relative critical sets: structure and applications
DESCRIPTION
A talk at the 2009 Joint Mathematics Meeting in Washington, D.C., on relative critical sets and their properties. The talk ends with an open question whose answer will help extend our understanding of the local generic structure of relative critical sets.TRANSCRIPT
Introduction Definition Structure Question References
Relative Critical Sets: Structure and application
Dr. Jason Miller
Truman State University
8 January 2009
[email protected] 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
[email protected] 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.
[email protected] 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.
[email protected] 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.
[email protected] 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.
[email protected] 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.
[email protected] 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.
[email protected] 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.
[email protected] 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.
[email protected] 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 .
[email protected] 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 .
[email protected] 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]
[email protected] 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]
[email protected] 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]
[email protected] 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]
[email protected] 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.
[email protected] 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.
[email protected] 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.
[email protected] 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.
[email protected] 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.
[email protected] 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.
[email protected] 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.
[email protected] 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.
[email protected] 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.
[email protected] 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.
[email protected] 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.
[email protected] 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.
[email protected] 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.
[email protected] 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.
[email protected] 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.
[email protected] 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.
[email protected] 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.
[email protected] 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.
[email protected] 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.
[email protected] 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.
[email protected] 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.
[email protected] 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.
[email protected] 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.
[email protected] 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.
[email protected] 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 [email protected] Truman State University
Relative Critical Sets: Structure and application