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

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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.

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Complex Numbers Absolute Value: Phase: a b i R

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Fourier Spectrum Fourier: Fourier Spectrum Fourier PhaseFourier:

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

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2D Discrete Fourier Fourier Transform Inverse Fourier Transform

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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)

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Decomposition

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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

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Derivatives I Inverse Fourier Transform

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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

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Translation

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Periodicity & Symmetry

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Rotation

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Linearity

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

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

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(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

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

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