Transforms: Basis to Basis

Transforms: Basis to Basis Normal Basis 2 1 1 0 0 1 0 0 0 0 1 0 2 0 0 1 0 0 1 1 0 0 0 1 Hadamard Basis 2 1 1 1 1 1 1 1 0 11 1 2 1 1 1 2

1 1 1 1 1 1 0 1 1 2 1 Hadamard Transform 1 1 0 1 2 1 2 0 Basis functions Method to find coefficients (Transform) Inverse Transform

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. 1 1 1 1 1 1 1 1 , 1 1 , 1 1 , 1 1 1 1 i

Complex Numbers R b a bi R e i a i e cos( ) i sin( ) Phase: Absolute Value: 2 b tan a 2

R (a b ) R1e i 1 R2 e i 2 R1 R2 e 1 i (1 2 ) Fourier Spectrum Fourier: Fourier Spectrum F (u ) R(u ) iI (u ) F (u ) R 2 (u ) I 2 (u ) Fourier Phase

(u ) tan 1 I (u ) / R (u ) Fourier: F (u ) F (u ) exp(i ) Discrete Fourier Transform Fourier Transform F (u ) N1 1 N f ( x) e 2iux N F (0) x 0

N1 1 N f ( x) e x 0 Inverse Fourier Transform N1 f ( x) 11 F (u ) e 2iux N u 0 Complexity: O(N2) (106 1012)

FFT: (106 107) O(N logN) 0 f 2D Discrete Fourier Fourier Transform N1 N1 F (u , v) N1 f ( x, y ) e 2i ( ux vy ) N x 0 y 0 Inverse Fourier Transform N1 N1

f ( x, y ) N1 F (u , v) e 2i ( ux vy ) N u 0 v 0 N1 N1 F (0,0) N1 f ( x, y ) e x 0 y 0 N f 2i ( 0 x 0 y ) N N1 N1 N1 f ( x, y ) x 0 y 0 Display Fourier Spectrum as Picture 1. Compute log F (u ) 1

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 0 100 10 Log (1+f) Scaled to 10 0

0 0.69 1 1.01 2 1.61 4 4.62 10 Decomposition N1 N1 F (u , v) N1 f ( x, y ) e 2i ( ux vy ) N x 0 y 0 N1

2iux N 1 2ivy F (u , v) N1 e N f ( x, y ) e N x 0 y 0 N1 2iux 1 N F (u, v) N e F ( x, v) x 0 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 f ( x) F (u ) e 2 iux N u f ' ( x) F (u ) e u ' 2 iux N

2 iux ' 2 iux N 2Ni uF (u ) e N F (u ) e u u 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

f ( x x0 , y y0 ) F (u , v) e F (u u0 , v v0 ) f ( x, y ) e 2i ( ux0 vy 0 ) N 2i ( u 0 x v0 y ) N Periodicity & Symmetry F (u, v) F (u N , v) F (u , v N ) F (u N , v N ) * F (u, v) F ( u, v) F (u, v) F ( u, v) Rotation x r cos y r sin u cos

v sin f ( x, y ) f (r , ) F (u , v) F ( , ) f (r , 0 ) F ( , 0 ) Linearity f1 ( x, y ) f 2 ( x, y ) f1 ( x, y ) f 2 ( x, y ) a f ( x, y ) a f ( x, y ) 1 u y f (ax, by ) F , a b ab Convolution Theorem f g F G f g F G Convolution by Fourier: f g

1 1 F G f g Complexity of Convolution: O(N logN) Filtering in the Frequency Domain Picture Fourier Low-Pass Filtering High-Pass Filtering Band-Pass Filtering Filter Filtered Fourier Filtered Picture

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) Sinc2 (0 1 4 6 4 1 0) = (0 0 1 1 0 0 ) 4 Sinc4 Fourier (Gaussian) Gaussian 1 g ( x) 2 e x2 2 2 Continuous Sampling :Image =

= T :Fourier * = T/1 * T/1 =

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