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Least Squares Curves, Rational Representations, Splines and Continuity

Explore degree reduction, rational representations, and spline continuity concepts with Dr. Scott Schaefer. Learn about least squares curves and the pseudo-inverse. Dive into constrained least squares optimization and error functions in this comprehensive resource.

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Least Squares Curves, Rational Representations, Splines and Continuity

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  1. Least Squares Curves, Rational Representations, Splines and Continuity Dr. Scott Schaefer

  2. Degree Reduction • Given a set of coefficients for a Bezier curve of degree n+1, find the best set of coefficients of a Bezier curve of degree n that approximate that curve

  3. Degree Reduction

  4. Degree Reduction

  5. Degree Reduction

  6. Degree Reduction

  7. Degree Reduction

  8. Degree Reduction • Problem: end-points are not interpolated

  9. Least Squares Optimization

  10. Least Squares Optimization

  11. Least Squares Optimization

  12. Least Squares Optimization

  13. Least Squares Optimization

  14. Least Squares Optimization

  15. The PseudoInverse • What happens when isn’t invertible?

  16. The PseudoInverse • What happens when isn’t invertible?

  17. The PseudoInverse • What happens when isn’t invertible?

  18. The PseudoInverse • What happens when isn’t invertible?

  19. The PseudoInverse • What happens when isn’t invertible?

  20. The PseudoInverse • What happens when isn’t invertible?

  21. The PseudoInverse • What happens when isn’t invertible?

  22. The PseudoInverse • What happens when isn’t invertible?

  23. The PseudoInverse • What happens when isn’t invertible?

  24. The PseudoInverse • What happens when isn’t invertible?

  25. The PseudoInverse • What happens when isn’t invertible?

  26. The PseudoInverse • What happens when isn’t invertible?

  27. The PseudoInverse • What happens when isn’t invertible?

  28. The PseudoInverse • What happens when isn’t invertible?

  29. Constrained Least Squares Optimization

  30. Constrained Least Squares Optimization Solution Constraint Space Error Function F(x)

  31. Constrained Least Squares Optimization

  32. Constrained Least Squares Optimization

  33. Constrained Least Squares Optimization

  34. Constrained Least Squares Optimization

  35. Constrained Least Squares Optimization

  36. Least Squares Curves

  37. Least Squares Curves

  38. Least Squares Curves

  39. Least Squares Curves

  40. Degree Reduction • Problem: end-points are not interpolated

  41. Degree Reduction

  42. Degree Reduction

  43. Rational Curves • Curves defined in a higher dimensional space that are “projected” down

  44. Rational Curves • Curves defined in a higher dimensional space that are “projected” down

  45. Rational Curves • Curves defined in a higher dimensional space that are “projected” down

  46. Rational Curves • Curves defined in a higher dimensional space that are “projected” down

  47. Why Rational Curves? • Conics

  48. Why Rational Curves? • Conics

  49. Why Rational Curves? • Conics

  50. Why Rational Curves? • Conics

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