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Fourier Transform (Chapter 4) CS474/674 – Prof. Bebis

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Mathematical Background: Complex Numbers A complex number x is of the form: α: real part, b: imaginary part Addition: Multiplication:

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Mathematical Background: Complex Numbers (cont’d) Magnitude-Phase (i.e.,vector) representation Magnitude: Phase: φ Magnitude-Phase notation:

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Mathematical Background: Complex Numbers (cont’d) Multiplication using magnitude-phase representation Complex conjugate Properties

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Mathematical Background: Complex Numbers (cont’d) Euler’s formula Properties j

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Mathematical Background: Sine and Cosine Functions Periodic functions General form of sine and cosine functions:

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Mathematical Background: Sine and Cosine Functions Special case: A=1, b=0, α=1 π π π/2 3π/2

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Mathematical Background: Sine and Cosine Functions (cont’d) Note: cosine is a shifted sine function: Shifting or translating the sine function by a const b

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Mathematical Background: Sine and Cosine Functions (cont’d) Changing the amplitude A

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Mathematical Background: Sine and Cosine Functions (cont’d) Changing the period T=2π/|α| consider A=1, b=0: y=cos(αt) period 2π/4=π/2 shorter period higher frequency (i.e., oscillates faster) α =4 Frequency is defined as f=1/T Alternative notation: cos(αt)=cos(2πt/T)=cos(2πft)

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Basis Functions Given a vector space of functions, S, then if any f(t) S can be expressed as the set of functions φ k (t) are called the expansion set of S. If the expansion is unique, the set φ k (t) is a basis.

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Image Transforms Many times, image processing tasks are best performed in a domain other than the spatial domain. Key steps: (1) Transform the image (2) Carry the task(s) in the transformed domain. (3) Apply inverse transform to return to the spatial domain.

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Transformation Kernels Forward Transformation Inverse Transformation inverse transformation kernel forward transformation kernel

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Kernel Properties A kernel is said to be separable if: A kernel is said to be symmetric if:

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Notation Continuous Fourier Transform (FT) Discrete Fourier Transform (DFT) Fast Fourier Transform (FFT)

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Fourier Series Theorem Any periodic function f(t) can be expressed as a weighted sum (infinite) of sine and cosine functions of varying frequency: is called the “fundamental frequency”

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Fourier Series (cont’d) α1α1 α2α2 α3α3

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Continuous Fourier Transform (FT) Transforms a signal (i.e., function) from the spatial (x) domain to the frequency (u) domain. where

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Why is FT Useful? Easier to remove undesirable frequencies. Faster perform certain operations in the frequency domain than in the spatial domain.

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Example: Removing undesirable frequencies remove high frequencies reconstructed signal frequencies noisy signal To remove certain frequencies, set their corresponding F(u) coefficients to zero!

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How do frequencies show up in an image? Low frequencies correspond to slowly varying information (e.g., continuous surface). High frequencies correspond to quickly varying information (e.g., edges) Original Image Low-passed

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Example of noise reduction using FT Input image Output image Spectrum Band-pass filter

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Frequency Filtering Steps 1. Take the FT of f(x): 2. Remove undesired frequencies: 3. Convert back to a signal: We’ll talk more about these steps later.....

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Definitions F(u) is a complex function: Magnitude of FT (spectrum): Phase of FT: Magnitude-Phase representation: Power of f(x): P(u)=|F(u)| 2 =

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Example: rectangular pulse rect(x) functionsinc(x)=sin(x)/x magnitude

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Example: impulse or “delta” function Definition of delta function: Properties:

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Example: impulse or “delta” function (cont’d) FT of delta function: 1 u x

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Example: spatial/frequency shifts Special Cases:

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Example: sine and cosine functions FT of the cosine function cos(2πu 0 x) 1/2 F(u)

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Example: sine and cosine functions (cont’d) FT of the sine function sin(2πu 0 x) -jF(u)

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Extending FT in 2D Forward FT Inverse FT

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Example: 2D rectangle function FT of 2D rectangle function 2D sinc()

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Discrete Fourier Transform (DFT)

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Discrete Fourier Transform (DFT) (cont’d) Forward DFT Inverse DFT 1/NΔx

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Example

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Extending DFT to 2D Assume that f(x,y) is M x N. Forward DFT Inverse DFT:

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Extending DFT to 2D (cont’d) Special case: f(x,y) is N x N. Forward DFT Inverse DFT u,v = 0,1,2, …, N-1 x,y = 0,1,2, …, N-1

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Extending DFT to 2D (cont’d) 2D cos/sin functions

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Visualizing DFT Typically, we visualize |F(u,v)| The dynamic range of |F(u,v)| is typically very large Apply streching: (c is const) before stretchingafter stretching original image |F(u,v)| |D(u,v)|

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DFT Properties: (1) Separability The 2D DFT can be computed using 1D transforms only: Forward DFT: kernel is separable:

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DFT Properties: (1) Separability (cont’d) Rewrite F(u,v) as follows: Let’s set: Then:

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How can we compute F(x,v)? How can we compute F(u,v)? DFT Properties: (1) Separability (cont’d) ) N x DFT of rows of f(x,y) DFT of cols of F(x,v)

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DFT Properties: (1) Separability (cont’d)

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DFT Properties: (2) Periodicity The DFT and its inverse are periodic with period N

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Symmetry Properties

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DFT Properties: (4) Translation f(x,y) F(u,v) ) N Translation in spatial domain: Translation in frequency domain:

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DFT Properties: (4) Translation (cont’d) Warning: to show a full period, we need to translate the origin of the transform at u=N/2 (or at (N/2,N/2) in 2D) |F(u-N/2)| |F(u)|

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DFT Properties: (4) Translation (cont’d) To move F(u,v) at (N/2, N/2), take ) N ) N

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DFT Properties: (4) Translation (cont’d) no translation after translation

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DFT Properties: (5) Rotation Rotating f(x,y) by θ rotates F(u,v) by θ

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DFT Properties: (6) Addition/Multiplication but …

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DFT Properties: (7) Scale

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DFT Properties: (8) Average value So: Average: F(u,v) at u=0, v=0:

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Magnitude and Phase of DFT What is more important? Hint: use the inverse DFT to reconstruct the input image using magnitude or phase only information magnitude phase

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Magnitude and Phase of DFT (cont’d) Reconstructed image using magnitude only (i.e., magnitude determines the strength of each component!) Reconstructed image using phase only (i.e., phase determines the phase of each component!)

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Magnitude and Phase of DFT (cont’d)

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