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# Relative Critical Sets: Structure and applications

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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.

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### Relative Critical Sets: Structure and applications

1. 1. 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
2. 2. Introduction Deﬁnition Structure Question References About the Talk Introduction 1 Deﬁnition 2 What’s known 3 Question 4 References 5 millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
3. 3. Introduction Deﬁnition Structure Question References The concept of d-dimensional relative critical set generalizes the concept of (zero dimensional) critical point of a diﬀerentiable function. 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 a unit eigenvector for λi so that {ei }n=1 an orthonormal basis of Rn . i millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
4. 4. Introduction Deﬁnition Structure Question References The concept of d-dimensional relative critical set generalizes the concept of (zero dimensional) critical point of a diﬀerentiable function. 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 a unit eigenvector for λi so that {ei }n=1 an orthonormal basis of Rn . i millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
5. 5. Introduction Deﬁnition Structure Question References The concept of d-dimensional relative critical set generalizes the concept of (zero dimensional) critical point of a diﬀerentiable function. 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 a unit eigenvector for λi so that {ei }n=1 an orthonormal basis of Rn . i millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
6. 6. 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
7. 7. 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
8. 8. 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
9. 9. 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
10. 10. 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
11. 11. 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
12. 12. 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
13. 13. 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’s d-dimensional height ridge. Structure Question What is the local generic structure of a function’s d-dimensional ridge in Rn (esp. near partial umbilics)? The d = 1 dimensional height ridge has applications in image analysis, so knowing its generic structure is important. [PE+ , PS, Ebe96] millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
14. 14. 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’s d-dimensional height ridge. Structure Question What is the local generic structure of a function’s d-dimensional ridge in Rn (esp. near partial umbilics)? The d = 1 dimensional height ridge has applications in image analysis, so knowing its generic structure is important. [PE+ , PS, Ebe96] millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
15. 15. 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’s d-dimensional height ridge. Structure Question What is the local generic structure of a function’s d-dimensional ridge in Rn (esp. near partial umbilics)? The d = 1 dimensional height ridge has applications in image analysis, so knowing its generic structure is important. [PE+ , PS, Ebe96] millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
16. 16. 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’s d-dimensional height ridge. Structure Question What is the local generic structure of a function’s d-dimensional ridge in Rn (esp. near partial umbilics)? The d = 1 dimensional height ridge has applications in image analysis, so knowing its generic structure is important. [PE+ , PS, Ebe96] millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
17. 17. 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 ) or at 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 where λn−2 = λn−1 = 0. millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
18. 18. 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 ) or at 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 where λn−2 = λn−1 = 0. millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
19. 19. 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 ) or at 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 where λn−2 = λn−1 = 0. millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
20. 20. 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 ) or at 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 where λn−2 = λn−1 = 0. millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
21. 21. 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 ) or at 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 where λn−2 = λn−1 = 0. millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
22. 22. 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 ) or at 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 where λn−2 = λn−1 = 0. millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
23. 23. 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 ) or at 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 where λn−2 = λn−1 = 0. millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
24. 24. Introduction Deﬁnition Structure Question References This and related genericity result is established by collecting closed submanifolds and stratiﬁed sets of jet space and 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 J k (M, N), let TΓ = {f ∈ C ∞ (M, N) | j k (f ) is transverse to Γ}. Then TΓ is a residual subset of C ∞ (M, N) in the Whitney C ∞ -topology. If Γ is closed, then TΓ is open. millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
25. 25. Introduction Deﬁnition Structure Question References This and related genericity result is established by collecting closed submanifolds and stratiﬁed sets of jet space and 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 J k (M, N), let TΓ = {f ∈ C ∞ (M, N) | j k (f ) is transverse to Γ}. Then TΓ is a residual subset of C ∞ (M, N) in the Whitney C ∞ -topology. If Γ is closed, then TΓ is open. millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
26. 26. Introduction Deﬁnition Structure Question References This and related genericity result is established by collecting closed submanifolds and stratiﬁed sets of jet space and 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 J k (M, N), let TΓ = {f ∈ C ∞ (M, N) | j k (f ) is transverse to Γ}. Then TΓ is a residual subset of C ∞ (M, N) in the Whitney C ∞ -topology. If Γ is closed, then TΓ is open. millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
27. 27. Introduction Deﬁnition Structure Question References This and related genericity result is established by collecting closed submanifolds and stratiﬁed sets of jet space and 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 J k (M, N), let TΓ = {f ∈ C ∞ (M, N) | j k (f ) is transverse to Γ}. Then TΓ is a residual subset of C ∞ (M, N) in the Whitney C ∞ -topology. If Γ is closed, then TΓ is open. millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
28. 28. Introduction Deﬁnition Structure Question References This and related genericity result is established by collecting closed submanifolds and stratiﬁed sets of jet space and 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 J k (M, N), let TΓ = {f ∈ C ∞ (M, N) | j k (f ) is transverse to Γ}. Then TΓ is a residual subset of C ∞ (M, N) in the Whitney C ∞ -topology. If Γ is closed, then TΓ is open. millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
29. 29. Introduction Deﬁnition Structure Question References The boundary of the d-dimensional ridge inherits its geometry from that of the geometry of the set of partial umbilic matrices (semialgebraic) geometry of singular (algebraic) as subsets of in S 2 Rn . Theorem (The ”ℓ chose two” Test) There is a closed semialgebraic set V (ℓ) ⊂ J 2 (n, 1) with the ℓ property that if d < 2 and another transversality condition holds, then the closure of a d-dimensional ridge of f misses the partial umbilics of order ℓ. Example The 3-dimensional ridge fails this test for the partial umbilics of order ℓ = 3. millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
30. 30. Introduction Deﬁnition Structure Question References The boundary of the d-dimensional ridge inherits its geometry from that of the geometry of the set of partial umbilic matrices (semialgebraic) geometry of singular (algebraic) as subsets of in S 2 Rn . Theorem (The ”ℓ chose two” Test) There is a closed semialgebraic set V (ℓ) ⊂ J 2 (n, 1) with the ℓ property that if d < 2 and another transversality condition holds, then the closure of a d-dimensional ridge of f misses the partial umbilics of order ℓ. Example The 3-dimensional ridge fails this test for the partial umbilics of order ℓ = 3. millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
31. 31. Introduction Deﬁnition Structure Question References The boundary of the d-dimensional ridge inherits its geometry from that of the geometry of the set of partial umbilic matrices (semialgebraic) geometry of singular (algebraic) as subsets of in S 2 Rn . Theorem (The ”ℓ chose two” Test) There is a closed semialgebraic set V (ℓ) ⊂ J 2 (n, 1) with the ℓ property that if d < 2 and another transversality condition holds, then the closure of a d-dimensional ridge of f misses the partial umbilics of order ℓ. Example The 3-dimensional ridge fails this test for the partial umbilics of order ℓ = 3. millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
32. 32. Introduction Deﬁnition Structure Question References The boundary of the d-dimensional ridge inherits its geometry from that of the geometry of the set of partial umbilic matrices (semialgebraic) geometry of singular (algebraic) as subsets of in S 2 Rn . Theorem (The ”ℓ chose two” Test) There is a closed semialgebraic set V (ℓ) ⊂ J 2 (n, 1) with the ℓ property that if d < 2 and another transversality condition holds, then the closure of a d-dimensional ridge of f misses the partial umbilics of order ℓ. Example The 3-dimensional ridge fails this test for the partial umbilics of order ℓ = 3. millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
33. 33. Introduction Deﬁnition Structure Question References The boundary of the d-dimensional ridge inherits its geometry from that of the geometry of the set of partial umbilic matrices (semialgebraic) geometry of singular (algebraic) as subsets of in S 2 Rn . Theorem (The ”ℓ chose two” Test) There is a closed semialgebraic set V (ℓ) ⊂ J 2 (n, 1) with the ℓ property that if d < 2 and another transversality condition holds, then the closure of a d-dimensional ridge of f misses the partial umbilics of order ℓ. Example The 3-dimensional ridge fails this test for the partial umbilics of order ℓ = 3. millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
34. 34. Introduction Deﬁnition Structure Question References The boundary of the d-dimensional ridge inherits its geometry from that of the geometry of the set of partial umbilic matrices (semialgebraic) geometry of singular (algebraic) as subsets of in S 2 Rn . Theorem (The ”ℓ chose two” Test) There is a closed semialgebraic set V (ℓ) ⊂ J 2 (n, 1) with the ℓ property that if d < 2 and another transversality condition holds, then the closure of a d-dimensional ridge of f misses the partial umbilics of order ℓ. Example The 3-dimensional ridge fails this test for the partial umbilics of order ℓ = 3. millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
35. 35. Introduction Deﬁnition Structure Question References Part of the boundary of the 3-dimensional ridge will coincide with partial umbilics of order 2 where λn−3 = λn−2 The ”ℓ chose two” Test implies the possibility that this part of the boundary also contains partial umbilics of order 3. Question Knowing the geometry of the set of umbilics of order 2 in a normal slice to the set of umbilics of order 3 will illuminate the boundary structure of the 3-dimensional ridge. Analogous information for higher umbilics could complete the structure theorem for ridges of all dimension. In [Arn72], Arnol’d remarks without proof that the structure is of a cone over projective space. millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
36. 36. Introduction Deﬁnition Structure Question References Part of the boundary of the 3-dimensional ridge will coincide with partial umbilics of order 2 where λn−3 = λn−2 The ”ℓ chose two” Test implies the possibility that this part of the boundary also contains partial umbilics of order 3. Question Knowing the geometry of the set of umbilics of order 2 in a normal slice to the set of umbilics of order 3 will illuminate the boundary structure of the 3-dimensional ridge. Analogous information for higher umbilics could complete the structure theorem for ridges of all dimension. In [Arn72], Arnol’d remarks without proof that the structure is of a cone over projective space. millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
37. 37. Introduction Deﬁnition Structure Question References Part of the boundary of the 3-dimensional ridge will coincide with partial umbilics of order 2 where λn−3 = λn−2 The ”ℓ chose two” Test implies the possibility that this part of the boundary also contains partial umbilics of order 3. Question Knowing the geometry of the set of umbilics of order 2 in a normal slice to the set of umbilics of order 3 will illuminate the boundary structure of the 3-dimensional ridge. Analogous information for higher umbilics could complete the structure theorem for ridges of all dimension. In [Arn72], Arnol’d remarks without proof that the structure is of a cone over projective space. millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
38. 38. Introduction Deﬁnition Structure Question References Part of the boundary of the 3-dimensional ridge will coincide with partial umbilics of order 2 where λn−3 = λn−2 The ”ℓ chose two” Test implies the possibility that this part of the boundary also contains partial umbilics of order 3. Question Knowing the geometry of the set of umbilics of order 2 in a normal slice to the set of umbilics of order 3 will illuminate the boundary structure of the 3-dimensional ridge. Analogous information for higher umbilics could complete the structure theorem for ridges of all dimension. In [Arn72], Arnol’d remarks without proof that the structure is of a cone over projective space. millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
39. 39. Introduction Deﬁnition Structure Question References Part of the boundary of the 3-dimensional ridge will coincide with partial umbilics of order 2 where λn−3 = λn−2 The ”ℓ chose two” Test implies the possibility that this part of the boundary also contains partial umbilics of order 3. Question Knowing the geometry of the set of umbilics of order 2 in a normal slice to the set of umbilics of order 3 will illuminate the boundary structure of the 3-dimensional ridge. Analogous information for higher umbilics could complete the structure theorem for ridges of all dimension. In [Arn72], Arnol’d remarks without proof that the structure is of a cone over projective space. millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
40. 40. Introduction Deﬁnition Structure Question References Part of the boundary of the 3-dimensional ridge will coincide with partial umbilics of order 2 where λn−3 = λn−2 The ”ℓ chose two” Test implies the possibility that this part of the boundary also contains partial umbilics of order 3. Question Knowing the geometry of the set of umbilics of order 2 in a normal slice to the set of umbilics of order 3 will illuminate the boundary structure of the 3-dimensional ridge. Analogous information for higher umbilics could complete the structure theorem for ridges of all dimension. In [Arn72], Arnol’d remarks without proof that the structure is of a cone over projective space. millerj@truman.edu Truman State University Relative Critical Sets: Structure and application
41. 41. 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 Rn and Applications to Image Analysis. PhD thesis, University of North Carolina, 1998. millerj@truman.edu Truman State University S. Pizer, D. Eberly, et al. Relative Critical Sets: Structure and application