Transforms Basis to Basis Normal Basis Hadamard Basis

  • Slides: 19
Download presentation
Transforms: Basis to Basis Normal Basis Hadamard Basis • Basis functions • Method to

Transforms: Basis to Basis Normal Basis Hadamard Basis • Basis functions • Method to find coefficients (“Transform”) • Inverse Transform

Basis for a Vector Space • Every vector in the space is a linear

Basis for a Vector Space • Every vector in the space is a linear combination of basis vectors. • n independent vectors from an nth dimensional vector space form a basis. • Orthogonal Basis: Every two basis vectors are Orthogonal. • Orthonormal Basis: The absolute value of all basis vectors is 1.

i Complex Numbers R b Absolute Value: a Phase:

i Complex Numbers R b Absolute Value: a Phase:

Fourier Spectrum Fourier: Fourier Spectrum Fourier Phase Fourier:

Fourier Spectrum Fourier: Fourier Spectrum Fourier Phase Fourier:

Discrete Fourier Transform Inverse Fourier Transform Complexity: O(N 2) (106 1012) FFT: (106 107)

Discrete Fourier Transform Inverse Fourier Transform Complexity: O(N 2) (106 1012) FFT: (106 107) O(N log. N)

2 D Discrete Fourier Transform Inverse Fourier Transform

2 D Discrete Fourier Transform Inverse Fourier Transform

Display Fourier Spectrum as Picture 1. Compute 2. Scale to full range 3. Move

Display Fourier Spectrum as Picture 1. Compute 2. Scale to full range 3. Move (0, 0) to center of image (Shift by N/2) Example for range 0. . 10: Original f Scaled to 10 0 0 1 0 2 0 4 0 10 Log (1+f) Scaled to 10 0 0 0. 69 1 1. 01 2 1. 61 4 4. 62 10

Decomposition

Decomposition

Decomposition (II) • 1 -D Fourier is sufficient to do 2 -D Fourier –

Decomposition (II) • 1 -D Fourier is sufficient to do 2 -D Fourier – Do 1 -D Fourier on each column. On result: – Do 1 -D Fourier on each row – (Multiply by N? ) • 1 -D Fourier Transform is enough to do Fourier for ANY dimension

Derivatives I Inverse Fourier Transform

Derivatives I Inverse Fourier Transform

Derivatives II • To compute the x derivative of f (up to a constant):

Derivatives II • To compute the x derivative of f (up to a constant): – Computer the Fourier Transform F – Multiply each Fourier coefficient F(u, v) by u – Compute the Inverse Fourier Transform • To compute the y derivative of f (up to a constant): – Computer the Fourier Transform F – Multiply each Fourier coefficient F(u, v) by v – Compute the Inverse Fourier Transform

Translation

Translation

Periodicity & Symmetry

Periodicity & Symmetry

Rotation

Rotation

Linearity

Linearity

Convolution Theorem Convolution by Fourier: Complexity of Convolution: O(N log. N)

Convolution Theorem Convolution by Fourier: Complexity of Convolution: O(N log. N)

Filtering in the Frequency Domain Picture Fourier Low-Pass Filtering High-Pass Filtering Band-Pass Filtering Filtered

Filtering in the Frequency Domain Picture Fourier Low-Pass Filtering High-Pass Filtering Band-Pass Filtering Filtered Fourier Filtered Picture

Low Pass: Frequency & Image • (0 0 1 1 0 0) Sinc •

Low Pass: Frequency & Image • (0 0 1 1 0 0) Sinc • (0 0 1 1 0 0) * (0 0 1 1 0 0 ) = (0 1 2 1 0 0) Sinc 2 • (0 1 4 6 4 1 0) = (0 0 1 1 0 0 ) 4 Sinc 4 • Fourier (Gaussian) Gaussian

Continuous Sampling Image: · = T Fourier: * = 1/T * 1/T =

Continuous Sampling Image: · = T Fourier: * = 1/T * 1/T =