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

2. Basis for a Vector Space Every vector in the space is a linear combination of basis vectors. n independent vectors from an n th 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.

3. Complex Numbers Absolute Value: Phase: a b  i R

4. Fourier Spectrum Fourier: Fourier Spectrum Fourier PhaseFourier:

5. Discrete Fourier Transform Fourier Transform Inverse Fourier Transform Complexity: O(N 2 ) (10 6 10 12 ) FFT: O(N logN) (10 6 10 7 )

6. 2D Discrete Fourier Fourier Transform Inverse Fourier Transform

7. Display Fourier Spectrum as Picture 1. Compute 2. Scale to full range Original f0124100 Scaled to 10000010 Log (1+f)00.691.011.614.62 Scaled to 10012410 Example for range 0..10: 3. Move (0,0) to center of image (Shift by N/2)

8. Decomposition

9. 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

10. Derivatives I Inverse Fourier Transform

11. 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

12. Translation

13. Periodicity & Symmetry

14. Rotation

15. Linearity

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

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

18. (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 Low Pass: Frequency & Image

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