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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

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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

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Announcements

⊲ Midterms graded

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

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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

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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

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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

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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

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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

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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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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

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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

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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

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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

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Sinusoid Review

f(x) = A cos(ωx + φ) f(x) = (A cos(φ))cos(ωx) − (A sin(φ))sin(ωx)

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

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Sinusoid Review

f(x) = A cos(ωx + φ) f(x) = (A cos(φ))cos(ωx) − (A sin(φ))sin(ωx) f(x) = a cos(ωx) + b sin(ωx) A =

  • a2 + b2, φ = tan−1(b

a)

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

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Sinusoid Review

f(x) = A cos(ωx + φ) f(x) = (A cos(φ))cos(ωx) − (A sin(φ))sin(ωx) f(x) = a cos(ωx) + b sin(ωx) A =

  • a2 + b2, φ = tan−1(b

a) In polar form f(x) = Aei(ωx+φ) = Aeiφeiωx = A(cos(θ) + i sin(θ))eiωx

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

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Sinusoid Example

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

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Sinusoid Example

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

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Sinusoid Example

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

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Fourier Decomposition

f[n] =

N−1

  • k=0

ak cos 2πk N n

  • + bk sin

2πk N n

  • Amplitude at frequency index k is
  • a2

k + b2 k

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

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Fourier Decomposition: Gaussian Examples

Show video frames

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

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Fourier Decomposition: Gaussian Examples

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

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Fourier Decomposition: Gaussian Examples

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

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Fourier Decomposition: Ramp Example

Show video frames

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

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Fourier Decomposition: Ramp Example

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

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Fourier Decomposition: Ramp Example

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

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Fourier Decomposition: Ramp Example

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

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Fourier Decomposition: Ramp Example

Continuous shifting videos

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

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Functions on The Circle

Show circle wrap video

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

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Functions on The Circle

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

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Phase as a rotation g(x) = f(x + φ) Show video

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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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2D Sinusoids (aka “Plane Waves”)

f(x, y) = cos(ωxx + ωyy + φ) f(x, y) = cos( ω · x + φ)

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

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2D Sinusoids: Example

f(x, y) = cos(x + y), ωx = 1, ωy = 1

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

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2D Sinusoids: Example

f(x, y) = cos(2x + 2y), ωx = 2, ωy = 2

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

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2D Sinusoids: Example

f(x, y) = cos(x + 2y), ωx = 1, ωy = 2

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

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2D Sinusoids: Example

f(x, y) = cos(x), ωx = 1, ωy = 0

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2D Sinusoids: Example

f(x, y) = cos(1.5y), ωx = 0, ωy = 1.5

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2D Sinusoids: Interference Pattern

f(x, y) = cos(x + y) + cos(x − y)

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2D Sinusoids: Interference Pattern

f(x, y) = cos(2x + y) + cos(2x − y)

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2D Sinusoids: Interference Pattern

f(x, y) = cos(3x + 2y) + cos(3x − 2y)

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2D Sinusoids: Interference

Why is this happening? g(x, y) = cos(ωxx + ωyy) + cos(ωxx − ωyy)

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2D Sinusoids: Interference

Why is this happening? g(x, y) = cos(ωxx + ωyy) + cos(ωxx − ωyy) g(x, y) = cos(ωxx) cos(ωyy) − sin(ωxx) sin(ωyy) + cos(ωxx) cos(ωyy) + sin(ωxx) sin(ωyy)

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

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2D Sinusoids: Interference

Why is this happening? g(x, y) = cos(ωxx + ωyy) + cos(ωxx − ωyy) g(x, y) = cos(ωxx) cos(ωyy) − sin(ωxx) sin(ωyy) + cos(ωxx) cos(ωyy) + sin(ωxx) sin(ωyy) g(x, y) = 2 cos(ωxx) cos(ωyy)

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Table of Contents

⊲ 1D Fourier Decomposition / Circle Functions ⊲ 2D Fourier Modes ◮ Spherical Harmonics

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Spherical Coordinates Review

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Spherical Harmonics

Y m

l

∝ Pm

l (cos(θ))eimφ

Interactive Demo

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Spherical Harmonic Shape Descriptors

Show Tom’s paper

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