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Lecture 18: Fourier Decomposition, Circular Functions, Spherical Harmonics COMPSCI/MATH 290-04 Chris Tralie, Duke University 3/22/2016 COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har Announcements

  1. Lecture 18: Fourier Decomposition, Circular Functions, Spherical Harmonics COMPSCI/MATH 290-04 Chris Tralie, Duke University 3/22/2016 COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  2. Announcements ⊲ Midterms graded COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  3. Announcements ⊲ Midterms graded ⊲ Group Assignment 1 Graded, Art contest up online (great work!!) COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  4. Announcements ⊲ Midterms graded ⊲ Group Assignment 1 Graded, Art contest up online (great work!!) ⊲ Group Assignment 2 Due Wednesday 3/30 ⊲ Piazza hours 8-9PM week nights (student Piazza answers encouraged) COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  5. Announcements ⊲ Midterms graded ⊲ Group Assignment 1 Graded, Art contest up online (great work!!) ⊲ Group Assignment 2 Due Wednesday 3/30 ⊲ Piazza hours 8-9PM week nights (student Piazza answers encouraged) ⊲ T.A.s grading mini assignment 3 COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  6. Announcements ⊲ Midterms graded ⊲ Group Assignment 1 Graded, Art contest up online (great work!!) ⊲ Group Assignment 2 Due Wednesday 3/30 ⊲ Piazza hours 8-9PM week nights (student Piazza answers encouraged) ⊲ T.A.s grading mini assignment 3 ⊲ No more extra credit for now COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  7. Announcements ⊲ Midterms graded ⊲ Group Assignment 1 Graded, Art contest up online (great work!!) ⊲ Group Assignment 2 Due Wednesday 3/30 ⊲ Piazza hours 8-9PM week nights (student Piazza answers encouraged) ⊲ T.A.s grading mini assignment 3 ⊲ No more extra credit for now ⊲ Final project directions sent out, first milestone due Friday 4/8 ⊲ Ditching Wikipedia entry, final project now worth 30 % COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  8. Table of Contents ◮ 1D Fourier Decomposition / Circle Functions ⊲ 2D Fourier Modes ⊲ Spherical Harmonics COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  9. Why Fourier?? (Interlude) Hey Chris, isn’t this a course on 3D geometry?? Why Fourier??? COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  10. Why Fourier?? (Interlude) Hey Chris, isn’t this a course on 3D geometry?? Why Fourier??? ⊲ Most CS majors don’t know about it, but extremely important COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  11. Why Fourier?? (Interlude) Hey Chris, isn’t this a course on 3D geometry?? Why Fourier??? ⊲ Most CS majors don’t know about it, but extremely important ⊲ Picks up on “shape” in a different way COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  12. Why Fourier?? (Interlude) Hey Chris, isn’t this a course on 3D geometry?? Why Fourier??? ⊲ Most CS majors don’t know about it, but extremely important ⊲ Picks up on “shape” in a different way ⊲ Entry point into harmonic analysis, nonrigid surface statistics COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  13. Sinusoid Review f ( x ) = A cos ( ω x + φ ) f ( x ) = ( A cos ( φ )) cos ( ω x ) − ( A sin ( φ )) sin ( ω x ) COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  14. Sinusoid Review f ( x ) = A cos ( ω x + φ ) f ( x ) = ( A cos ( φ )) cos ( ω x ) − ( A sin ( φ )) sin ( ω x ) f ( x ) = a cos ( ω x ) + b sin ( ω x ) a 2 + b 2 , φ = tan − 1 ( b � A = a ) COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  15. Sinusoid Review f ( x ) = A cos ( ω x + φ ) f ( x ) = ( A cos ( φ )) cos ( ω x ) − ( A sin ( φ )) sin ( ω x ) f ( x ) = a cos ( ω x ) + b sin ( ω x ) a 2 + b 2 , φ = tan − 1 ( b � A = a ) In polar form f ( x ) = Ae i ( ω x + φ ) = Ae i φ e i ω x = A ( cos ( θ ) + i sin ( θ )) e i ω x COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  16. Sinusoid Example COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  17. Sinusoid Example COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  18. Sinusoid Example COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  19. Fourier Decomposition N − 1 � 2 π k � � 2 π k � � f [ n ] = a k cos N n + b k sin N n k = 0 � a 2 k + b 2 Amplitude at frequency index k is k COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  20. Fourier Decomposition: Gaussian Examples Show video frames COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  21. Fourier Decomposition: Gaussian Examples COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  22. Fourier Decomposition: Gaussian Examples COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  23. Fourier Decomposition: Ramp Example Show video frames COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  24. Fourier Decomposition: Ramp Example COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  25. Fourier Decomposition: Ramp Example COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  26. Fourier Decomposition: Ramp Example COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  27. Fourier Decomposition: Ramp Example Continuous shifting videos COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  28. Functions on The Circle Show circle wrap video COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  29. Functions on The Circle COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  30. Phase as a rotation g ( x ) = f ( x + φ ) Show video COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  31. Table of Contents ⊲ 1D Fourier Decomposition / Circle Functions ◮ 2D Fourier Modes ⊲ Spherical Harmonics COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  32. 2D Sinusoids (aka “Plane Waves”) f ( x , y ) = cos ( ω x x + ω y y + φ ) ω · � f ( x , y ) = cos ( � x + φ ) COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  33. 2D Sinusoids: Example f ( x , y ) = cos ( x + y ) , ω x = 1 , ω y = 1 COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  34. 2D Sinusoids: Example f ( x , y ) = cos ( 2 x + 2 y ) , ω x = 2 , ω y = 2 COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  35. 2D Sinusoids: Example f ( x , y ) = cos ( x + 2 y ) , ω x = 1 , ω y = 2 COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  36. 2D Sinusoids: Example f ( x , y ) = cos ( x ) , ω x = 1 , ω y = 0 COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  37. 2D Sinusoids: Example f ( x , y ) = cos ( 1 . 5 y ) , ω x = 0 , ω y = 1 . 5 COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  38. 2D Sinusoids: Interference Pattern f ( x , y ) = cos ( x + y ) + cos ( x − y ) COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  39. 2D Sinusoids: Interference Pattern f ( x , y ) = cos ( 2 x + y ) + cos ( 2 x − y ) COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  40. 2D Sinusoids: Interference Pattern f ( x , y ) = cos ( 3 x + 2 y ) + cos ( 3 x − 2 y ) COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  41. 2D Sinusoids: Interference Why is this happening? g ( x , y ) = cos ( ω x x + ω y y ) + cos ( ω x x − ω y y ) COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  42. 2D Sinusoids: Interference Why is this happening? g ( x , y ) = cos ( ω x x + ω y y ) + cos ( ω x x − ω y y ) g ( x , y ) = cos ( ω x x ) cos ( ω y y ) − sin ( ω x x ) sin ( ω y y ) + cos ( ω x x ) cos ( ω y y ) + sin ( ω x x ) sin ( ω y y ) COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  43. 2D Sinusoids: Interference Why is this happening? g ( x , y ) = cos ( ω x x + ω y y ) + cos ( ω x x − ω y y ) g ( x , y ) = cos ( ω x x ) cos ( ω y y ) − sin ( ω x x ) sin ( ω y y ) + cos ( ω x x ) cos ( ω y y ) + sin ( ω x x ) sin ( ω y y ) g ( x , y ) = 2 cos ( ω x x ) cos ( ω y y ) COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

  44. Table of Contents ⊲ 1D Fourier Decomposition / Circle Functions ⊲ 2D Fourier Modes ◮ Spherical Harmonics COMPSCI/MATH 290-04 Lecture 18: Fourier Decomposition, Circular Functions, Spherical Har

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