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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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Transcript of "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
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