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### FourierTransform

Course: CS 474, Fall 2008
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Transform Fourier (Chapter 4) CS474/674 Fall 2008 Prof. Bebis Mathematical Background: Complex Numbers A complex number x has the form: a: real part, b: imaginary part Addition Multiplication Mathematical Background: Complex Numbers (cont'd) Magnitude-Phase (i.e.,vector) representation Magnitude: Phase: Mathematical Background: Complex Numbers (cont'd) Multiplication using magnitude-phase...

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Transform Fourier (Chapter 4) CS474/674 Fall 2008 Prof. Bebis Mathematical Background: Complex Numbers A complex number x has the form: a: real part, b: imaginary part Addition Multiplication Mathematical Background: Complex Numbers (cont'd) Magnitude-Phase (i.e.,vector) representation Magnitude: Phase: Mathematical Background: Complex Numbers (cont'd) Multiplication using magnitude-phase representation Complex conjugate Properties Mathematical Background: Complex Numbers (cont'd) Euler's formula Properties j Mathematical Background: Sine and Cosine Functions Periodic functions General form of sine and cosine functions: Mathematical Background: Sine and Cosine Functions Special case: A=1, b=0, =1 Period equal to T=2 Frequency is defined as f=1/T Mathematical Background: Sine and Cosine Functions (cont'd) Shifting or translating the sine function by a const b Cosine is a shifted sine function: i.e., shift to the right and take reflection along x-axis Mathematical Background: Sine and Cosine Functions (cont'd) Changing the amplitude A Mathematical Background: Sine and Cosine Functions (cont'd) Changing the period 2/|| Replace t with t: y=sin(t) =4 period 2/4=/2 shorter period higher frequency (i.e., oscillates faster) Fourier Series Theorem: Any periodic signal can be expressed as a weighted sum (infinite) of sine and cosine waves of varying frequency is called the "fundamental frequency" and are the weights of the expansion Fourier Series (cont'd) Illustration 1 2 3 Continuous Fourier Transform (FT) Transforms a signal (i.e., function) from the spatial domain to the frequency domain. (IFT) where Why is FT Useful? Remove undesirable frequencies from a signal. Easier and faster to perform certain operations in the frequency domain than in the spatial domain. Example: Removing undesirable frequencies noisy signal frequencies To remove certain frequencies, set their corresponding F(u) coefficients to zero! remove high frequencies reconstructed signal How do frequencies show up in an image? High frequencies correspond to quickly varying information (i.e., edges) Low frequencies correspond to slowly varying information (i.e., continuous surface) Original Image High-passed Low passed Steps in Frequency Filtering 1. Take the FT of f(x): 2. Remove undesired frequencies: 3. Convert back to a signal: We'll talk more about this later ..... Definitions F(u) is a complex function: of Magnitude FT (spectrum): Phase of FT: Magnitude-Phase representation: Power of f(x): P(u)=|F(u)|2= Example: rectangular pulse rect(x) function sinc(x) function or sin(x)/x Example: impulse or "delta" function Definition of delta function: Properties: Example: impulse or "delta" function (cont'd) FT of delta function Example: sine and cosine functions FT of the sine function jF(u) sin(2u0x) Example: sine and cosine functions (cont'd) FT of the cosine function cos(2u0x) 1/2 F(u) Extending FT in 2D Forward FT Inverse FT Example: 2D rectangle function FT of 2D rectangle function 2D sinc() Discrete Fourier Transform (DFT) Discrete Fourier Transform (DFT) (cont'd) Forward DFT Inverse DFT 1/Nx Example Extending DFT to 2D Assume that f(x,y) is M x N image. Forward DFT Inverse DFT: Extending DFT to 2D (cont'd) Special case: f(x,y) is N x N image. Forward DFT Inverse DFT What is the complexity of 2D DFT? O(N2) Visualizing DFT Typically, we display |F(u,v)| The dynamic range of |F(u,v)| is typically very large Apply scaling: (c is const) original image before scaling after scaling FT Properties: (1) Separability The 2D DFT can be computed using 1D transforms only: Forward DFT: Inverse DFT: FT Properties: (1) Separability (cont'd) Rewrite F(u,v) as follows: Let's set: Then: FT Properties: (1) Separability (cont'd) ) FT Properties: (2) Periodicity and Symmetry The DFT and its inverse are per...

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