7- 1 Chapter 7: Fourier Analysis Fourier analysis = Series + Transform ◎ Fourier Series -- A periodic (T) function f(x) can be written as the sum of sines and cosines of varying amplitudes and frequencies

7- 2 ○ Some function is formed by a finite number of sinuous functions

7- 3 Some function requires an infinite number of sinuous functions to compose

7- 4 Spectrum The spectrum of a periodic function is discrete, consisting of components at dc, 1/T, and its multiples, e.g., For non-periodic functions, i.e., The spectrum of the function is continuous

7- 5 ○ In complex form: Compare with

7- 6 Euler ’ s formula:

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7- 9 Continuous case

7- 10 Discrete case: ◎ Fourier Transform

7- 11 Matrix form Let

7- 12 。 Example: f = {1,2,3,4}. Then, N = 4,

7- 13

7- 14 ○ Inverse DFT Let

7- 15 。 Example:

7- 16 ◎ Properties ○ Linearity: e.g., Noise removal f’ = f + n, n: additive noise,

8-17 Fourier spectrum noise Corresponding spatial noise

○ Scaling : Show:

7- 19 ○ Periodicity:

7- 20 ○ Shifting:

7- 21 。 Example:

7- 22 ◎ Convolution: Convolution theorem: Correlation theorem ◎ Correlation

7- 23 。 Discrete Case: A = 4, B = 5, A + B – 1 = 8, e.g.,

7- 24 ＊ Convolution can be defined in terms of polynomial product Extend f, g to if f, g have different numbers of sample points Let Compute The coefficients of to form the result of convolution

7- 25 。 Example: Let The coefficients of form the convolution

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7- 27 ○ Fast Fourier Transform (FFT) -- Successive doubling method

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7- 29 。 Time complexity : the length of the input sequence FT: FFT: Times of speed increasing: N FT FFT Ratio

7- 30 ○ Inverse FFT ← Given ← compute i. Input into FFT. The output is ii. Taking the complex conjugate and multiplying by N, yields the f(x)

7- 31 ◎ 2D Fourier Transform ○ FT: IFT:

7- 32 ◎ Properties ○ Filtering: every F(u,v) is obtained by multiplying every f(x,y) by a fixed value and adding up the results. DFT can be considered as a linear filtering ○ DC coefficient:

7- 33 ○ Separability:

7- 34 ○ Conjugate Symmetry: F(u,v) = F*(-u,-v)

7- 35 ○ Shifting

7- 36 ○ Rotation Polor coordinates:

7- 37 ○ Display: effect of log operation

7- 38

7- 39 ◎ Image Transform

7- 40 ◎ Filtering in Frequency Domain ○ Low pass filtering I FT m IFT

7- 41 D = 5 D = 30 ○ High pass filtering

7- 42 Different Ds

7- 43 ◎ Butterworth Filtering ○ Low pass filter ○ High pass filter

7- 44 ○ Low pass filter ○ High pass filter

7- 45 ◎ Homomorphic Filtering -- Deals with images with large variation of illumination, e.g., sunshine + shadows -- Both reduce intensity range and increases local contrast ○ Idea: I = LR, L: illumination, R: Reflectance logI = logL + logR low frequency high frequency

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7- 48 ○ Fast Fourier Transform (FFT) -- Successive doubling method Assume Let Let N = 2M.

7- 49 = ] ∵ = ]

7- 50 Let (B) Consider

7- 51

(C)

7- 53 F(u+M) = Recursively divide F(u) and F(u+M), ○ Analysis : The Fourier sequence F(u), u = 0, …, N-1 of f(x), x = 0, …, N-1 can be formed from sequences F(u) = Eventually, each contains one element F(w), i.e., w = 0, and F(w) = f(x). u = 0, ……, M-1

7- 54

7- 55 ○ Example: needs { f(0), f(2), f(4), f(6) } needs { f(1), f(3), f(5), f(7) } Computing Input { f(0), f(1), ……, f(7) }

7- 56 Reorder the input sequence into {f(0), f(4), f(2), f(6), f(1), f(5), f(3), f(7)} Computing ＊ Bit-Reversal Rule