theboundary also contains partial umbilics of order 3.

Question

Knowing the geometry of the set of umbilics of order 2 in a normalslice to the set of umbilics of order 3 will illuminate the boundarystructure of the 3-dimensional ridge. Analogous information forhigher umbilics could complete the structure theorem for ridges ofall dimension.

In [Arn72], Arnol’d remarks without proof that the structure is of acone over projective space.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

Part of the boundary of the 3-dimensional ridge will coincide withpartial umbilics of order 2 where λn−3 = λn−2

The ”ℓ chose two” Test implies the possibility that this part of theboundary also contains partial umbilics of order 3.

Question

Knowing the geometry of the set of umbilics of order 2 in a normalslice to the set of umbilics of order 3 will illuminate the boundarystructure of the 3-dimensional ridge. Analogous information forhigher umbilics could complete the structure theorem for ridges ofall dimension.

In [Arn72], Arnol’d remarks without proof that the structure is of acone over projective space.

millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application

Introduction Definition Structure Question References

V.I. Arnol’d.Modes and quasimodes.Funct. Anal. and Appl., 6(2):94–101, 1972.

James Damon.Properties of ridges and cores for two-dimensional images.Journal of Mathematical Imaging and Vision, 10:163–174,1999.

D. Eberly.Ridges in Image and Data Analysis, volume 7 of Series Comp.Imaging and Vision.Kluwer, 1996.

Jason Miller.Relative Critical Sets in R

n and Applications to ImageAnalysis.PhD thesis, University of North Carolina, 1998.

S. Pizer, D. Eberly, et al.millerj@truman.edu Truman State University

Relative Critical Sets: Structure and application